{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-05396660135b61cc5088","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v05\n- v02 --refutes--> v01\n- v06 --refutes--> v02\n- v03 --refutes--> v06\n- v01 --refutes--> v00\n- v01 --refutes--> v04\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7e12fa540b1d4dac1e67196a3f7013db56aba6a38be47fdd687f384a397b7a26","schema":"livegraph.boolean-propagation.v1","spec_hash":"05396660135b61cc50887e3206ed6fe0a67dcf7313f88084949dbc4bc28aadbf","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-6631e46e6068cfac63ee","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v02\n- v00 --refutes--> v03\n- v05 --supports--> v01\n- v05 --refutes--> v04\n- v03 --refutes--> v05\n- v04 --supports--> v06\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e3a9fbc04cd0ea1e9fc93a2644ed60bdd56cc934a301e1b7eba522898856bbc4","schema":"livegraph.boolean-propagation.v1","spec_hash":"6631e46e6068cfac63eeba5e787e9d8b4da7afa8cad23c8589085853992b1cbd","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-31c5ab6ab6bc8803d962","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v00\n- v03 --supports--> v01\n- v04 --refutes--> v05\n- v04 --supports--> v02\n- v02 --supports--> v03\n- v03 --supports--> v06\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"87ca3a24b9c729fd339d28f5f7815fb79b66814cac7584b2190c32aa932a8472","schema":"livegraph.boolean-propagation.v1","spec_hash":"31c5ab6ab6bc8803d9620e7aac898f92645c5cbdd1c99cfbe2a0a7c5aa15f1ed","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-ed331ec38b12b31343dd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v00\n- v00 --supports--> v01\n- v02 --supports--> v06\n- v05 --supports--> v04\n- v01 --refutes--> v02\n- v05 --supports--> v03\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6951196ed5b5a6efceb3c753540c05d797d1b82b36ca19931ea00b00fbffc6f8","schema":"livegraph.boolean-propagation.v1","spec_hash":"ed331ec38b12b31343dd2dc35df283bcb20da3a00da7b8e461d867f053ed5c1c","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-cdd6b68429bf7f7ce4be","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v02\n- v03 --supports--> v06\n- v06 --supports--> v00\n- v02 --refutes--> v01\n- v06 --refutes--> v04\n- v02 --supports--> v05\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d4a82da47e8bb02b25f31de8975f665ff18469ca130f75825a49ae3f4789fbf6","schema":"livegraph.boolean-propagation.v1","spec_hash":"cdd6b68429bf7f7ce4be5f30413e9055162e9acd424dfe6a6b20b92114888356","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-e19ee7a2932c9307eeae","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v04\n- v00 --supports--> v05\n- v05 --refutes--> v06\n- v04 --supports--> v02\n- v04 --refutes--> v03\n- v06 --supports--> v01\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"970821880c26042bbc8e04bc09a404ec1a020f65b348f4394c298e63cbb4f1e5","schema":"livegraph.boolean-propagation.v1","spec_hash":"e19ee7a2932c9307eeaef8596df29f0899c03a857cf8a9ccdb5a8e6415ed8b8d","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-44d94aab6139082f766b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v01\n- v01 --supports--> v02\n- v03 --supports--> v04\n- v02 --supports--> v05\n- v06 --supports--> v00\n- v06 --refutes--> v03\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"78d36d9e01dfb316b76cc158db903749a63cc314ad76797c1abd8432c702d1f4","schema":"livegraph.boolean-propagation.v1","spec_hash":"44d94aab6139082f766b95e113820879625333eaeffb7887436f2dc580bb1281","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-df584826a2595cab563c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v00\n- v01 --supports--> v03\n- v04 --refutes--> v02\n- v03 --refutes--> v06\n- v02 --supports--> v01\n- v01 --refutes--> v05\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"95f28a4ae44003111e27e89652c313fcfca6f0cdbde0bba3ade2ca7466aa2042","schema":"livegraph.boolean-propagation.v1","spec_hash":"df584826a2595cab563cb01a6b58f9869a2f2975aa4dcb223824a95430c3da83","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"}],\"hop\":4,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-fd9b0bf9b3a1dd8c9385","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v00\n- v00 --refutes--> v04\n- v03 --supports--> v05\n- v00 --supports--> v01\n- v03 --refutes--> v02\n- v06 --refutes--> v03\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"67cce788685bb31140566428edc40a1b23bcc86597300d5a73ab54ff86ca7dd1","schema":"livegraph.boolean-propagation.v1","spec_hash":"fd9b0bf9b3a1dd8c938518eabe53b2f353aa07f48979a1028b2c94ce12ca8f15","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-2ae94efaef369924f48d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v00\n- v05 --refutes--> v03\n- v04 --supports--> v02\n- v00 --supports--> v05\n- v00 --refutes--> v01\n- v02 --supports--> v06\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"189a06e6f2b3864bf293247f4f32c37699e97acafea2a57d966a5da7d868f37b","schema":"livegraph.boolean-propagation.v1","spec_hash":"2ae94efaef369924f48d6b3572e5016894564ca1886bb82b4e1d618ed461183f","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-47ddc94b408677072d42","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v01\n- v02 --refutes--> v00\n- v05 --refutes--> v04\n- v04 --supports--> v02\n- v06 --supports--> v05\n- v02 --supports--> v03\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0094e7fac118328e43f650e7f315db137f8db8ae35aa71235989f20fcd64d85a","schema":"livegraph.boolean-propagation.v1","spec_hash":"47ddc94b408677072d4223ff8238a59097de5f13fed9a1dab1e682a3fddf0ed4","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-aac7ac03def4cc282b7b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v06\n- v01 --supports--> v00\n- v06 --refutes--> v01\n- v01 --supports--> v05\n- v02 --refutes--> v04\n- v01 --refutes--> v03\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fb444c8083a8b71441d3b0a927c95fe9c55972912d9d0e92481d5d00c2eeba2a","schema":"livegraph.boolean-propagation.v1","spec_hash":"aac7ac03def4cc282b7bc0c5b8367ccc8aabf7a5ef77ce162dbfaad596f906e3","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-85ca39ddaadf7cfb73f9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v02\n- v05 --refutes--> v06\n- v02 --supports--> v05\n- v05 --supports--> v00\n- v05 --refutes--> v01\n- v03 --refutes--> v04\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d67f0f8171691115fbf9e5f3e03bf0cd11c328aa5df26c998e0595dce4c53461","schema":"livegraph.boolean-propagation.v1","spec_hash":"85ca39ddaadf7cfb73f9de084cfd2a5891d0ef5eae8d38fa24f5a41e598a8b5f","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-06e7117a6059ed4c71dc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v06\n- v00 --refutes--> v05\n- v03 --supports--> v04\n- v04 --refutes--> v01\n- v01 --refutes--> v00\n- v04 --refutes--> v02\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"59b4040a1d364d6e87bfb4261e1e961de8551c4cd9406abc590a4170b62264c3","schema":"livegraph.boolean-propagation.v1","spec_hash":"06e7117a6059ed4c71dc1424237de7eba059c74df86784124360764f892f7e97","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-9ffbe857fea84e29daac","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v06\n- v04 --supports--> v05\n- v03 --supports--> v04\n- v03 --refutes--> v00\n- v05 --refutes--> v02\n- v01 --supports--> v03\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fac112461c9b3287d06a94dde07b248c11b1aae2de1a1a66ddc7ec2d9eae9a7d","schema":"livegraph.boolean-propagation.v1","spec_hash":"9ffbe857fea84e29daac8778ec028b136f9032541f9e15f8727172909411066d","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-b4cb30d0931e182afb86","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v03\n- v01 --refutes--> v00\n- v04 --supports--> v05\n- v05 --supports--> v06\n- v00 --refutes--> v02\n- v05 --refutes--> v01\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6cbb6bffefe1bed00d7260b40c71de4b4d1f5ab0592aceaf0f169b5281e3cef4","schema":"livegraph.boolean-propagation.v1","spec_hash":"b4cb30d0931e182afb866720140b633f8ef575589257b65c613192ba19792fd5","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-41bcd9b3dca92e4a519d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v04\n- v04 --refutes--> v02\n- v01 --refutes--> v03\n- v00 --supports--> v01\n- v01 --supports--> v05\n- v04 --supports--> v06\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bac85a1b418982d3b56111cf327354b72dee84763eff4f5727ef692a7482f822","schema":"livegraph.boolean-propagation.v1","spec_hash":"41bcd9b3dca92e4a519d443b051d051539ec2786744b6c7d0f68d41f082cb1e4","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-7ba0a38b99417942c9f2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v02\n- v01 --supports--> v03\n- v02 --refutes--> v04\n- v02 --refutes--> v06\n- v01 --refutes--> v05\n- v03 --supports--> v00\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6dd950d055bef7d6ac3fc0dd15cd5ec4140c7aa8200cb1e1d5220e69510f38bf","schema":"livegraph.boolean-propagation.v1","spec_hash":"7ba0a38b99417942c9f2490cc726fcbe82d334cc8527a08031bbcf48cc1cda25","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-d53d6476c1f74d2fb624","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v01\n- v02 --supports--> v04\n- v04 --supports--> v03\n- v00 --supports--> v02\n- v03 --refutes--> v06\n- v02 --refutes--> v05\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"eeaf8d61ff99c62106b8bb4e183fe97d8d50b4cacda23ae6945aeb63c43faa4b","schema":"livegraph.boolean-propagation.v1","spec_hash":"d53d6476c1f74d2fb624c39eea9a5c3f391b252d49eabbd76a5362687eca4072","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-d074545a0462b0b94f79","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v03\n- v05 --refutes--> v06\n- v02 --refutes--> v04\n- v04 --refutes--> v01\n- v02 --supports--> v00\n- v01 --supports--> v05\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"013f261cebb3a54cd9ed0f13f17f499f22ba5d6bcc176f311a4d88ce83bb897b","schema":"livegraph.boolean-propagation.v1","spec_hash":"d074545a0462b0b94f7936e8f9b3b055da031fcf21e5f4b229be7b9f56cdf2fe","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-92b2a7778e1d20f49d55","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v04\n- v02 --refutes--> v01\n- v05 --refutes--> v06\n- v06 --supports--> v03\n- v00 --refutes--> v02\n- v06 --refutes--> v00\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"396a77080dfd9fe1ef127891e2ee10cac3068c58b0f0f8e8b283aa14ca6cf314","schema":"livegraph.boolean-propagation.v1","spec_hash":"92b2a7778e1d20f49d55fa3c693d9b5f1c61095276f986d59fcfdcc6f8e1f1dd","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-8aa2f9f40386a11ca516","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v03\n- v01 --refutes--> v04\n- v04 --refutes--> v06\n- v06 --refutes--> v00\n- v01 --supports--> v02\n- v05 --refutes--> v01\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"517668b563b77c03922d63bef56e92167030aa860e24551d94059f935e509216","schema":"livegraph.boolean-propagation.v1","spec_hash":"8aa2f9f40386a11ca5163a5ca6034eb10904c0ce1fe81cbbd5c7d8105f0039b4","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-889a8865f97d8edb20d8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v00\n- v01 --supports--> v03\n- v00 --refutes--> v06\n- v00 --supports--> v04\n- v05 --supports--> v01\n- v03 --supports--> v02\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d2b953768116e74e7f61da629a93e056343e641b0fceb65896b2523df7a2a2fa","schema":"livegraph.boolean-propagation.v1","spec_hash":"889a8865f97d8edb20d8958f1486a9aa419c3f13fcd13f68501fd31a96b15e8c","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-451f82baad6386186461","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v06\n- v03 --refutes--> v01\n- v05 --supports--> v02\n- v06 --refutes--> v05\n- v03 --supports--> v00\n- v03 --refutes--> v04\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8ea015a21b307d3fba63827d4d201d2428b2800688c0e0e416f627b9ef5d86f9","schema":"livegraph.boolean-propagation.v1","spec_hash":"451f82baad638618646143421f9968ce2b96f2f6c7ae56f3b7fa40ff50bfa74b","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-3d51155d8ad138d891ff","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v00\n- v04 --supports--> v02\n- v04 --refutes--> v03\n- v06 --refutes--> v01\n- v01 --refutes--> v04\n- v05 --refutes--> v06\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bf5b8ddcd71de1703679ce3aa183aea42713b3fb93672c49b94e96cf3e7e92a6","schema":"livegraph.boolean-propagation.v1","spec_hash":"3d51155d8ad138d891ff919f5f63562edc54c161976d81722d6f41509a5815e4","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-2e08cc4e9f4ec7c033b4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v03\n- v05 --refutes--> v00\n- v05 --refutes--> v04\n- v02 --refutes--> v06\n- v06 --refutes--> v05\n- v00 --refutes--> v01\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3efe171fef1ff672041ffce7e447da10b8c5db009bd490014c51aebaffcf3266","schema":"livegraph.boolean-propagation.v1","spec_hash":"2e08cc4e9f4ec7c033b40bbc27e190d49ad6c7315b3501f7c2c58a3b23133182","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-3552b84f589e3741402b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v02\n- v02 --refutes--> v06\n- v03 --supports--> v04\n- v00 --refutes--> v03\n- v00 --refutes--> v01\n- v05 --refutes--> v00\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5a2a56e072b2384353b207492c35468e99c1b01415ed387f4b175ce88004ceb5","schema":"livegraph.boolean-propagation.v1","spec_hash":"3552b84f589e3741402b04a17e298a2487a3a92340ca51d8de803dce00e134d4","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-e96e26b95385572f65d6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v02\n- v00 --refutes--> v03\n- v05 --refutes--> v01\n- v02 --refutes--> v06\n- v02 --refutes--> v05\n- v01 --refutes--> v04\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"337018660601a5125420cc2047dd661dc3ee956e28cc2f231cb65238f1b857c3","schema":"livegraph.boolean-propagation.v1","spec_hash":"e96e26b95385572f65d6318d234edf3e92d5a93cd5aa06e4333dc721e8d4bc48","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-2781f7b1a3edbbd8f5c8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v04\n- v02 --refutes--> v01\n- v02 --refutes--> v06\n- v05 --supports--> v00\n- v00 --supports--> v03\n- v01 --refutes--> v05\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cbb17c33533e0ee4e0bbfac07faaea10ef4d319d4a3b2c94455c5899af5820db","schema":"livegraph.boolean-propagation.v1","spec_hash":"2781f7b1a3edbbd8f5c8764d30a1fe42d33fa7d26c240e194a0a9ae1a65e1227","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-4cd6dfa1d0bd49584643","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v05\n- v01 --supports--> v04\n- v06 --supports--> v01\n- v03 --refutes--> v02\n- v04 --supports--> v03\n- v04 --refutes--> v00\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7512673ffbeddee8c0514ee79321444c13ec99b34f03c945c6cff462211b7cb8","schema":"livegraph.boolean-propagation.v1","spec_hash":"4cd6dfa1d0bd49584643ac5b859591c31cb0b73530e777c0f400483b02063614","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-84b0a6495d4bd3f7c121","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v04\n- v00 --refutes--> v02\n- v04 --refutes--> v00\n- v00 --refutes--> v05\n- v00 --supports--> v01\n- v01 --refutes--> v06\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f0dc3d44d66204540d08f87463a73e925c8a129e2a15a6849d72a9ecfb1f6544","schema":"livegraph.boolean-propagation.v1","spec_hash":"84b0a6495d4bd3f7c121974f6e7881f463dcac6ad07c54582112a8edba6a40ca","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-96528a3fff1d72267efc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v02\n- v04 --refutes--> v01\n- v06 --refutes--> v05\n- v03 --supports--> v06\n- v00 --supports--> v03\n- v04 --supports--> v00\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"207edd776c0f2f11ac05a5c82374f834667da4eb88adbdb1bf30340b448b1bc2","schema":"livegraph.boolean-propagation.v1","spec_hash":"96528a3fff1d72267efce76c5dfb4f28aa7ef693c32b8e1f88a086b632c7baa6","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-3f69634874cc347793c7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v00\n- v06 --refutes--> v05\n- v01 --refutes--> v06\n- v06 --refutes--> v02\n- v02 --refutes--> v03\n- v04 --supports--> v01\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1bc874654d566531d2d3e71c1c3cd2b38c9723c59128c75df29affac31e746be","schema":"livegraph.boolean-propagation.v1","spec_hash":"3f69634874cc347793c7db1533d60a29b3e49f8892a6795c5955fbe8f2dac5eb","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-c5e5e74e2a37fc7c1c04","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v05\n- v01 --supports--> v06\n- v04 --refutes--> v03\n- v00 --refutes--> v02\n- v03 --supports--> v00\n- v01 --supports--> v04\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d4dee6a2f2634e91b7c0f8275a879953f13e87b0d88926948577ac507c66f906","schema":"livegraph.boolean-propagation.v1","spec_hash":"c5e5e74e2a37fc7c1c04b1de8b6cd1058d5513d8136ae46c5dc350351f746aa1","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-1ecc608a0c67879c6485","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v06\n- v05 --refutes--> v03\n- v01 --supports--> v05\n- v06 --refutes--> v01\n- v02 --refutes--> v04\n- v05 --supports--> v00\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2ad97d2cb3f43e81a7b6d0c2c51aa8ba43126415b97b44b5df7a98b9637f4293","schema":"livegraph.boolean-propagation.v1","spec_hash":"1ecc608a0c67879c64859ac2d49266c211a7454be5dafeec06910ea2015a689e","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-06797bde61467bfda541","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v02\n- v03 --supports--> v06\n- v01 --refutes--> v03\n- v00 --refutes--> v01\n- v05 --supports--> v00\n- v01 --refutes--> v04\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"edc0a47607ca5cc77616a959f530a2955d88698a54b4864451386ca0640b32db","schema":"livegraph.boolean-propagation.v1","spec_hash":"06797bde61467bfda541aac47258ebc08555c97d654502b10e89a16701f2cc08","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-aa55ff744ee105009bd2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v04\n- v06 --supports--> v01\n- v03 --supports--> v06\n- v02 --supports--> v05\n- v04 --refutes--> v00\n- v04 --refutes--> v02\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"39bef55b6ddcf5c06444cbcc1ca17b77be19f768362c845f4b956410d17b7104","schema":"livegraph.boolean-propagation.v1","spec_hash":"aa55ff744ee105009bd26d7cff0e276dbcd5806ac7b00d50ee11f8fdd4430f99","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-dc00835b562101bb8388","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v06\n- v03 --supports--> v02\n- v04 --refutes--> v05\n- v06 --supports--> v01\n- v01 --refutes--> v00\n- v03 --refutes--> v04\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"80d79167032de274915c41ded698dbac2202b8a90143a0b9318564bf31b48ec4","schema":"livegraph.boolean-propagation.v1","spec_hash":"dc00835b562101bb838802a33190688fe76ec6fcf1bdbd07b095c436f88ce31c","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-7655845f42819feedce9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v00\n- v03 --supports--> v04\n- v01 --refutes--> v03\n- v03 --refutes--> v06\n- v03 --refutes--> v05\n- v02 --refutes--> v01\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"af7bf90f5e2877b2b91d23289fae68a5dcf4a1bb3962e29d6ac91ebb839abd11","schema":"livegraph.boolean-propagation.v1","spec_hash":"7655845f42819feedce9effc8da3ac55ef1c5a7d8918aa196f2e80acbd757506","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-d1d564976e8087507f2b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v02\n- v00 --supports--> v06\n- v05 --supports--> v03\n- v05 --supports--> v01\n- v06 --refutes--> v04\n- v04 --refutes--> v05\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"001f8720214edeed83fa14b7e50a94060699b5d0d7dc933fe744c1b40d755551","schema":"livegraph.boolean-propagation.v1","spec_hash":"d1d564976e8087507f2bbb318428cfea6d57b60ce30f448850d5b2d30cc29c32","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-e4a13ca98f4aee719efd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v05\n- v00 --supports--> v02\n- v05 --supports--> v01\n- v02 --supports--> v04\n- v02 --supports--> v06\n- v05 --supports--> v03\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"55860e413ed66f8dbd278e5a93ab52d18cef5a3ff4f4dbd030e66232c84bd023","schema":"livegraph.boolean-propagation.v1","spec_hash":"e4a13ca98f4aee719efd3ee185b6863bd9047a2bf7d00ec30e8b1b310530921a","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-dc765f9e133925f5b725","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v00\n- v00 --refutes--> v02\n- v01 --supports--> v04\n- v05 --supports--> v03\n- v06 --supports--> v05\n- v05 --refutes--> v01\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4cfb56fb700af60718f28fa60446094efca5a1bddb224eaff382ff58820a4740","schema":"livegraph.boolean-propagation.v1","spec_hash":"dc765f9e133925f5b725b6103c1d888b1fc4144d4d7849cc40e10025c57b6724","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-d31215089257e5aaadf8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v04\n- v04 --refutes--> v03\n- v04 --supports--> v00\n- v05 --supports--> v01\n- v06 --refutes--> v02\n- v03 --supports--> v05\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cb48e97c7bfd8a31d7a1fbdd800635e47d31a9268d93c2b51915c1be7ee083a1","schema":"livegraph.boolean-propagation.v1","spec_hash":"d31215089257e5aaadf8aa8a83e58b56838ce707c23e706c1eb99146c797643d","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-885f0f2e38ca79b5159f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v00\n- v03 --refutes--> v05\n- v03 --supports--> v01\n- v03 --refutes--> v04\n- v00 --supports--> v02\n- v02 --supports--> v06\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9ed30701edf3304fc774e0fdf7d1117604d05b34e4b8e12fc9d6a7f5f4d11113","schema":"livegraph.boolean-propagation.v1","spec_hash":"885f0f2e38ca79b5159f55a271ad9b3926392b7eb43dd7341d897ef2a94b59bb","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-cc5f2931da9083ef1848","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v00\n- v03 --supports--> v02\n- v00 --refutes--> v04\n- v04 --refutes--> v06\n- v06 --refutes--> v01\n- v03 --supports--> v05\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"656575bbe902201140edffe3d8129bd6ce0e436c7ae7b735a742292ba80cfb79","schema":"livegraph.boolean-propagation.v1","spec_hash":"cc5f2931da9083ef1848ab76aa651b3a40423ea5bf1815c3a7e408fd193cc2c6","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-59ffa4a28e0c4d97e18c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v02\n- v00 --supports--> v04\n- v05 --refutes--> v00\n- v01 --refutes--> v03\n- v00 --supports--> v06\n- v04 --refutes--> v01\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2217872b13198fecb447218a164a7b07d0ee4109be37570ab6ed3bb91f10179f","schema":"livegraph.boolean-propagation.v1","spec_hash":"59ffa4a28e0c4d97e18c03407f9d70513f771f7b7c382998a957d6fbfcaad282","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-f1f33ed7e8821088f4a3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v06\n- v00 --refutes--> v02\n- v00 --supports--> v05\n- v06 --supports--> v01\n- v02 --supports--> v04\n- v03 --supports--> v00\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3ccb8f7b7f759a200472d50dfa1a9e0e77ab7aed02319443782235635a6c78f1","schema":"livegraph.boolean-propagation.v1","spec_hash":"f1f33ed7e8821088f4a3949aefa74e725c3af767dc04c8f8f05314fff4e1ff45","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-226b9cd0d97841d28ad5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v05\n- v02 --supports--> v04\n- v06 --supports--> v03\n- v04 --supports--> v00\n- v05 --supports--> v01\n- v05 --refutes--> v06\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e2474fe4108d47598a8524f6c60ae0df1ae5f9da1517ca8990b099fca2fb5315","schema":"livegraph.boolean-propagation.v1","spec_hash":"226b9cd0d97841d28ad522f0c0331454a2540d2f9de459208f0bf42cf05a9c1e","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-7b5c1398a4b5e9829b81","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v04\n- v05 --supports--> v02\n- v05 --supports--> v00\n- v06 --supports--> v03\n- v01 --supports--> v05\n- v02 --refutes--> v06\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3a66c896eb4e1b0965a998998d925f0b8bd121b9e187e7780dda044fa11e9eba","schema":"livegraph.boolean-propagation.v1","spec_hash":"7b5c1398a4b5e9829b81510fcad3e9d7575eb59b04c312e62ddebdbddfb78620","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-1f2d3633977e64170be0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v03\n- v05 --supports--> v00\n- v02 --refutes--> v04\n- v05 --supports--> v06\n- v03 --refutes--> v05\n- v04 --supports--> v01\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"10d48ce174c8aac045c0fc3433e1fa6733386428f18bba0065d01caf64fcf476","schema":"livegraph.boolean-propagation.v1","spec_hash":"1f2d3633977e64170be0024b444343e70d6e280761b7913a7e385422ae03899e","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-5c526708a443847aa1b1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v01\n- v00 --supports--> v05\n- v03 --supports--> v02\n- v02 --refutes--> v06\n- v05 --supports--> v03\n- v02 --supports--> v04\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6c182e2ff84026f07adbfe67ee62047f5773ed724046a2d1d9ac57f000cfd5b1","schema":"livegraph.boolean-propagation.v1","spec_hash":"5c526708a443847aa1b1baaaecfd40566e4212a7c328f6c0bdbc1f4390d6c0b0","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-885c44cbb234da8e3ad9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v01\n- v00 --supports--> v02\n- v02 --supports--> v06\n- v02 --supports--> v03\n- v03 --supports--> v04\n- v04 --supports--> v05\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ed6cdc94e4a2cdda6fd0c4bb1540760b8ff5c8395323abe8c7f9ab0f7b976ca6","schema":"livegraph.boolean-propagation.v1","spec_hash":"885c44cbb234da8e3ad9d2443840abe1505833cb78ef0e90b8f71a518499fb46","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-b90cddcf32d1fc341f1e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v04\n- v01 --refutes--> v00\n- v05 --supports--> v02\n- v04 --refutes--> v03\n- v04 --refutes--> v06\n- v00 --refutes--> v05\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"32e9bd6387599194e95abce9635327cdebf7e2dcfb15131033f00461f0ff7e89","schema":"livegraph.boolean-propagation.v1","spec_hash":"b90cddcf32d1fc341f1e3eac4844f82c2adad30e8d713afdc786109598eb24c0","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-674bbeac3b882ba2c773","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v00\n- v04 --refutes--> v02\n- v02 --refutes--> v03\n- v02 --refutes--> v05\n- v06 --refutes--> v01\n- v01 --supports--> v04\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d2331866bcb813f1361600c878dab2b148e706bce87557e5cd2e604ad0c555c6","schema":"livegraph.boolean-propagation.v1","spec_hash":"674bbeac3b882ba2c773c5999b848f52974f8ff6ebb9b2a2a85661581ccfc390","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-51d41c06001945805693","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v00\n- v04 --supports--> v01\n- v02 --refutes--> v05\n- v06 --supports--> v04\n- v04 --refutes--> v03\n- v05 --refutes--> v06\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"36ed603febb99ce6ea7348b07e71f727553f027a00f671f1f1ff42e75678fcfc","schema":"livegraph.boolean-propagation.v1","spec_hash":"51d41c06001945805693d818627aa085d9abf858689803c2227889cf19d63b5c","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-a2f484c8a06d538febeb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v00\n- v01 --refutes--> v02\n- v02 --supports--> v03\n- v05 --supports--> v04\n- v03 --supports--> v05\n- v02 --refutes--> v06\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4ffc69097aadeea38d9265b49e3b3de03597c2dbaa82cb642e7c352fa745c0b2","schema":"livegraph.boolean-propagation.v1","spec_hash":"a2f484c8a06d538febeb847aa7eba7530f8ccc3ee65034333b2b40df1b9903f0","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-a0a117c9b26e572aeab5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v05\n- v02 --refutes--> v01\n- v06 --supports--> v02\n- v01 --refutes--> v03\n- v06 --refutes--> v04\n- v03 --supports--> v00\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"90e888376b0171982893eb4e6bf003d8cf6aa016f8c3caf7bbc9da74e6451dc3","schema":"livegraph.boolean-propagation.v1","spec_hash":"a0a117c9b26e572aeab5e8f89789293a2f52a73251d081d5449e6cb7eae6d970","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-c8dbd70066e456d0b543","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v01\n- v01 --refutes--> v02\n- v02 --refutes--> v00\n- v02 --supports--> v05\n- v04 --supports--> v06\n- v00 --supports--> v03\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"463f835234782a42c8d13554d8bdfd41fadd54ff71195caec8f6f5821baebdf5","schema":"livegraph.boolean-propagation.v1","spec_hash":"c8dbd70066e456d0b543837030127d7834d2742c396d486b828b1a64af835cfb","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"}],\"hop\":4,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-fdc149bb89bd980d76dd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v04\n- v01 --refutes--> v00\n- v05 --refutes--> v06\n- v03 --supports--> v01\n- v00 --refutes--> v02\n- v05 --refutes--> v03\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c2816034c61b15232ddf8b5b49fc0f8ba86a357ecb5c9616e94ea8479c0c9b5d","schema":"livegraph.boolean-propagation.v1","spec_hash":"fdc149bb89bd980d76dd72c63b0b7c891b4b1e108e3dfea1e9e5c71c639a84ae","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-25db01f5c473d0fd19e0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v04\n- v02 --supports--> v01\n- v02 --refutes--> v05\n- v04 --supports--> v03\n- v04 --supports--> v06\n- v03 --refutes--> v00\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"626e4269888acb14a66afa7ee68bdfb6926cb249975e70da48775e48e850d329","schema":"livegraph.boolean-propagation.v1","spec_hash":"25db01f5c473d0fd19e0ac51780b5d1e0411132e55e738f5dfe19c93c6218e83","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-6bd565baf6e2b5908fb5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v05\n- v01 --refutes--> v04\n- v04 --refutes--> v02\n- v02 --supports--> v00\n- v00 --refutes--> v03\n- v02 --supports--> v06\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"350eaf38de05377156f10bd702a9967717bca862764b920addc3ce79ee7f8b28","schema":"livegraph.boolean-propagation.v1","spec_hash":"6bd565baf6e2b5908fb5149f4f9657eeb0bd835691a0e6cb880419fd7dd51a4e","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-3a67d401b0726e33de48","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v00\n- v02 --supports--> v06\n- v06 --supports--> v01\n- v04 --supports--> v03\n- v06 --supports--> v05\n- v03 --refutes--> v02\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a41a41882e312ad8371a25f04053be5be147bde53470ecc6baedd3219eb99aa8","schema":"livegraph.boolean-propagation.v1","spec_hash":"3a67d401b0726e33de48a679656077dc7a311c37210270a5492fbf6a9b9aa909","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-b6488b0a3350cded8b4f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v06\n- v05 --supports--> v01\n- v00 --refutes--> v05\n- v03 --supports--> v00\n- v02 --supports--> v03\n- v02 --refutes--> v04\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a141232f74e7332609dcf15c8a3f217d3e5e210075b8bc059985e6f9e069e110","schema":"livegraph.boolean-propagation.v1","spec_hash":"b6488b0a3350cded8b4fcaedde38a1717626186c425f177bcb1995522637774f","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-839fb23494c98d0c2b28","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v02\n- v01 --supports--> v06\n- v05 --supports--> v01\n- v00 --refutes--> v05\n- v06 --refutes--> v04\n- v00 --supports--> v03\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dbc3cfbf6979bab771fa0e61523adb764eae1c8f571e09efb39b7bf101d2ff18","schema":"livegraph.boolean-propagation.v1","spec_hash":"839fb23494c98d0c2b286f58db8792b6d3260ed0c8c8dd3c942777551f275946","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-e87379b7d42b9e242a65","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v01\n- v01 --supports--> v03\n- v00 --refutes--> v02\n- v01 --supports--> v06\n- v01 --supports--> v05\n- v02 --refutes--> v04\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d27ce823aba3eb90a24fc10d94d9fd39cb7cc8af08c5c33cd676b55446024ca9","schema":"livegraph.boolean-propagation.v1","spec_hash":"e87379b7d42b9e242a65d1bc2dec7c32e412edd02281ad17747350771976dfa7","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-91bc7cda24188949a014","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v00\n- v03 --refutes--> v04\n- v03 --supports--> v01\n- v06 --supports--> v02\n- v00 --supports--> v05\n- v01 --refutes--> v06\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"55105fcbb53ebf3892387b81a1168479058c5c567a491cf9f523a04d2fabdcef","schema":"livegraph.boolean-propagation.v1","spec_hash":"91bc7cda24188949a014b13306c90bc680e26ce2055591bb6d9e66513756ea45","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-d2de4797035f2cafd76e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v05\n- v03 --supports--> v00\n- v01 --supports--> v02\n- v03 --supports--> v06\n- v04 --refutes--> v03\n- v05 --supports--> v04\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4813a2db650ba1976cb78399af0e9491951def9c569f0fbc2706b605f30f0c21","schema":"livegraph.boolean-propagation.v1","spec_hash":"d2de4797035f2cafd76e66e53f7032f7d04a9ca8eb9144c5d6f46727c9923485","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-007ddfedf5f0133694db","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v06\n- v04 --supports--> v02\n- v01 --refutes--> v03\n- v04 --refutes--> v05\n- v00 --supports--> v04\n- v03 --refutes--> v00\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f9eb00217a9d800d88da406981c751d3e287a35b203991dee47515c5e2d0b538","schema":"livegraph.boolean-propagation.v1","spec_hash":"007ddfedf5f0133694db7b2239758d62b2ad29b26bbbd5c47dcc1a5f368e91ce","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-2710d9127a377557566c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v06\n- v01 --refutes--> v03\n- v06 --refutes--> v02\n- v05 --supports--> v00\n- v05 --refutes--> v04\n- v04 --supports--> v01\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4bf5f6df849d76561aebbea16e0871fb4594ba74d0b0e8f99b66b59b6f6f142f","schema":"livegraph.boolean-propagation.v1","spec_hash":"2710d9127a377557566c3426e45f899ed16c0a29cb8b5496dc13ae7c03313ada","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-d6a056938acce3ec1774","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v01\n- v00 --refutes--> v06\n- v00 --refutes--> v02\n- v05 --supports--> v03\n- v05 --refutes--> v04\n- v03 --refutes--> v00\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5db8c6348010184fd06e03bff7c37d3485ec4b4d8c33f2f8bf74268bddef61da","schema":"livegraph.boolean-propagation.v1","spec_hash":"d6a056938acce3ec1774609ff4541d60f04f048f2ab28e5b945639a44b499b71","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-bfed0cdc3a27321a4e89","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v02\n- v06 --supports--> v00\n- v02 --supports--> v05\n- v06 --refutes--> v04\n- v05 --supports--> v06\n- v06 --supports--> v01\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0fb13973b29237e55fc473fc6407a1ba38b7c5582bd5f99129cb87fde5bfcb76","schema":"livegraph.boolean-propagation.v1","spec_hash":"bfed0cdc3a27321a4e89347729a7936d968852b55b00661e694b36d58db07de0","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-2dc88979450aa9593742","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v02\n- v03 --supports--> v06\n- v00 --supports--> v04\n- v03 --refutes--> v00\n- v01 --refutes--> v03\n- v04 --refutes--> v05\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"938a787b23ada907954495cb5f34ca750b45ba7636e4637b21a2eb21e0d03322","schema":"livegraph.boolean-propagation.v1","spec_hash":"2dc88979450aa9593742d922393bafe18d807bb140c8b141725f4e98ed21a587","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-9a46890927877421f995","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v01\n- v03 --refutes--> v06\n- v03 --supports--> v04\n- v04 --refutes--> v00\n- v00 --supports--> v05\n- v02 --refutes--> v03\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"83cbbef4591d227f5266f0fdb97266f31b35c4c2d32ad6ef1f5276e379b258db","schema":"livegraph.boolean-propagation.v1","spec_hash":"9a46890927877421f99569e75a0b0fefdf56532ce5feb421e96274e2e849fdee","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-fcf95353f31ca0bd37bb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v04\n- v00 --supports--> v03\n- v06 --refutes--> v02\n- v06 --supports--> v01\n- v05 --refutes--> v00\n- v04 --refutes--> v05\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1c052754c75618507e666ef8b4e758280177859e698884058b40fe415c0bf209","schema":"livegraph.boolean-propagation.v1","spec_hash":"fcf95353f31ca0bd37bb74c772ca507effc0d5a66b9cf075ba68184b0519df9f","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-8db7df613d13a523f0ee","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v02\n- v00 --refutes--> v06\n- v01 --supports--> v04\n- v01 --refutes--> v00\n- v02 --supports--> v01\n- v05 --refutes--> v03\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d70f9410763039c09995e758727595c3b0e1272535c12a19951c1ba4d5130b6e","schema":"livegraph.boolean-propagation.v1","spec_hash":"8db7df613d13a523f0eeb40c844f1a4e4f5c706e61282d28c994d2a7a977e9a6","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-fa9d5582b78783a3c966","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v05\n- v01 --supports--> v04\n- v05 --supports--> v02\n- v03 --refutes--> v06\n- v02 --refutes--> v01\n- v03 --refutes--> v00\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"300943781a9e4ce640fd10e4b0c54f9c3aceae4ffa95c0e9c03437001b73c41a","schema":"livegraph.boolean-propagation.v1","spec_hash":"fa9d5582b78783a3c9667206e0d27250e9a0ab0c41fa1f71681666ded13967af","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-ae4fdb4d5d004a26e9c6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v06\n- v03 --refutes--> v02\n- v03 --refutes--> v05\n- v04 --supports--> v01\n- v02 --refutes--> v04\n- v00 --refutes--> v03\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f1deefe0a94216218e06e8d939660ef7ff13eaf803a0510d4cf752965a59216f","schema":"livegraph.boolean-propagation.v1","spec_hash":"ae4fdb4d5d004a26e9c6fa86ae133eb9970128d012139ff0b6fadc99cc8ec887","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-2131ab0d7ca75dedcfb9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v06\n- v05 --refutes--> v00\n- v01 --supports--> v05\n- v02 --supports--> v03\n- v02 --supports--> v04\n- v06 --refutes--> v02\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"06f30da73c8b0c635ed010bc3a9ed4a0823793d54e345ac1ceb46194b6e83417","schema":"livegraph.boolean-propagation.v1","spec_hash":"2131ab0d7ca75dedcfb952f7dc967aa1b650b9f5d9af2fc6d4ee742cc8d7cb2e","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-519ec650a96001b2cd47","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v01\n- v02 --supports--> v03\n- v03 --refutes--> v06\n- v00 --supports--> v02\n- v03 --refutes--> v05\n- v05 --supports--> v04\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1c3d97688386ddb26ecdad8f85aac4dbd6c8c58dfdb3fa1c7864a66d8e6f1f36","schema":"livegraph.boolean-propagation.v1","spec_hash":"519ec650a96001b2cd47d463a1241a9bc9295fd3c5cf2acc46a5e70ec984ddb5","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-897312915c772a0bfbb0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v04\n- v04 --supports--> v01\n- v03 --refutes--> v02\n- v00 --refutes--> v05\n- v01 --supports--> v03\n- v04 --refutes--> v06\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3e88f57b86c7da80c426e380efc9646590a05a7ac85dc391258e7c309b1fb58a","schema":"livegraph.boolean-propagation.v1","spec_hash":"897312915c772a0bfbb02f28e51e401916d30e1a4c3a9ccd97e7e0b24724ae16","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-06e5d6703036a98d62a1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v04\n- v00 --supports--> v06\n- v06 --refutes--> v02\n- v02 --refutes--> v05\n- v03 --refutes--> v00\n- v06 --supports--> v01\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"32de6c19bccc564db4c650b2fd3d0fd019f666106cf1ccf5185dac790bee7564","schema":"livegraph.boolean-propagation.v1","spec_hash":"06e5d6703036a98d62a1de2f7c3032c7ab44a49af7ad871bc2bf9ef0912848f1","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-152585616dce5f4c3dee","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v03\n- v01 --supports--> v05\n- v02 --refutes--> v01\n- v02 --refutes--> v04\n- v03 --refutes--> v00\n- v06 --supports--> v02\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3e3ee2298bd2b509fb73d04ae09b84d3aac0f9187c3cd5e177238f13827a9c84","schema":"livegraph.boolean-propagation.v1","spec_hash":"152585616dce5f4c3dee1f3f53c98f6c95c0eac911678e68693cc363e741aeb5","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-e4f1e8678ada0425a179","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v02\n- v00 --supports--> v03\n- v02 --refutes--> v05\n- v01 --supports--> v04\n- v01 --refutes--> v06\n- v03 --supports--> v01\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"85b95a32ff600d9163f21abffbf8600291adad2cc96165cdf247e3e3e9c6cc32","schema":"livegraph.boolean-propagation.v1","spec_hash":"e4f1e8678ada0425a17906dffdcb23ee086af81d1390aad49d66825a7fa723dc","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-b1aef08729b80d9302dc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v06\n- v04 --refutes--> v05\n- v04 --refutes--> v00\n- v00 --refutes--> v02\n- v00 --refutes--> v01\n- v06 --refutes--> v03\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0f69bff8f2fc91b9f50b4d155e4f5f21d77fb3770808b38d208680889a9f7c5a","schema":"livegraph.boolean-propagation.v1","spec_hash":"b1aef08729b80d9302dc3ac26b7aa297bb34343cb86d9c03f17be8f1465324b7","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-dd9a89ea0eb362ce21f2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v02\n- v05 --refutes--> v06\n- v04 --supports--> v01\n- v06 --refutes--> v04\n- v06 --supports--> v03\n- v01 --refutes--> v00\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"92ff3f42451c72431501eb7d479a056d5daea75c2eb491aa636049e69ff1ba3e","schema":"livegraph.boolean-propagation.v1","spec_hash":"dd9a89ea0eb362ce21f2b369980a4a9069d9bc56d444d9be71c4c65b07a5f829","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-620ae51ce4663572cdd3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v06\n- v01 --supports--> v04\n- v01 --refutes--> v05\n- v00 --supports--> v01\n- v05 --supports--> v03\n- v05 --supports--> v02\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"56f0ac3b83dc2caddd214ca4b9f8acf84f3b1e7a07a64e709b30cbd6a353ce0c","schema":"livegraph.boolean-propagation.v1","spec_hash":"620ae51ce4663572cdd38372602616426dfcf5ec20485c1e0b39ea596281d54e","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-48552bae341662aa60f1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v00\n- v00 --refutes--> v02\n- v06 --refutes--> v05\n- v03 --refutes--> v06\n- v00 --refutes--> v01\n- v03 --refutes--> v04\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"431f96fe9c9a62c79531585f9dcc9bfc6ef929dd1686262571c79d7e1afda43d","schema":"livegraph.boolean-propagation.v1","spec_hash":"48552bae341662aa60f1584f6950cde23d8a1ee4c4455e519a9011abe8d996fe","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-0e8353ee80788ab0a076","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v01\n- v05 --refutes--> v03\n- v03 --supports--> v00\n- v06 --refutes--> v05\n- v04 --supports--> v06\n- v03 --refutes--> v02\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"50b80a7ec7ca416a38c7dc5ae5de8e4c22e169e6f67f9baaaabc232386f3deef","schema":"livegraph.boolean-propagation.v1","spec_hash":"0e8353ee80788ab0a0762df8eb0c09982ebe0a45c5abf5ae5bfa17bdfdaf9efd","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-bf2ab4a5ea5e8c81c598","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v05\n- v02 --refutes--> v00\n- v06 --supports--> v02\n- v01 --refutes--> v03\n- v00 --supports--> v01\n- v02 --refutes--> v04\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"290916e18ab492a6408e5c80938d974316f4e71efc28fa83810c9d261c75887b","schema":"livegraph.boolean-propagation.v1","spec_hash":"bf2ab4a5ea5e8c81c59824582fa17a41e0c42f7f28572a511877891213f72dae","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-665f496a7403bd86c00e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v06\n- v05 --supports--> v00\n- v02 --supports--> v05\n- v04 --refutes--> v01\n- v04 --supports--> v03\n- v06 --refutes--> v04\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4c0acaa747fb5b11b633aff1f5638070e6e1ea21c43d6b9727897d2a685a2a0c","schema":"livegraph.boolean-propagation.v1","spec_hash":"665f496a7403bd86c00e3dc59e1cd846824eb4fc75ef6ba6ee5da6b83511030d","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-9b7f1ad704b9d201cc55","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v03\n- v00 --supports--> v05\n- v05 --supports--> v04\n- v03 --supports--> v02\n- v05 --supports--> v06\n- v02 --refutes--> v01\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"87e0f8c28fdfe01ee455e459403ac7220a478cc5193e477e51a1e76c9f8860b4","schema":"livegraph.boolean-propagation.v1","spec_hash":"9b7f1ad704b9d201cc5513092dccd28ba9272152eba9e8459747d44cb7b1bd69","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-f4cc56b5d6a902f1f9b9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v01\n- v04 --refutes--> v02\n- v00 --refutes--> v05\n- v04 --supports--> v06\n- v03 --refutes--> v04\n- v05 --supports--> v03\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"142f2aac5ae45d3f8824f6eace6896e51f89e9c2c292e7d446bc30fb459cc3ba","schema":"livegraph.boolean-propagation.v1","spec_hash":"f4cc56b5d6a902f1f9b9b62858fc2b1a9b78d01403b05fca24326f2ead3c3cb3","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-d7d1f6e82d87b0000ace","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v02\n- v04 --refutes--> v06\n- v00 --supports--> v03\n- v02 --supports--> v01\n- v01 --supports--> v00\n- v00 --refutes--> v05\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bf1db0b0da0f694dc6766a05b4ac73c0042ce391429d5815c36d3286eeb8f595","schema":"livegraph.boolean-propagation.v1","spec_hash":"d7d1f6e82d87b0000ace4b0e1ac19726661388fcf8ef95692320f26680fc8ba7","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"}],\"hop\":4,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-0272d46eebf922cbf939","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v02\n- v03 --refutes--> v00\n- v03 --refutes--> v06\n- v04 --refutes--> v01\n- v03 --supports--> v05\n- v01 --supports--> v03\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b607e5fdf3c0f837942f3c90afc0fb3e820082254f5510fb27da54cc429f0fa9","schema":"livegraph.boolean-propagation.v1","spec_hash":"0272d46eebf922cbf939778c89fee3f306a1d074580d076be7c6c41ef6ebccc7","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"}],\"hop\":4,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-369744a0a0c73a2c6a22","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v05\n- v04 --refutes--> v02\n- v05 --refutes--> v03\n- v04 --refutes--> v01\n- v05 --refutes--> v06\n- v06 --supports--> v04\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fff4b85b5a7a70009a9410a69971a515d682a5aebd3aa25f6022801fb7be33f1","schema":"livegraph.boolean-propagation.v1","spec_hash":"369744a0a0c73a2c6a221bbf13dcd087a3183ce9b590badc074afdecf95a318b","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-38ea4ebd74ea108ff10f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v01\n- v02 --supports--> v03\n- v02 --refutes--> v00\n- v06 --refutes--> v02\n- v03 --supports--> v05\n- v05 --refutes--> v04\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"52bcb8bcde0ecec04f1b2d828d763611199e6edf4e6642527c6d4708d7c9b05f","schema":"livegraph.boolean-propagation.v1","spec_hash":"38ea4ebd74ea108ff10f298bccf72c5c54c5f27b12614c51abe6cde9aeac0ba8","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-c501110f4e5808290639","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v00\n- v05 --supports--> v03\n- v03 --supports--> v02\n- v00 --supports--> v05\n- v05 --supports--> v04\n- v03 --supports--> v01\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fa012241ce733af59832203174d6aa363ab122a75ede85fd899336572f95be17","schema":"livegraph.boolean-propagation.v1","spec_hash":"c501110f4e5808290639a016ca13ecd78b74fe143e35585e9bbf7b7adc5276e8","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-938e2ab0df1574d2425f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v03\n- v01 --supports--> v05\n- v02 --supports--> v06\n- v06 --supports--> v01\n- v05 --supports--> v00\n- v05 --supports--> v04\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fed87d8ae9d962f7ece70f5a186adbbac61aaa38ba1faa0c9cb863ebcaa0acba","schema":"livegraph.boolean-propagation.v1","spec_hash":"938e2ab0df1574d2425fcea5bb8eb6bad4bcef1e96d8af4d3c73958d7e3c084e","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-295aef65ce6200f60b3f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v03\n- v04 --supports--> v01\n- v06 --supports--> v00\n- v03 --refutes--> v04\n- v00 --supports--> v02\n- v06 --supports--> v05\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dc69b212e71da99e101dacba316db034ba656dfc796cb3f05e6412d583083bf6","schema":"livegraph.boolean-propagation.v1","spec_hash":"295aef65ce6200f60b3fcba7b354c2a575b5f4444182af399e6db89680b054c9","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-47a8e1b23bc0e2f52e18","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v00\n- v02 --supports--> v03\n- v00 --supports--> v04\n- v05 --supports--> v02\n- v05 --refutes--> v01\n- v00 --refutes--> v06\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"347753462ed225e889f7ca10864fc0934ae737205eebdb16f241d4820fdd575b","schema":"livegraph.boolean-propagation.v1","spec_hash":"47a8e1b23bc0e2f52e18ad66327cf4e0de84cd9d4f965b3d24660359b603561f","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-9573271d800c715e95b3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v06\n- v00 --supports--> v04\n- v06 --supports--> v05\n- v00 --refutes--> v02\n- v06 --refutes--> v01\n- v03 --supports--> v00\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a802fbaf4822deb8fbfdd75ebe89860480b000e713080e577e802c8592f8d23f","schema":"livegraph.boolean-propagation.v1","spec_hash":"9573271d800c715e95b3fc67603d318b688e01678564e77266e121c4ea92256b","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-aa86556f367e8b18a51b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v06\n- v02 --refutes--> v05\n- v05 --supports--> v01\n- v04 --supports--> v02\n- v03 --refutes--> v04\n- v02 --supports--> v00\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d98828c74ad23f8817cd66384cc2a91b1f2a1837e29ff76265b6eb9cbf474d26","schema":"livegraph.boolean-propagation.v1","spec_hash":"aa86556f367e8b18a51b3e14f237d739d5e569097bd6b5b8c2d6bd9276f122a0","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-3df088473880dc1d579a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v03\n- v00 --refutes--> v01\n- v02 --refutes--> v05\n- v06 --refutes--> v02\n- v04 --supports--> v06\n- v02 --supports--> v00\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ed24b59d8090a85a08f86072542b1033d984cf0dc97e8deba59466d4dee34749","schema":"livegraph.boolean-propagation.v1","spec_hash":"3df088473880dc1d579ad07d3fbffacc554de7cef70277683307bc7e5a9eadfd","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-b3c382d553e398444f8b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v03\n- v03 --refutes--> v01\n- v03 --refutes--> v02\n- v00 --refutes--> v05\n- v05 --refutes--> v06\n- v03 --refutes--> v04\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"985682d6c4b1b30fdd6de88e8946eaac334ef5413b0743a46a68ff0054e3c600","schema":"livegraph.boolean-propagation.v1","spec_hash":"b3c382d553e398444f8b470a2e2837e4b676cf227fe4a0ace68bd36a6a5bdae1","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-d3dcc3b2fbbd51547a20","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v04\n- v03 --refutes--> v06\n- v05 --supports--> v03\n- v01 --supports--> v00\n- v04 --refutes--> v05\n- v03 --refutes--> v02\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7572b680d5c1d2ae9f4bcf77d5b173f1403c8d6b05cc4c892bb24504a80adf52","schema":"livegraph.boolean-propagation.v1","spec_hash":"d3dcc3b2fbbd51547a206b5a52716c562be9ac506cc46e56662b14298c2c606a","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-8d7027645f91d3eb8c60","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v06\n- v03 --refutes--> v00\n- v02 --refutes--> v04\n- v00 --supports--> v02\n- v03 --refutes--> v05\n- v02 --refutes--> v01\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f064b7a78a4f31e8d0c42b8b8bd842c3019cb17838ee1860d2b15db6230a61f9","schema":"livegraph.boolean-propagation.v1","spec_hash":"8d7027645f91d3eb8c602c422e851d194c018b8baa9b10e7ab7b212dc70b7e40","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-f7a848f8157f82854e4e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v02\n- v06 --refutes--> v01\n- v06 --refutes--> v03\n- v02 --supports--> v06\n- v05 --refutes--> v00\n- v04 --refutes--> v05\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"19d4977eda8fdb444f35cecbcbc709a1c06b46b1198110862d3b79e77ed0f3f7","schema":"livegraph.boolean-propagation.v1","spec_hash":"f7a848f8157f82854e4ead054600f202db12f5abc48f6d2597719ea523ac2cc9","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-d9a8d61e1941617fbeda","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v02\n- v03 --supports--> v04\n- v04 --supports--> v00\n- v05 --supports--> v06\n- v00 --refutes--> v01\n- v00 --supports--> v05\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7666e86fd117952616424f573903d446c61de283419978f0caf8086a9560b9f3","schema":"livegraph.boolean-propagation.v1","spec_hash":"d9a8d61e1941617fbeda6272488032ff32f7c767602a57d37ee87b95fb4fe74f","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-36e2446957ae07a46f84","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v06\n- v00 --supports--> v02\n- v01 --supports--> v04\n- v01 --refutes--> v00\n- v02 --refutes--> v03\n- v03 --refutes--> v05\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6ab5dfa471197c86bec25f49ece91b6a0ffbd3e61766164204417d86815bd05e","schema":"livegraph.boolean-propagation.v1","spec_hash":"36e2446957ae07a46f84cbc10f70b632312e8e5c7bbb6a0c028929beb7a36c89","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-3dd5423030cfd6ebe5d8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v02\n- v03 --supports--> v04\n- v06 --supports--> v05\n- v00 --refutes--> v01\n- v02 --supports--> v06\n- v06 --supports--> v00\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5d43b9cbc10f9dbf8d0d0bca171610403e549eeba952b8d021e2aa5e0be87694","schema":"livegraph.boolean-propagation.v1","spec_hash":"3dd5423030cfd6ebe5d8b1643bf6cbbbf121fdeb5c51bba57f458c5d5ee3dcc6","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-44ee89fa5435429cb06e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v01\n- v01 --refutes--> v06\n- v03 --supports--> v04\n- v03 --supports--> v00\n- v06 --supports--> v03\n- v05 --refutes--> v02\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"394aef2135106ac4b795107a6be8e16e420961a890e601fac5ea440094fd43b1","schema":"livegraph.boolean-propagation.v1","spec_hash":"44ee89fa5435429cb06e0b3c3189081127e67573427d1af60792953d9e13fa33","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-057fd9b5994e2284b2ab","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v00\n- v06 --supports--> v05\n- v04 --refutes--> v02\n- v01 --supports--> v04\n- v02 --supports--> v06\n- v01 --supports--> v03\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1c1dae366066d788b669dd2bc950cfff91f318926b238b1a0a2ad14db23ca5d3","schema":"livegraph.boolean-propagation.v1","spec_hash":"057fd9b5994e2284b2ab6443a3aabc9b7cd090a6ab944be50bb1143039c177ff","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-90275e888ae1146c750c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v06\n- v00 --refutes--> v01\n- v04 --supports--> v00\n- v00 --refutes--> v02\n- v04 --refutes--> v03\n- v06 --supports--> v05\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1a8ceba53b476863339311e7a7a226a7821a8e79d86037f53bdfeb707f0ac30c","schema":"livegraph.boolean-propagation.v1","spec_hash":"90275e888ae1146c750c8933f328f66e57721d746f3829feecadba0e4f1ce41a","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-153fdad25637291166f9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v06\n- v05 --supports--> v03\n- v02 --refutes--> v01\n- v05 --supports--> v04\n- v00 --refutes--> v05\n- v03 --refutes--> v02\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f3c4c5a0ba72c1c4c401dfe2a257f4969b1bcbeb64b9daada10063002628f2a8","schema":"livegraph.boolean-propagation.v1","spec_hash":"153fdad25637291166f95efb7b6f4c53822f2b711f23aecf6a38ea5c977c9880","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-9c0f3e87995d01199144","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v00\n- v01 --supports--> v02\n- v02 --refutes--> v03\n- v03 --refutes--> v04\n- v02 --supports--> v06\n- v05 --supports--> v01\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"55d13fb3342e2a371e8bad045e08308988f457577f579902fd22113ba0dfe0e3","schema":"livegraph.boolean-propagation.v1","spec_hash":"9c0f3e87995d01199144b2f64a03d07f46182a0267702e25c72ba754905a8689","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-a6d0bbae0b2b83dc12d8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v02\n- v04 --refutes--> v00\n- v00 --supports--> v05\n- v00 --supports--> v01\n- v01 --supports--> v06\n- v03 --refutes--> v04\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4e13e5c0d26fdf80f7a7a0f0ce4fdaf4903e69e4827973987c8c8ab3afd35edf","schema":"livegraph.boolean-propagation.v1","spec_hash":"a6d0bbae0b2b83dc12d89bbb899f33b4f42afb86b276d8864f5c7b9417f9619e","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-f54ac370cdfd1645552c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v00\n- v00 --refutes--> v03\n- v01 --supports--> v05\n- v01 --supports--> v06\n- v04 --refutes--> v02\n- v03 --supports--> v04\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"20171825612fe5435f88924b76f70c3089a4dcdb8d14906e9094a3d6cb310dcc","schema":"livegraph.boolean-propagation.v1","spec_hash":"f54ac370cdfd1645552c6b50fd284989cc45f258ffa59a2fb69e9552e1ea745a","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-2f41bb698ca6bbe306d0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v06\n- v03 --refutes--> v02\n- v00 --refutes--> v03\n- v03 --refutes--> v01\n- v05 --supports--> v04\n- v01 --refutes--> v05\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5f303f901a7573c32e3a3c16e4a07c05ac79cae10bae238e617666bdd086134d","schema":"livegraph.boolean-propagation.v1","spec_hash":"2f41bb698ca6bbe306d056da494f8153d356266e9b41d4608286fe23f431a66f","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-b6b1f88fb201731a519f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v05\n- v02 --supports--> v00\n- v05 --refutes--> v01\n- v05 --refutes--> v04\n- v05 --refutes--> v06\n- v03 --supports--> v02\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bce6805a95cdb44a6affe3472ef6db4633f6bb6cf935aac2c9c193605327f158","schema":"livegraph.boolean-propagation.v1","spec_hash":"b6b1f88fb201731a519ff7b03e65922c0dfbb00c5e786258440d734816031981","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-4e1f37fc6a3f2ad5892e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v06\n- v05 --supports--> v02\n- v03 --supports--> v04\n- v02 --refutes--> v01\n- v02 --supports--> v00\n- v03 --refutes--> v05\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ee310ea276efce5d6217bc85826877285ba070153d39368950b2bb8fff8eb8dd","schema":"livegraph.boolean-propagation.v1","spec_hash":"4e1f37fc6a3f2ad5892e4310a5396cc8d4e89070a59a8fe18288bf9c41402f15","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-d8a275961a6df3b3d285","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v03\n- v01 --supports--> v05\n- v02 --refutes--> v04\n- v05 --refutes--> v02\n- v02 --refutes--> v00\n- v04 --refutes--> v06\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f8fa8c92a13fceaca1d3342dd70cc7681d5b426134ebfafe898c91b0b8e397ae","schema":"livegraph.boolean-propagation.v1","spec_hash":"d8a275961a6df3b3d285702a0af42df200893f5026eb90306c5d36508ba60250","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-9bffefb670c55bcced2d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v01\n- v00 --refutes--> v02\n- v05 --supports--> v03\n- v05 --supports--> v06\n- v03 --supports--> v00\n- v03 --refutes--> v04\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f4aae83f2b6f1504ce87d693051471e174b853d65b732bda84f32e09f1df1775","schema":"livegraph.boolean-propagation.v1","spec_hash":"9bffefb670c55bcced2db1103ba917786e31432694baf6dc58451658c5a1de1c","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-75e356406a4b6407a272","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v03\n- v06 --refutes--> v00\n- v00 --refutes--> v04\n- v05 --supports--> v06\n- v06 --supports--> v01\n- v05 --supports--> v02\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"15f688b9d9ba828d4e7e2a38e5e1b8b691adbe8431894e3745abca01b75ca80d","schema":"livegraph.boolean-propagation.v1","spec_hash":"75e356406a4b6407a2720cf0aaac3cdf5367b2b6802cb71d2a1260027e53d3ac","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-310527dee6a467e6513c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v02\n- v01 --supports--> v03\n- v05 --refutes--> v00\n- v01 --refutes--> v06\n- v06 --supports--> v05\n- v02 --refutes--> v04\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5a2d84a911271b94f9c4350b619f13e283fffe2179f07ae4d1f20df000a5f139","schema":"livegraph.boolean-propagation.v1","spec_hash":"310527dee6a467e6513cbe343478bbc0c0ba24b02d1237358e6b987e4bddf9a7","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-925aa8a2e90246bc082e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v06\n- v06 --refutes--> v02\n- v00 --supports--> v01\n- v02 --refutes--> v05\n- v04 --supports--> v03\n- v04 --supports--> v00\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f7b4c8cda922465dd5c2a4a82554c055996dd48e87b1f32606868c3247de15de","schema":"livegraph.boolean-propagation.v1","spec_hash":"925aa8a2e90246bc082e2b50037475b2bc37e9352a7d5259cdba502395341198","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-02155672ad9aa50a389e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v02\n- v01 --supports--> v04\n- v05 --supports--> v06\n- v02 --refutes--> v00\n- v00 --supports--> v01\n- v01 --refutes--> v03\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a221ceb835dd059a85e1151567f6982565c198e8c1d812c507a5cd51768ca317","schema":"livegraph.boolean-propagation.v1","spec_hash":"02155672ad9aa50a389ebb237021db2d54ab51e49fe2b873c8651c71f2af373d","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-5048974d2339225d705f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v05\n- v05 --refutes--> v01\n- v01 --refutes--> v04\n- v01 --supports--> v02\n- v05 --refutes--> v00\n- v03 --supports--> v06\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cb2d5317473a267c968390237817fcf96c83558ca881e01cd04aac524a106ed0","schema":"livegraph.boolean-propagation.v1","spec_hash":"5048974d2339225d705f1c3f2b92afcadfc3c5cca818ebdf1afb89587acaac40","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-2c7e0fa84a022acce62c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v01\n- v02 --supports--> v04\n- v01 --supports--> v02\n- v02 --supports--> v03\n- v02 --supports--> v05\n- v06 --supports--> v00\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8d4450b6ffd1dde14c8e9124febe29ab79cf25356f8066a7fd0b6846f8feeb00","schema":"livegraph.boolean-propagation.v1","spec_hash":"2c7e0fa84a022acce62c9fa6aa144375f9dd0cd27126feadb82f7dc80e99b474","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-a4421de306bccbda2686","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v01\n- v04 --supports--> v06\n- v00 --refutes--> v03\n- v00 --supports--> v05\n- v02 --supports--> v00\n- v05 --supports--> v04\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4451a8df9ada32d96410ed3cfaf856063bc446d95918fdb2ed15a3964bb18c97","schema":"livegraph.boolean-propagation.v1","spec_hash":"a4421de306bccbda26868031b0ba3aaccf6e2272f13f6f2087814e7e02311551","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-4efe8af4f40652f7e2d6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v03\n- v05 --refutes--> v01\n- v06 --refutes--> v04\n- v02 --supports--> v00\n- v05 --refutes--> v06\n- v03 --refutes--> v02\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4a4c1d73d85a7ee80a68a9a5d468716c2759a87239e67c1b0b2974c7db825cb6","schema":"livegraph.boolean-propagation.v1","spec_hash":"4efe8af4f40652f7e2d66e225d4e731bed6ccb88f9f060fa7d6807d362e836f9","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-a655ffd198c8cc354a9c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v06\n- v06 --refutes--> v02\n- v01 --refutes--> v05\n- v04 --refutes--> v01\n- v06 --supports--> v00\n- v06 --supports--> v03\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"102efe2155b76936ff8ed2a21714683d9a5ca54e1631fc559a05624def2aea01","schema":"livegraph.boolean-propagation.v1","spec_hash":"a655ffd198c8cc354a9c3d18a82722fc77d223e5d1e8b62e60ed772e8d68f830","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-491bddf11f61e2ab305b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v00\n- v00 --refutes--> v03\n- v03 --supports--> v02\n- v03 --supports--> v04\n- v02 --refutes--> v01\n- v06 --supports--> v05\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1eacf33bf001d7e17ced3edb24f4467fb604c91d12765eb6956058a19f1ff998","schema":"livegraph.boolean-propagation.v1","spec_hash":"491bddf11f61e2ab305b7873d579a9e614ff4d565c05e233a067788be0274070","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-98c90641a8b9c6dd8448","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v02\n- v02 --refutes--> v04\n- v02 --refutes--> v06\n- v03 --refutes--> v00\n- v05 --refutes--> v03\n- v05 --refutes--> v01\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e10aee14e5e42d99532ad40845e10755a7f5900f8bddbdcb98ef0ffd3fa65d7c","schema":"livegraph.boolean-propagation.v1","spec_hash":"98c90641a8b9c6dd8448daf5eda44fe910f50ea06e1d3acd9649a41dac86972d","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-56ee138c74ac07eba9d3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v02\n- v00 --refutes--> v03\n- v06 --supports--> v04\n- v03 --supports--> v06\n- v03 --supports--> v01\n- v05 --supports--> v00\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"02cd89ccc8d4356da1dff0d7de7149a9fc1b7501b61159b8f067cfeac6b3a4e7","schema":"livegraph.boolean-propagation.v1","spec_hash":"56ee138c74ac07eba9d3380599b8ba0d6283c3264e0a073ffe90c6a4a5bb814f","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-0dd296a26e44be8223ea","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v01\n- v04 --refutes--> v02\n- v00 --supports--> v06\n- v00 --supports--> v03\n- v02 --supports--> v05\n- v06 --supports--> v04\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2b090dbd015cc6bc218fbfa89af16a9e14192604eb5bc313e6711fb7cfcb2b0a","schema":"livegraph.boolean-propagation.v1","spec_hash":"0dd296a26e44be8223ea148c4a35865eea228d61ab66471bc058f868a6633a2c","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-6309c6a5b10b68fa5d9f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v03\n- v01 --refutes--> v00\n- v06 --supports--> v04\n- v05 --supports--> v06\n- v01 --supports--> v02\n- v04 --supports--> v01\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5d61cb066d16df4ef394492e6735c27b80934e06d38e056346f24ca92642645d","schema":"livegraph.boolean-propagation.v1","spec_hash":"6309c6a5b10b68fa5d9f5363a082afc995286d871864ed3d2b11935d1893cb74","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-9f57e859017842511f3a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v05\n- v04 --supports--> v03\n- v04 --supports--> v06\n- v05 --refutes--> v01\n- v00 --refutes--> v02\n- v00 --refutes--> v04\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6b5ce2ada69627c25758221774c3fed6677bfcd601d7144225f2e0ec6836d01d","schema":"livegraph.boolean-propagation.v1","spec_hash":"9f57e859017842511f3acda2801d768e5e14dba98e97465d0cc4fe9232fdd35e","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-49f1e5195dde17bb74fc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v03\n- v02 --refutes--> v06\n- v00 --refutes--> v02\n- v06 --supports--> v01\n- v05 --refutes--> v04\n- v06 --refutes--> v05\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9aa458071421cd90853c3ffa05bdf6d23a1769319da19acacd5f47cbac9c6402","schema":"livegraph.boolean-propagation.v1","spec_hash":"49f1e5195dde17bb74fc00eab9b2bfd8527d32550489820d585b1932d412d9b3","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-f2571ecf4c156c891103","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v02\n- v05 --supports--> v04\n- v02 --supports--> v03\n- v02 --supports--> v06\n- v04 --refutes--> v00\n- v06 --supports--> v01\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bb0f7160e3ea5f6cb1efb6bcebff9e8abe6bef38e52138ec7123ef6b3e3b3fd4","schema":"livegraph.boolean-propagation.v1","spec_hash":"f2571ecf4c156c8911030bec2302eedf89891443ebadcccf7ee17417f26bfeeb","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-fe21a1d3de1271b00fc3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v04\n- v05 --refutes--> v00\n- v02 --refutes--> v06\n- v06 --supports--> v03\n- v01 --supports--> v05\n- v05 --supports--> v02\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f924e5313954c5cc830a34d2a9864f12de12aaeb93b6b6ffa2df02aff0c98edd","schema":"livegraph.boolean-propagation.v1","spec_hash":"fe21a1d3de1271b00fc380595cee88f663faa941b99ccf61d1db554195017f2a","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-e874f34532ed10ce6f7d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v06\n- v05 --refutes--> v00\n- v00 --refutes--> v02\n- v06 --refutes--> v05\n- v06 --refutes--> v03\n- v00 --supports--> v01\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a2d361d1c64d331526a4aad682f21956f10837acc819cb8a281793a2e7102b21","schema":"livegraph.boolean-propagation.v1","spec_hash":"e874f34532ed10ce6f7d25f50ac32ac2a88440ab0e1abb29a872662bd6ec8399","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-b24bb868dc5380c2cdd3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v04\n- v01 --supports--> v00\n- v06 --refutes--> v01\n- v01 --supports--> v02\n- v05 --refutes--> v03\n- v02 --supports--> v05\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4984f515d92a36a3115affe1edb553fa2ae15b64f3f047a2258fafeb511f8130","schema":"livegraph.boolean-propagation.v1","spec_hash":"b24bb868dc5380c2cdd3574902177f99ce4cdf4d732c07c4bc6cd8a16af2ac2f","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-d4bef54ae9e069c2e48a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v02\n- v02 --supports--> v06\n- v00 --refutes--> v03\n- v02 --refutes--> v04\n- v00 --supports--> v01\n- v01 --supports--> v05\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e795f9baa78a657f26e03fc2db1a8d70c6832b3d8babbb77ff294c6fc35e77cc","schema":"livegraph.boolean-propagation.v1","spec_hash":"d4bef54ae9e069c2e48a91df877fe8c92290bf9164402d56cbe8723bb555b89a","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-eab7c06310e6f793eebc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v01\n- v03 --refutes--> v04\n- v06 --refutes--> v00\n- v00 --refutes--> v05\n- v04 --refutes--> v06\n- v00 --refutes--> v02\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9e135daa8afd2f07f4037f2ce56e5a9c3987f445121ea2db115dd19cf8217f97","schema":"livegraph.boolean-propagation.v1","spec_hash":"eab7c06310e6f793eebc48ecbc6fdfbb4f53227e8a1367e9234cb7ad438894cf","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-f8040b70bc2ae237ba8f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v00\n- v05 --supports--> v03\n- v04 --refutes--> v01\n- v03 --refutes--> v02\n- v05 --supports--> v06\n- v02 --supports--> v04\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"298d431722b96866b11b26c4605b9a015f56643464c5a63b2f8133192e4c7096","schema":"livegraph.boolean-propagation.v1","spec_hash":"f8040b70bc2ae237ba8fa6f3beed47212faf21534258bdb04c0a8c36bd7a6c8e","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-5d62b5d54582b669eca0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v04\n- v03 --refutes--> v01\n- v03 --supports--> v06\n- v04 --supports--> v03\n- v00 --supports--> v05\n- v00 --supports--> v02\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"898b0c51a9c93f0c9b7f1c0d3869793428d00b5b488c55f28d35eb8688e04747","schema":"livegraph.boolean-propagation.v1","spec_hash":"5d62b5d54582b669eca06cd933bd4a3b443a5ecf14956a5d8ae81fe5768caa23","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-3cb9e633ac4849b41538","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v01\n- v01 --refutes--> v03\n- v02 --refutes--> v06\n- v03 --supports--> v05\n- v03 --refutes--> v04\n- v06 --refutes--> v00\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"54aaa50cde1573c172bce2fe2f4cbddbd781ad22c71daa7a582a0540e7680452","schema":"livegraph.boolean-propagation.v1","spec_hash":"3cb9e633ac4849b4153834bbaa25b98970c6a13c3ef22e52b6f153561b657e93","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-50c248467b005739f56c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v00\n- v00 --supports--> v01\n- v04 --supports--> v06\n- v02 --supports--> v05\n- v02 --supports--> v04\n- v02 --supports--> v03\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2bfa450f5440b65060dd4b7d881320ab61b9ad647618c6229bb7d016634e8d59","schema":"livegraph.boolean-propagation.v1","spec_hash":"50c248467b005739f56cd5b632919b926b7230bd08c046953c88cc585b9f7361","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-b2ef581cb126b2091ef5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v05\n- v00 --refutes--> v02\n- v03 --refutes--> v00\n- v00 --refutes--> v01\n- v04 --refutes--> v03\n- v01 --supports--> v06\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"98a4e941d6143be8a7e6bdcc952bb90b2599ddeba021ce656cb558b41068afac","schema":"livegraph.boolean-propagation.v1","spec_hash":"b2ef581cb126b2091ef5a54ab9df9cfeaa9dde6e17537a868c182a82c0fd2b3a","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"}],\"hop\":4,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-d4284df982b97988461a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v02\n- v01 --supports--> v03\n- v03 --refutes--> v04\n- v02 --supports--> v01\n- v03 --supports--> v00\n- v01 --supports--> v06\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5ac4ab68c81840f64f2cb2c6d6a58a26f06945703ac89cf1f9b7f9d7d81ba40e","schema":"livegraph.boolean-propagation.v1","spec_hash":"d4284df982b97988461a7e6cf304f050a1925a55c8302d502ba2a85055fceb3a","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-59254c103c52522323aa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v01\n- v05 --supports--> v03\n- v06 --refutes--> v00\n- v02 --supports--> v05\n- v00 --refutes--> v02\n- v02 --supports--> v04\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"500f56f93b776dc375ff65c9234d243af3bf862253441fd679492d949afdc0d3","schema":"livegraph.boolean-propagation.v1","spec_hash":"59254c103c52522323aa8b41682f145f9bdef3d1b9e7330deb9bd73ab7dbf5c5","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-d3335ffb92b1fc974536","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v01\n- v00 --refutes--> v04\n- v06 --refutes--> v05\n- v03 --refutes--> v00\n- v05 --supports--> v03\n- v03 --refutes--> v02\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bf81b6be59eb4e50d14ddd2232330a30cbc58590e67c4cadb3932a169e29b724","schema":"livegraph.boolean-propagation.v1","spec_hash":"d3335ffb92b1fc974536ebc4275ef811efa8df528626b337a6da97ea8a26def8","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-431389f0fccdc27ba97d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v00\n- v05 --supports--> v01\n- v02 --refutes--> v05\n- v03 --refutes--> v02\n- v02 --refutes--> v04\n- v03 --refutes--> v06\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5f89c360fd8844dd625208d34579e175c76aab8b0fbd21a5f01cb99f45d6322e","schema":"livegraph.boolean-propagation.v1","spec_hash":"431389f0fccdc27ba97da2999c71bf4c896946a7cdee50be6e77bad228182c49","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-4538edfe034353e3c764","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v01\n- v01 --supports--> v06\n- v01 --refutes--> v05\n- v01 --supports--> v02\n- v05 --refutes--> v04\n- v00 --supports--> v03\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"407bc3e01b956ca3972e4e55d5fc3ab7cb53904d20128bdc9c62e6056f1d922b","schema":"livegraph.boolean-propagation.v1","spec_hash":"4538edfe034353e3c764c4f776e606111d37b57248ea5f24bf44c740d28e74b0","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-59a5f3fddab40c03396e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v02\n- v04 --supports--> v01\n- v00 --refutes--> v04\n- v06 --supports--> v00\n- v06 --supports--> v05\n- v03 --supports--> v06\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"485c19e47fbbaacc7325eba53e781ab3c552f28f2f6fa3a31460c3e8b10cce62","schema":"livegraph.boolean-propagation.v1","spec_hash":"59a5f3fddab40c03396e4615ae2666b5710a75aa6e70d87483896cb7601f72ab","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-406257cadf1fcda8d2b7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v06\n- v02 --supports--> v05\n- v02 --refutes--> v00\n- v06 --supports--> v04\n- v03 --supports--> v01\n- v03 --refutes--> v02\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c974c080fb647f046b1d7087cc48c3e027a9846a7b7c67f4f7cecbaf4533017e","schema":"livegraph.boolean-propagation.v1","spec_hash":"406257cadf1fcda8d2b7cfeccaeabecdd6eef7709611fb4e6bef1bf91cd42cbd","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-d2091b3eb394162a9bb2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v05\n- v03 --refutes--> v00\n- v04 --supports--> v02\n- v01 --supports--> v06\n- v05 --refutes--> v01\n- v01 --supports--> v04\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9915f92b8d8d02cd9d0acbcf19ecb3cf7683d46cacf8d5f496f72fe640309317","schema":"livegraph.boolean-propagation.v1","spec_hash":"d2091b3eb394162a9bb2eec73e26a3604b3c6da2513866426d6725b20926847a","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-b9666822ab5bca5a0670","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v02\n- v06 --refutes--> v00\n- v00 --supports--> v01\n- v06 --refutes--> v03\n- v01 --supports--> v04\n- v01 --refutes--> v05\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6f1465f379c7f1a67652ca4d0b3dc5f7af251cf67860115915a23b1986a0ede2","schema":"livegraph.boolean-propagation.v1","spec_hash":"b9666822ab5bca5a0670f4d74859b9817a325ac4225137afa28da3a7fa7a3755","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-b744bf45cad24da7c64d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v02\n- v06 --supports--> v05\n- v03 --supports--> v06\n- v04 --refutes--> v01\n- v06 --supports--> v00\n- v00 --supports--> v04\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"57f4cb6dbf83bfff0ab210f0fc9aff279faadb475d91193fe3bd2639e1849f68","schema":"livegraph.boolean-propagation.v1","spec_hash":"b744bf45cad24da7c64d8d05a76c03c98dbd206e12d358e1de5e36562191b3a3","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-3f7dfdd53bc02621b8eb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v03\n- v01 --refutes--> v05\n- v00 --refutes--> v02\n- v06 --supports--> v00\n- v01 --supports--> v06\n- v02 --supports--> v04\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e47f6630701dc1412dae2eca2a619e7e8cf8abe5d4e6c1766e5a0a63e29a50a7","schema":"livegraph.boolean-propagation.v1","spec_hash":"3f7dfdd53bc02621b8eb91c59a03e6b4f68eaf6d3f39919669bf239633285786","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-63b49932331a4acec216","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v05\n- v00 --refutes--> v04\n- v00 --supports--> v03\n- v06 --supports--> v02\n- v02 --supports--> v01\n- v03 --refutes--> v06\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"41e9d78ee110eca1587e00ff0e07183a94ba04ea63cf34ede0a0ab3d498d1854","schema":"livegraph.boolean-propagation.v1","spec_hash":"63b49932331a4acec216530d98b786ac9eb443c2e9a3ad0ef79c181f536f013b","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-a2137c173fc3a8358b6f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v06\n- v06 --refutes--> v03\n- v06 --refutes--> v01\n- v05 --supports--> v04\n- v03 --supports--> v02\n- v03 --refutes--> v00\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5d4c16ca29a1cc4ba53eddb77ff5aea2797e20a55decb879862a57d1078cb977","schema":"livegraph.boolean-propagation.v1","spec_hash":"a2137c173fc3a8358b6fdf3e51ab6246cf1a5b8b656e238297fe35334fdbcfea","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-7204ead00736cc34c61e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v05\n- v01 --supports--> v06\n- v06 --refutes--> v03\n- v03 --refutes--> v02\n- v03 --refutes--> v00\n- v03 --refutes--> v04\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"058dcfd84cc04e6d6e8daf5dd88821ac930abd8a3e0e525f193cdf6badd4344e","schema":"livegraph.boolean-propagation.v1","spec_hash":"7204ead00736cc34c61ef33e208e15c25a158743cd1dfc59f3ddb4b608661573","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-55091b276b392f5d4348","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v00\n- v05 --supports--> v01\n- v02 --refutes--> v06\n- v04 --refutes--> v03\n- v03 --refutes--> v02\n- v05 --refutes--> v04\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9ad83e28e2e1608a14a41b5fb01f58c9f2c0ec12179d6785b3a928a889b5a0bc","schema":"livegraph.boolean-propagation.v1","spec_hash":"55091b276b392f5d434861722ad01771aa9c4be6842547df5c5a742280b71c2d","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-f51343ba31115c1f8405","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v05\n- v03 --supports--> v06\n- v04 --supports--> v03\n- v01 --refutes--> v00\n- v00 --supports--> v04\n- v03 --refutes--> v02\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a26b17aafe266b08f294dd0a4d8acd237ee52499c75572f4aa92f69d337ac138","schema":"livegraph.boolean-propagation.v1","spec_hash":"f51343ba31115c1f84052750e0c1e4d7e5bddc749ec46329114e33f6a28affb2","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-c9dfc14eb16c784f525c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v06\n- v00 --supports--> v03\n- v01 --refutes--> v04\n- v02 --supports--> v01\n- v04 --refutes--> v00\n- v01 --refutes--> v05\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f2f550d053aa93a6d9e56d18551e21d9f968a6ed73eb88b4d9a3158ea9abad4c","schema":"livegraph.boolean-propagation.v1","spec_hash":"c9dfc14eb16c784f525c4fe6f8d793ba2da891b7a53d97d1f5226a3f141a28f1","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-25a75a96aa5191c0eb64","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v02\n- v01 --supports--> v00\n- v03 --supports--> v04\n- v00 --refutes--> v06\n- v04 --supports--> v01\n- v04 --refutes--> v05\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1fd82839876a2eb19136b8a40cc33c9225c327cb58c8c79daaf528fb47ca6149","schema":"livegraph.boolean-propagation.v1","spec_hash":"25a75a96aa5191c0eb6404ab4d42d4c0b5ced704f3145150d860d6d33144834f","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-ad0f0c3b12b647a4151d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v01\n- v01 --supports--> v05\n- v04 --supports--> v00\n- v00 --supports--> v03\n- v02 --supports--> v06\n- v00 --supports--> v02\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8ffa7363dc5e990fb3a7561c39fa9d61e166a81a94870acc5b3a6e2086801542","schema":"livegraph.boolean-propagation.v1","spec_hash":"ad0f0c3b12b647a4151d5d13cdabb0456a7760523f2fbfe12bb0e84ad57bd875","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"}],\"hop\":4,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-a12bd5b6cbe9e79a569a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v00\n- v06 --refutes--> v05\n- v02 --supports--> v01\n- v00 --supports--> v04\n- v04 --supports--> v06\n- v06 --supports--> v03\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e2848584e95341f1844c6d731f943392db5404113ebaa94b4a6e4f83374732a6","schema":"livegraph.boolean-propagation.v1","spec_hash":"a12bd5b6cbe9e79a569aa978e911fb05dc9925122fd3ca05344eb49318e4c023","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-0acc6858715f754f126a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v02\n- v00 --refutes--> v03\n- v06 --supports--> v04\n- v04 --supports--> v00\n- v00 --refutes--> v01\n- v05 --refutes--> v06\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"897ec36e164b714f7e92eb8fd66942182e5f1547ab45664757f7fdae9d52c391","schema":"livegraph.boolean-propagation.v1","spec_hash":"0acc6858715f754f126a49c20b5fbf6e990feb8b4553e360911c7ae7a9418b84","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-558562eb2c4965df198d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v02\n- v06 --refutes--> v05\n- v01 --refutes--> v03\n- v02 --supports--> v04\n- v06 --refutes--> v01\n- v00 --supports--> v06\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9a3f1943de489b2778ecfe92b586d80f7fe287a45b72302f8e074b2d5f9e2531","schema":"livegraph.boolean-propagation.v1","spec_hash":"558562eb2c4965df198da6e5009ee48432c0739b9d4c44820edb7159fdbf1a3a","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-5005439cc1eb2fea977b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v00\n- v01 --supports--> v03\n- v05 --refutes--> v04\n- v01 --refutes--> v06\n- v02 --refutes--> v01\n- v06 --supports--> v05\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"41396f9bd36865ab3d157ebee5e85d62e1fd5f90532f4e6c6162346cacebb533","schema":"livegraph.boolean-propagation.v1","spec_hash":"5005439cc1eb2fea977b838dcd6aeab4822c4a5a128d3227443da1d1ed0396a0","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-c99c4af78b56be977922","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v00\n- v03 --refutes--> v05\n- v05 --refutes--> v06\n- v02 --refutes--> v03\n- v00 --supports--> v04\n- v00 --supports--> v01\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"88054a8d7e2d1def458622f3127e0b2748e886243f3adb58d2b215039e9f2510","schema":"livegraph.boolean-propagation.v1","spec_hash":"c99c4af78b56be97792291c8df480ac1ca3fb262e2f67aff01540bcfbeb4a510","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-165a41c98bdb4d0a30c7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v01\n- v04 --refutes--> v02\n- v01 --refutes--> v03\n- v04 --refutes--> v06\n- v04 --supports--> v05\n- v03 --supports--> v04\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"da079e0644edea06ae4723905f30524fe17a607366b27646b00740233f768857","schema":"livegraph.boolean-propagation.v1","spec_hash":"165a41c98bdb4d0a30c7c7a86816caabe90b4833713b6805386c22c78f8b3e14","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-3ed251181809add5da92","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v03\n- v00 --refutes--> v01\n- v02 --refutes--> v05\n- v01 --refutes--> v06\n- v04 --refutes--> v00\n- v01 --supports--> v02\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0e9756d20ed5d635932d0109fad9dc0bf3e26e7ea8bceb7a6c1815caf2946d52","schema":"livegraph.boolean-propagation.v1","spec_hash":"3ed251181809add5da92e368cb6057abe8d7ec776eb79f792a142f0de137d02c","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"}],\"hop\":4,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-31668056c83b736c2ade","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v04\n- v05 --supports--> v03\n- v00 --supports--> v05\n- v03 --supports--> v06\n- v05 --refutes--> v01\n- v03 --refutes--> v02\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d0da860e79d4cc9b82deeef263adee38abbfa36b0815124a3e561e2fc0a459c8","schema":"livegraph.boolean-propagation.v1","spec_hash":"31668056c83b736c2ade282317f8252b2c6787ab192b4511ef7cc7ca06f6c461","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-18dd7c6949c8eb25a240","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v02\n- v02 --refutes--> v04\n- v05 --supports--> v06\n- v02 --refutes--> v01\n- v02 --refutes--> v05\n- v06 --supports--> v03\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"22ff2c9586e8a159216f3b348dd56ffc120cdd53692604a83aedc057ff3a8784","schema":"livegraph.boolean-propagation.v1","spec_hash":"18dd7c6949c8eb25a24085f0a75e6169d278cbf5da0062ff0ad7929ba6c9565d","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-418d22e4c741bc6c1d6c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v00\n- v04 --refutes--> v02\n- v06 --supports--> v04\n- v01 --supports--> v05\n- v03 --refutes--> v01\n- v01 --refutes--> v06\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"02a2916890d445112d1718f7e06596648be965b77fcd361ec05fb7cf7bdda031","schema":"livegraph.boolean-propagation.v1","spec_hash":"418d22e4c741bc6c1d6c23d91082727bac91da0fe3760ced9b24e38cdc72271f","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-7a1c15f05c65f951af19","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v06\n- v05 --supports--> v02\n- v01 --supports--> v04\n- v01 --supports--> v03\n- v02 --refutes--> v00\n- v00 --supports--> v01\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"91cfb923608b743915d46a6f7ddbdc15c547de7a4e369d23c857b2a872a816f9","schema":"livegraph.boolean-propagation.v1","spec_hash":"7a1c15f05c65f951af19e1352e9ce6062aaffa5b794682b3f322b4c93cdfd4ce","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-11c7fc9407232b37b63a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v01\n- v02 --supports--> v04\n- v06 --supports--> v05\n- v04 --refutes--> v00\n- v03 --refutes--> v06\n- v04 --supports--> v03\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"39108f632d3927c44b289a26cc90f53aca696fec6faf319916f32de3c931609d","schema":"livegraph.boolean-propagation.v1","spec_hash":"11c7fc9407232b37b63a141f50a27ffda167ab06fde5a5c95a4bfcdf71a714d0","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-d752417be3dc83c0b5f9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v03\n- v05 --supports--> v06\n- v04 --supports--> v02\n- v01 --refutes--> v04\n- v06 --supports--> v01\n- v01 --refutes--> v00\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c840071a15ce8d0fb03d1a2ac6c2bdc25428d850a17c3a24ba51a33311645663","schema":"livegraph.boolean-propagation.v1","spec_hash":"d752417be3dc83c0b5f9bd77d525deeaa1e02ed364827bcd29cab09146df54a2","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-3d175ccf86d72e1b8f0f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v05\n- v02 --refutes--> v01\n- v00 --supports--> v06\n- v01 --refutes--> v04\n- v01 --refutes--> v03\n- v03 --refutes--> v00\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bd2b31fd9edba2f0587f326fe2b3f0b5b85e293747e7a424221b3e03d5fdf7f5","schema":"livegraph.boolean-propagation.v1","spec_hash":"3d175ccf86d72e1b8f0f83d1361e651066d92a500cc61331cf553173a90304e1","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-4082e23014f56159f71a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v06\n- v05 --supports--> v04\n- v00 --refutes--> v02\n- v04 --refutes--> v01\n- v04 --supports--> v00\n- v02 --supports--> v03\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"efcb4724368da5bdfa5521259f4e7b4b90a96643097203dcec90c12d5606ee12","schema":"livegraph.boolean-propagation.v1","spec_hash":"4082e23014f56159f71aea72518c9b70797506b60d84be3860e8909282322e27","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-f29e58fc7c73b9dca483","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v04\n- v06 --supports--> v05\n- v04 --supports--> v00\n- v01 --supports--> v03\n- v04 --supports--> v06\n- v01 --supports--> v02\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5ef2c1d8306e4d506753586996f82837356f52d52ca22e9e88fb260a1bf9de3a","schema":"livegraph.boolean-propagation.v1","spec_hash":"f29e58fc7c73b9dca483f5ec6a0673446107a921e86d09496cbba3d2c5873e38","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-b5c8de0392e1b21ac27e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v04\n- v02 --supports--> v03\n- v00 --refutes--> v06\n- v03 --supports--> v05\n- v05 --supports--> v00\n- v02 --supports--> v01\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f42a3de12630b70e8ff244013d486e7952073df2a4fe69cbee4fe6aaffc33bb6","schema":"livegraph.boolean-propagation.v1","spec_hash":"b5c8de0392e1b21ac27e6d9bd22d8e0434328235bb79ed6f6b264319c6672f88","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-f737fdd83becf2573aa3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v04\n- v04 --refutes--> v01\n- v04 --refutes--> v00\n- v02 --refutes--> v03\n- v02 --supports--> v06\n- v05 --supports--> v02\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"23cc7ece768cfc42bafac22008eaf90a22128f3d244b5737e8d6c5dbe8a50e01","schema":"livegraph.boolean-propagation.v1","spec_hash":"f737fdd83becf2573aa3fe495881ddd8b7ad37d6fcc2862a0b9417226e6459ae","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-dea86d525cf01a179ee4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v03\n- v03 --supports--> v02\n- v06 --refutes--> v01\n- v02 --refutes--> v00\n- v02 --supports--> v04\n- v06 --supports--> v05\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"35194844c68f56ee91e3ccf35f08e2e2cbae56256f7896fb04138688d38afb2d","schema":"livegraph.boolean-propagation.v1","spec_hash":"dea86d525cf01a179ee45ad0b0bf64b55cc63b5522c96517bf1bfc8f4d3809f0","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-72e62a21e61d6359582a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v04\n- v04 --refutes--> v02\n- v05 --refutes--> v06\n- v06 --refutes--> v00\n- v00 --refutes--> v01\n- v06 --refutes--> v03\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"af107872382ec80d2049acd2075c8ecdae16da69a2997ab2d8f63b86c549d2ea","schema":"livegraph.boolean-propagation.v1","spec_hash":"72e62a21e61d6359582adb28188abdbefe36d9a97ae370bf7ead7ea2f2b83a16","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-acb2da7d46c6275be68f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v02\n- v00 --supports--> v03\n- v05 --supports--> v01\n- v05 --supports--> v06\n- v00 --refutes--> v04\n- v01 --supports--> v00\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"faa86581ff194df437dc11831c0fe719393aac6b702765d05976c0413676c4fb","schema":"livegraph.boolean-propagation.v1","spec_hash":"acb2da7d46c6275be68fefb2a3258f8a6e1672eb3345b1853eb4e8cd87423f08","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-392e9a7b6cb42e4beebe","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v05\n- v04 --supports--> v01\n- v05 --supports--> v06\n- v02 --supports--> v00\n- v05 --refutes--> v04\n- v05 --refutes--> v03\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c7adc9cea22b9757030df8ffb1b9e34c775994491f90ed40c17ed122558d3a38","schema":"livegraph.boolean-propagation.v1","spec_hash":"392e9a7b6cb42e4beebe96d662121ce50d02ac3f27f6615322311c45d7c0f276","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-8ac0b3a5d1550642914f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v05\n- v02 --supports--> v00\n- v01 --supports--> v03\n- v02 --supports--> v04\n- v06 --supports--> v02\n- v03 --refutes--> v06\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b5bb39c7838f4130cc593ee7d13145a8dbcc5dea72820ebb940bb5a6f96a261a","schema":"livegraph.boolean-propagation.v1","spec_hash":"8ac0b3a5d1550642914f98bb5919173ac702406c7a16ba91a3fe23209266e22e","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-3f804615d550d7283a01","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v05\n- v00 --supports--> v02\n- v02 --supports--> v06\n- v02 --supports--> v03\n- v02 --refutes--> v01\n- v05 --supports--> v00\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4b6693d820f9cc9b1b619c577409e718bbae526327d8c102cca21fb46a16dbbd","schema":"livegraph.boolean-propagation.v1","spec_hash":"3f804615d550d7283a0188ea360943cf5ac6918cb5a8c18a7a60a254d9324db9","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-b6c20a3e1a73ccf4ef78","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v05\n- v02 --refutes--> v00\n- v00 --supports--> v03\n- v03 --supports--> v06\n- v00 --refutes--> v04\n- v01 --supports--> v02\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e67dcdc60adfaca3ae4cccbd040b42bc3f815d3caf00ca2f1759aa10e6e54354","schema":"livegraph.boolean-propagation.v1","spec_hash":"b6c20a3e1a73ccf4ef787f07a6360088ed19d454a45bd6cdf529b50536ed8ac8","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-490aa967d90f68daade1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v05\n- v04 --refutes--> v03\n- v02 --supports--> v01\n- v00 --supports--> v02\n- v03 --refutes--> v00\n- v04 --supports--> v06\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d05e64c65c7023c5762c5f34868dd6fc9acc559832caa5fe2ad4682beb2d49eb","schema":"livegraph.boolean-propagation.v1","spec_hash":"490aa967d90f68daade15d26d3ebb41716a958edc7fbc2ce91e37cc61f4899d9","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"}],\"hop\":4,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-1a9e0e6058ea01e418db","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v00\n- v03 --supports--> v06\n- v01 --supports--> v04\n- v01 --supports--> v03\n- v06 --refutes--> v02\n- v02 --supports--> v05\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4f8ba055642571619c8382e9c4be5c320de0453428d72fdc93d1516770761297","schema":"livegraph.boolean-propagation.v1","spec_hash":"1a9e0e6058ea01e418db4d9f45ccd22fa68535c08cbe27854fd3fb8eb2c60432","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-5e9dfdc7b212d6991f4c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v04\n- v02 --refutes--> v06\n- v03 --refutes--> v02\n- v05 --refutes--> v03\n- v05 --refutes--> v00\n- v05 --supports--> v01\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3c49de68b683e96c67797dc53074a8ee611a46f15680185260ab558f4a516a9e","schema":"livegraph.boolean-propagation.v1","spec_hash":"5e9dfdc7b212d6991f4ce859c48d6bd4e20a9f629588f448410723d8b002072b","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-45fb33872f6cdb02fcb7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v06\n- v01 --supports--> v00\n- v05 --supports--> v04\n- v02 --refutes--> v03\n- v04 --supports--> v01\n- v02 --supports--> v05\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"31c5d258f08b3e6d157e930fcaec493870746855606a233819c4874f222ff365","schema":"livegraph.boolean-propagation.v1","spec_hash":"45fb33872f6cdb02fcb71b9068c9d544ce9cdbeb4395fd1eb2c759c018300173","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-9ea3875db35150eff8d6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v05\n- v00 --supports--> v04\n- v04 --supports--> v03\n- v02 --supports--> v00\n- v03 --supports--> v01\n- v04 --refutes--> v06\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"275ffdb71081eb8d61fb765a698198a2c8068ab06032c5adfc3eabebf9c61ec9","schema":"livegraph.boolean-propagation.v1","spec_hash":"9ea3875db35150eff8d6a7cd380f7a9bf95812000525f6dcf42865e701552d04","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-ee0aa964b448416305fd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v05\n- v01 --refutes--> v02\n- v04 --supports--> v06\n- v01 --supports--> v03\n- v06 --refutes--> v01\n- v02 --refutes--> v00\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"74ce9ed580e30619564caa3618a73119f36cf5b32115939081339da96f9f01de","schema":"livegraph.boolean-propagation.v1","spec_hash":"ee0aa964b448416305fdccf6d57146f14e45c2b9e5839f7364c3205bddeb7710","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-50ef6076fc2ae2cad5b4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v03\n- v06 --supports--> v05\n- v06 --refutes--> v00\n- v03 --refutes--> v01\n- v02 --supports--> v06\n- v03 --supports--> v04\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"81374957dba4083a5b06bf8a449b8c0b24fb67c89afc730b1009dfa55741ddc2","schema":"livegraph.boolean-propagation.v1","spec_hash":"50ef6076fc2ae2cad5b4ed2f8224e56bcc09c6a586a6191d20cf1e6ece2f2b65","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-e13d47fef6388e45a3c4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v00\n- v02 --refutes--> v04\n- v00 --refutes--> v05\n- v06 --supports--> v01\n- v02 --refutes--> v03\n- v01 --supports--> v02\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4044176f321638ab6b84ceaf2d2f8f6146253fc0796ebf6d16077641537d195f","schema":"livegraph.boolean-propagation.v1","spec_hash":"e13d47fef6388e45a3c4e2d1c8c368e2025851447a1039b9de139fef354abb3b","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-57c497c604af94809607","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v01\n- v04 --refutes--> v03\n- v00 --refutes--> v04\n- v00 --supports--> v05\n- v01 --supports--> v00\n- v04 --refutes--> v02\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3541f989edcafa41a16922930301f1ea401537640eca695a48bb912d52dc9ace","schema":"livegraph.boolean-propagation.v1","spec_hash":"57c497c604af948096077fe9b32d53a26013d38f41c3ba39d2de745bc996ddc3","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-ccb19a5feb7181279063","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v04\n- v05 --supports--> v02\n- v03 --supports--> v05\n- v04 --refutes--> v03\n- v05 --refutes--> v00\n- v05 --refutes--> v01\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f96440413822b7648321bb5b424f58fcbf0263b2873aec7af6dc32eae9a1bd85","schema":"livegraph.boolean-propagation.v1","spec_hash":"ccb19a5feb7181279063149702948c71929e07888f09a37b7a799277dd429871","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-cc829992e4301a61ea97","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v02\n- v01 --refutes--> v04\n- v04 --refutes--> v00\n- v05 --refutes--> v06\n- v00 --refutes--> v03\n- v05 --supports--> v01\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6f3d42a15b8c909c7eabf9605c8072c9dd253c27fa22a40b2a5340e9dd52ee12","schema":"livegraph.boolean-propagation.v1","spec_hash":"cc829992e4301a61ea9772d1904ee06c623ea8224e9bc32af1aa6ea7f705a147","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-60f2c110566a070e0ff9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v06\n- v03 --refutes--> v01\n- v03 --refutes--> v05\n- v04 --supports--> v02\n- v03 --refutes--> v00\n- v01 --refutes--> v04\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2d70a2f617b858792e707d1a602d2c76e016dc36041f326828e7012f44b4e140","schema":"livegraph.boolean-propagation.v1","spec_hash":"60f2c110566a070e0ff95e469022c95aeb10caf69b7d1ffcec77983b3a2f3a55","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-fca35b42eff1c7dd8494","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v03\n- v03 --supports--> v06\n- v03 --refutes--> v00\n- v05 --supports--> v02\n- v05 --refutes--> v04\n- v00 --supports--> v05\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a9016bc1d69574fc9f50a2781e47f38461bd1c87ede19edabb11553476c5f338","schema":"livegraph.boolean-propagation.v1","spec_hash":"fca35b42eff1c7dd84945f1656ff83cb6bb6083bb5fb217bc9e0ee9f5e1c2c62","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-c3f63add47af1aa12bf9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v00\n- v00 --refutes--> v01\n- v06 --supports--> v04\n- v01 --supports--> v06\n- v00 --supports--> v02\n- v06 --refutes--> v03\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8f27712b2d38c5e549c29d8e5e85c75a4db7e6536d4d61f097182a57cc151047","schema":"livegraph.boolean-propagation.v1","spec_hash":"c3f63add47af1aa12bf99b0d36b8f752f95d535d8d7a8ee907d9dd017b731815","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-83084c6a9662766284fe","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v04\n- v00 --supports--> v05\n- v01 --refutes--> v00\n- v02 --refutes--> v03\n- v00 --refutes--> v02\n- v00 --supports--> v06\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c75b841074150f6ad5e349ca52d5817292a09635c27ebadd60bdb26412f8d8b8","schema":"livegraph.boolean-propagation.v1","spec_hash":"83084c6a9662766284fe33e095c8691ad9c63945989b251eb7e6cc52d4885f1a","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-c453c7e819b7a35a973e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v05\n- v01 --refutes--> v06\n- v01 --supports--> v00\n- v05 --refutes--> v01\n- v06 --refutes--> v03\n- v06 --supports--> v04\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0f45a7abcdd8f1b79dd431c4da8698f17adcde1831c72c50534f3c2f56726959","schema":"livegraph.boolean-propagation.v1","spec_hash":"c453c7e819b7a35a973e64645c71957b8d9d21f28384d7213b35e745f95e6a1a","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-f951101f2259074a2caa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v03\n- v06 --supports--> v05\n- v05 --supports--> v00\n- v02 --supports--> v06\n- v01 --refutes--> v02\n- v01 --refutes--> v04\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c9861292142d68f5552b14273dda350a8641881ea2d166cd26c186696b5640b8","schema":"livegraph.boolean-propagation.v1","spec_hash":"f951101f2259074a2caa62d93779d330a42d62d2b747d0c19c5a791b7784c0cb","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-be6c7a30a3da869216ed","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v05\n- v04 --supports--> v01\n- v05 --refutes--> v02\n- v03 --supports--> v04\n- v03 --supports--> v00\n- v03 --refutes--> v06\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"df282b5dcc8d8e01f7e40a608e85b29facdcefe6b66fae936f3bd6e17639a5fb","schema":"livegraph.boolean-propagation.v1","spec_hash":"be6c7a30a3da869216ed11adf81b8db534cb13c81ed5f2f6e16b3bb537ab2aad","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-d0e3b9a965c6f3fa5156","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v06\n- v01 --supports--> v00\n- v03 --supports--> v01\n- v03 --supports--> v02\n- v04 --refutes--> v03\n- v03 --supports--> v05\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"107e266b8efec77381df30b1efbccecfa9380e1ac6977537944917497861b128","schema":"livegraph.boolean-propagation.v1","spec_hash":"d0e3b9a965c6f3fa515659656b8709c11d10e030d77e33e76c32a9a0f54b6fb0","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"}],\"hop\":4,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-b2747d3559155a693ac6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v00\n- v03 --supports--> v04\n- v01 --supports--> v06\n- v03 --refutes--> v02\n- v03 --refutes--> v01\n- v00 --refutes--> v05\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"69fc4a1257b18335a6c84c4306cd6f5d0682bf4a5bbf02de86d51ebf2959c550","schema":"livegraph.boolean-propagation.v1","spec_hash":"b2747d3559155a693ac6246fd52344662521f5679f5730dd7dbff2ed143ce23a","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-5606ddab5e188bee8fc0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v05\n- v00 --supports--> v06\n- v03 --refutes--> v02\n- v00 --supports--> v01\n- v03 --supports--> v04\n- v05 --supports--> v03\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5a19c439e53d354ce98336bfa3d415178117741a407c1d71da881015947a6929","schema":"livegraph.boolean-propagation.v1","spec_hash":"5606ddab5e188bee8fc0ce7ed975623b2c8684fae1b548bf35f8e5314133a941","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-7a453a1274e37b50a9b2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v02\n- v01 --supports--> v05\n- v02 --supports--> v04\n- v02 --supports--> v03\n- v02 --supports--> v06\n- v05 --supports--> v00\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3256f27ff03e0fa3e6b2df0f6847f02040daf1a22cc7e415402f54fde0b4aad6","schema":"livegraph.boolean-propagation.v1","spec_hash":"7a453a1274e37b50a9b2332d02a5caa21b8e33d6ba1f48f289f1c8fb9b5cbe14","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-15ef8ca60c703f30a36d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v02\n- v05 --refutes--> v01\n- v06 --refutes--> v05\n- v03 --supports--> v06\n- v03 --refutes--> v04\n- v01 --refutes--> v00\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dece4ff9ab5fec64279c7c42ffde8406b9248cd36c6320f5782286c5b3beed4a","schema":"livegraph.boolean-propagation.v1","spec_hash":"15ef8ca60c703f30a36de51541117a09b6e07c86f6a44249f7dd6d3ba5e08937","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-931c01d03b7ea7f807a4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v01\n- v06 --supports--> v05\n- v03 --supports--> v02\n- v00 --supports--> v06\n- v03 --refutes--> v04\n- v04 --refutes--> v00\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9578bddfdde5de6abbd0ca81261fa0c729762b7b06adb3a0a842c934e447c5eb","schema":"livegraph.boolean-propagation.v1","spec_hash":"931c01d03b7ea7f807a41676438718668f6a91dea90b9f03097b509e8ee8d094","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-4955bd58075998c5ccba","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v03\n- v05 --refutes--> v00\n- v06 --refutes--> v01\n- v00 --supports--> v06\n- v06 --refutes--> v02\n- v04 --supports--> v05\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3f86ef9fe26ea209d7e74d0f6a0d8d52829546d61cab847c50d54874451c5dd0","schema":"livegraph.boolean-propagation.v1","spec_hash":"4955bd58075998c5ccba871d62c5dcb093a372d3cbd39f1597d8bf3e868b734a","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-3f4564dc9bab994f3243","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v04\n- v05 --supports--> v06\n- v04 --supports--> v05\n- v01 --refutes--> v02\n- v05 --supports--> v03\n- v05 --refutes--> v00\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"21078325805542a4f6254cafd092ec64ab0b405f075b3f7f58756dc8751a5291","schema":"livegraph.boolean-propagation.v1","spec_hash":"3f4564dc9bab994f324323c0dab9d8bc9c15461d64a5a25a6813dbd5e6a48b42","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-7098195298f847532c06","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v02\n- v04 --refutes--> v06\n- v04 --refutes--> v01\n- v02 --refutes--> v05\n- v05 --refutes--> v00\n- v03 --supports--> v04\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c5290eb84f5d2dd7cf980ff4d4423d2abc8e8a7e9b8adb510a667d8ac071bcf1","schema":"livegraph.boolean-propagation.v1","spec_hash":"7098195298f847532c068a3ab71c81b59a764981b33ffea315bc2a9f25364cb0","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-1859e248771703245cc1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v05\n- v06 --supports--> v00\n- v05 --supports--> v02\n- v00 --supports--> v04\n- v05 --supports--> v01\n- v00 --refutes--> v03\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"198ca256264f6432d40282737b973adaf30a2ade26aac74fa8b328db2735a3cf","schema":"livegraph.boolean-propagation.v1","spec_hash":"1859e248771703245cc1ac2f4cfa66761cd056cfcec29e9d7dfbad0711b4c96e","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-350ae4c36fe21b5f3837","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v00\n- v02 --supports--> v01\n- v02 --refutes--> v04\n- v04 --refutes--> v05\n- v00 --supports--> v03\n- v04 --supports--> v06\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a5b993f4c103babc21737ab954b7999eeedbfb72a3c1ba39637b908ac37f6384","schema":"livegraph.boolean-propagation.v1","spec_hash":"350ae4c36fe21b5f38375d851ade2c91965bc031b2288caca3ac13fcc09674db","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-ff3fb32aa938c8e34771","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v02\n- v04 --refutes--> v01\n- v03 --refutes--> v05\n- v03 --refutes--> v06\n- v01 --supports--> v03\n- v03 --refutes--> v00\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"89ba2020bf43f6fee01c2a0e92a2ba4bf12c1a915860dec542a11a47a6b10405","schema":"livegraph.boolean-propagation.v1","spec_hash":"ff3fb32aa938c8e34771880713efaf9d6ed6265ced9fbc02cbc453787e5ec568","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-dea02a40e31819fab0bb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v02\n- v02 --refutes--> v06\n- v05 --supports--> v04\n- v04 --supports--> v03\n- v02 --refutes--> v05\n- v04 --refutes--> v00\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b65ae95dda66a3b1aef7c5ba19e7075bfa2db952e46fcce7e24ad9de9f203d3a","schema":"livegraph.boolean-propagation.v1","spec_hash":"dea02a40e31819fab0bbcbefbfe3610c2360c37602ad964b0a22e6b3d36c809b","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-a267b48416eb2dec711e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v03\n- v03 --supports--> v01\n- v06 --supports--> v04\n- v01 --refutes--> v06\n- v06 --supports--> v00\n- v06 --supports--> v02\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c748776b31f553d3a880a3b062f0a4e50cd110967928206a0efdb45baf2c766a","schema":"livegraph.boolean-propagation.v1","spec_hash":"a267b48416eb2dec711e228a5de565a7c3bec5b24efb5acdbde8fb340716b7b4","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-9c4417dbc633079885b0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v01\n- v06 --refutes--> v03\n- v03 --supports--> v00\n- v04 --refutes--> v06\n- v05 --supports--> v04\n- v03 --refutes--> v02\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"786fe5383c5a7c25369faca95a5be17541d84d30d043f1eb9509c84f568010ab","schema":"livegraph.boolean-propagation.v1","spec_hash":"9c4417dbc633079885b0f63bd8f70b82457c26d4270b217d3c78c10b8ec79275","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-6de55d3aca098b836e75","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v06\n- v05 --refutes--> v00\n- v02 --supports--> v05\n- v00 --refutes--> v04\n- v01 --supports--> v02\n- v01 --supports--> v03\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"34f04421eefdb466447e0cc7b69478f6215162d197a85279ac1acc383eebe29b","schema":"livegraph.boolean-propagation.v1","spec_hash":"6de55d3aca098b836e754e9563cf5cfd32b510cfac60d20721f91d29ba8b115d","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-53b6f19c8c473dc45d0a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v03\n- v01 --refutes--> v06\n- v06 --refutes--> v05\n- v00 --refutes--> v02\n- v00 --supports--> v04\n- v05 --refutes--> v00\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f3ce32573b35fdce7016878b90a499693a797387943d1a05481e2e31741f2f0d","schema":"livegraph.boolean-propagation.v1","spec_hash":"53b6f19c8c473dc45d0a9598433b01da10d4178a93992d7b203461b967d6376f","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-f541529050a81f2eca46","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v03\n- v04 --refutes--> v01\n- v04 --refutes--> v02\n- v06 --refutes--> v05\n- v00 --supports--> v06\n- v02 --supports--> v00\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"50f0eb985cde2dfa9ac95c6968dcf9413a509e95cd3fdcdb6f3fba1677021d70","schema":"livegraph.boolean-propagation.v1","spec_hash":"f541529050a81f2eca46f4232ac019f3fdcbbd37a73973fc780d8620d205878f","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-87fc0f96982a8aa46c20","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v01\n- v03 --supports--> v05\n- v00 --supports--> v04\n- v04 --refutes--> v02\n- v04 --refutes--> v03\n- v03 --refutes--> v06\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"993517319d1a2175c8a532b20c13943096f6ff73b43bfa8ffcf263cc46f8a485","schema":"livegraph.boolean-propagation.v1","spec_hash":"87fc0f96982a8aa46c20a04a5c9a00fd4e9bd7f37d65db9a7f628cb1f91f1e15","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-05a7140e415175f7ff8b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v06\n- v05 --refutes--> v01\n- v04 --supports--> v00\n- v04 --supports--> v02\n- v05 --refutes--> v03\n- v03 --refutes--> v04\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"df07f4d88a9f93ebeb76b2bbe245671f190a76fdde5cb9b0f8c00db5b54f2084","schema":"livegraph.boolean-propagation.v1","spec_hash":"05a7140e415175f7ff8bf4003444cf9a14bbcc3544e7c8d616945953c88d115e","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-229af66d3fc24ebe40b4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v05\n- v03 --supports--> v06\n- v02 --refutes--> v03\n- v06 --refutes--> v00\n- v01 --supports--> v02\n- v03 --refutes--> v04\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6c968286ad6c0961912bf6241ac4f49372d2e6738c6012c0b632e4761eb3aaba","schema":"livegraph.boolean-propagation.v1","spec_hash":"229af66d3fc24ebe40b4ae91c153d2bcf902c975dc1bf50b257b5d469629e488","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-281740b78f25bd09db28","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v06\n- v05 --refutes--> v04\n- v02 --supports--> v05\n- v05 --refutes--> v00\n- v06 --refutes--> v01\n- v05 --refutes--> v03\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"300a0d7e9a1a98f8166a1df2c6b4b2a9289bdcf00c52e2dc7c0083cbd64d3660","schema":"livegraph.boolean-propagation.v1","spec_hash":"281740b78f25bd09db2814b3e632b3b1ec8e69253f6563c0b7db9d538e50a792","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-c13865d2144487bc4893","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v06\n- v00 --supports--> v04\n- v06 --supports--> v00\n- v00 --refutes--> v02\n- v04 --refutes--> v03\n- v05 --refutes--> v01\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5a3a0e58cbcdda5d44afe5f4cef9d9b8bae07e42817d13136813a39d9039eda5","schema":"livegraph.boolean-propagation.v1","spec_hash":"c13865d2144487bc489341ac32abb2363efed3a9ac908e4fb9c88d808ac89729","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-0d14c5c495b3b166c00c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v02\n- v00 --supports--> v06\n- v01 --refutes--> v04\n- v02 --refutes--> v05\n- v01 --refutes--> v03\n- v02 --refutes--> v01\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"35c5f02fdd58fc142f5950aaeb27ac8d4b24f2dc7568e916a0eb3be9ef9d81b4","schema":"livegraph.boolean-propagation.v1","spec_hash":"0d14c5c495b3b166c00cf4d39f4865109942556e85fcfb4c16d52b1671bd6817","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-2832959c6d5f3346a0df","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v04\n- v04 --supports--> v01\n- v04 --refutes--> v06\n- v05 --supports--> v02\n- v05 --supports--> v03\n- v06 --supports--> v05\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2df29b09b2ecfebcd32f15ec2479589407bd125df800e41ebb7928bf726e9bbc","schema":"livegraph.boolean-propagation.v1","spec_hash":"2832959c6d5f3346a0df418054d972ae3e92db46a9cb7eeede5150246c4b0fde","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-56b0af3c5f70ce84c821","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v04\n- v03 --supports--> v06\n- v03 --refutes--> v01\n- v04 --supports--> v05\n- v04 --supports--> v03\n- v00 --refutes--> v02\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6e666e1e0dce7f0ad7620222862539ecfae7be164d8dd4205c626faf56b2e8d3","schema":"livegraph.boolean-propagation.v1","spec_hash":"56b0af3c5f70ce84c8210bf4e9900d975e9e995cdeb6e4e38bd983d46073bf1d","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-a556f649bd3b57cdb6b6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v05\n- v00 --refutes--> v06\n- v02 --refutes--> v00\n- v01 --supports--> v02\n- v06 --supports--> v03\n- v06 --supports--> v04\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7ca76cc4935c9cfed27b32ec32ff27efb1ae7c18f63b7d043612fac7082ffff5","schema":"livegraph.boolean-propagation.v1","spec_hash":"a556f649bd3b57cdb6b605c60cb8647ed6b280cc7e6eff196f323ac8ae719cb3","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-fbff4ab0396166c0b7cd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v03\n- v05 --supports--> v02\n- v01 --refutes--> v06\n- v05 --supports--> v00\n- v03 --refutes--> v05\n- v05 --refutes--> v04\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0b2a805724f14eedfe7092a7a551ec132839c2c509c09a536ac89eb96cd411ba","schema":"livegraph.boolean-propagation.v1","spec_hash":"fbff4ab0396166c0b7cd5bfd8262879d54baf9c9478c40fbc62a5529fbbc7271","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-16695c0977e8fbdbb128","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v00\n- v03 --refutes--> v04\n- v02 --supports--> v06\n- v02 --refutes--> v05\n- v01 --supports--> v02\n- v03 --refutes--> v01\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dd7d4df1058341bbb7e2eeb4cc697b06c411a4977043b09784b2def0c609b304","schema":"livegraph.boolean-propagation.v1","spec_hash":"16695c0977e8fbdbb1287596b9828996b4a0d640dba14c8851d68f18f14bff7b","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-977db84a26b9b22c9f39","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v02\n- v06 --supports--> v01\n- v03 --refutes--> v06\n- v05 --refutes--> v00\n- v05 --refutes--> v03\n- v01 --refutes--> v04\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"988baeb6a6ce1bf38dd82b51184c1816136a233a84e2bf9865d3eadb64923105","schema":"livegraph.boolean-propagation.v1","spec_hash":"977db84a26b9b22c9f3944930642dd99bec93bdafdc726a5d16b18bad06619bb","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-84ae49290e5b2cd59bb9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v04\n- v04 --supports--> v02\n- v01 --supports--> v06\n- v02 --refutes--> v00\n- v02 --refutes--> v03\n- v01 --refutes--> v05\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c22ec5ae72b3d4d4c839d0d3e08c8a06df6cdcd341393d8e9cc0978c03b5979e","schema":"livegraph.boolean-propagation.v1","spec_hash":"84ae49290e5b2cd59bb9de201a50b27ba795f164f0575621edb462d784a813e2","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-6d14de6e72f90814ccb9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v05\n- v04 --supports--> v03\n- v02 --refutes--> v01\n- v03 --supports--> v00\n- v02 --supports--> v04\n- v00 --supports--> v06\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c0a60b8554153fbe1e4ca35812c3db5a946f7e3c0b241a05e76282020dc94993","schema":"livegraph.boolean-propagation.v1","spec_hash":"6d14de6e72f90814ccb949796b0ce0e58fa28ae178e7f3530d504ac505dafb3b","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-4d0133411d28a1abf8dc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v03\n- v05 --supports--> v01\n- v03 --refutes--> v04\n- v02 --supports--> v00\n- v04 --refutes--> v05\n- v03 --supports--> v06\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d7974de4f50313bc51fae1789cc29573af24347201f93f0ccde17ce202b709aa","schema":"livegraph.boolean-propagation.v1","spec_hash":"4d0133411d28a1abf8dc28b625ebeab159639ba1814e59d37390268f7d4a6537","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-49f0a5a4a824c665d88a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v02\n- v06 --supports--> v03\n- v02 --supports--> v01\n- v06 --supports--> v04\n- v01 --refutes--> v05\n- v03 --refutes--> v00\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f7d401972e6daf2bef82c539117fbd22361f76dee2f0eb25c7bcedbdd8acec68","schema":"livegraph.boolean-propagation.v1","spec_hash":"49f0a5a4a824c665d88a290f74e08de955edbe3b6976b9e666c265ec4443e56d","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-6f811b34f357c5843d09","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v04\n- v02 --supports--> v05\n- v02 --refutes--> v03\n- v06 --supports--> v02\n- v03 --refutes--> v00\n- v03 --supports--> v01\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a41bb079edfaf9b52b64a1d25d3b8d4dce90e43f1a2d095e4c0c0199464c671c","schema":"livegraph.boolean-propagation.v1","spec_hash":"6f811b34f357c5843d0990734125b92fcbfb6d039ed6a281ea6f3a2f85b0cfa6","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-7bc5cceee80ca136ba38","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v00\n- v01 --supports--> v04\n- v03 --supports--> v06\n- v05 --supports--> v01\n- v01 --refutes--> v02\n- v02 --refutes--> v03\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"abaff68994b462f9da5d806bd916e2977152374580ed8d16311dedf87fce6729","schema":"livegraph.boolean-propagation.v1","spec_hash":"7bc5cceee80ca136ba385dcd7b85cbee271dd2e4f7d50b1dd5b65bd87d1232f8","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-668e195b2465e38a7ebf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v06\n- v01 --supports--> v00\n- v06 --supports--> v04\n- v06 --refutes--> v02\n- v05 --refutes--> v03\n- v03 --supports--> v01\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"440f5c5ea8ae285c55f41236c7404c76428893e177eccfd875a35b9bc4fa25c6","schema":"livegraph.boolean-propagation.v1","spec_hash":"668e195b2465e38a7ebff520ddc240fb1b6edfcf5fc8a6cbabb8eeb66b0dd1e7","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-9046282cbb1a4c1342d8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v03\n- v03 --supports--> v02\n- v05 --refutes--> v06\n- v02 --refutes--> v00\n- v06 --refutes--> v01\n- v03 --refutes--> v04\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2e6bbf6e796467b736d57e040051c4e9d592d8c4b66f837c8479bca698b5c252","schema":"livegraph.boolean-propagation.v1","spec_hash":"9046282cbb1a4c1342d82b8ba18addb9944b26cc975d41eefb7c3b6e57d78a9e","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-556ef7753fb975404e25","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v01\n- v02 --refutes--> v06\n- v05 --supports--> v04\n- v06 --supports--> v03\n- v06 --supports--> v00\n- v03 --refutes--> v05\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0e12d3ba9c12d39d08ee75e71af72ecd987ee27ddf7a86f589a21dd2f105bfa4","schema":"livegraph.boolean-propagation.v1","spec_hash":"556ef7753fb975404e25eed8e288f9fd6392e906646c0415bcbc75e4ca1ea5ac","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-053c6b3c83dc41a3b1ec","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v00\n- v02 --supports--> v01\n- v05 --refutes--> v04\n- v01 --supports--> v05\n- v04 --supports--> v06\n- v04 --refutes--> v03\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7075cd0311278def498abc97032307e4874d61a763eadc0c0f4247634010732b","schema":"livegraph.boolean-propagation.v1","spec_hash":"053c6b3c83dc41a3b1ecebaa36546b6bdf4db91c3f099c189e4bab078eac9b16","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-898832aa5b637b467675","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v04\n- v06 --refutes--> v00\n- v02 --supports--> v03\n- v04 --supports--> v05\n- v04 --refutes--> v06\n- v00 --supports--> v01\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"592b107768950b79c37796fafffa23c5763e4c64ed3f56c1382bfa94b8024975","schema":"livegraph.boolean-propagation.v1","spec_hash":"898832aa5b637b4676756d1ffde9136a443761188fa3b55cae11645adb4aebcb","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-4aedfbff4e1bc1295433","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v03\n- v06 --refutes--> v05\n- v02 --refutes--> v01\n- v03 --refutes--> v06\n- v04 --supports--> v02\n- v06 --supports--> v00\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d0ef2bfc4d7b3b61aa6f2485141ffe8647ad9bd0a3989382c636252fdeb79e77","schema":"livegraph.boolean-propagation.v1","spec_hash":"4aedfbff4e1bc129543315931eb485a32dd3003e174ab359a05bdb9ff991269e","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-7b94d59dfb6817f8ca40","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v06\n- v03 --refutes--> v05\n- v06 --supports--> v03\n- v02 --refutes--> v04\n- v03 --refutes--> v01\n- v02 --supports--> v00\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9015ecf32245f6183d8fcfa8586b58758f474f2a3ef2abe759dd407e1cf8c8fe","schema":"livegraph.boolean-propagation.v1","spec_hash":"7b94d59dfb6817f8ca408927f6bb666e99fbf9758c2e71b3eb01038902b5ee29","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-8b1e6c20d9b0c258d290","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v01\n- v03 --supports--> v02\n- v02 --refutes--> v00\n- v01 --supports--> v06\n- v01 --supports--> v03\n- v03 --supports--> v05\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d20b6981cab9a312d2da56305a085244684e6ee3ee8a706accf5bd7da5647a4b","schema":"livegraph.boolean-propagation.v1","spec_hash":"8b1e6c20d9b0c258d290bc4cbb79e6574c9443b14ca9d972cda9863690ec5939","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"}],\"hop\":4,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-d180d263950d305bd84a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v02\n- v02 --supports--> v03\n- v02 --refutes--> v01\n- v05 --supports--> v00\n- v01 --refutes--> v04\n- v02 --supports--> v06\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4d9ca637231215dab8a8fa55c1d7f82d208019665fbfeaa9a398693a1a381891","schema":"livegraph.boolean-propagation.v1","spec_hash":"d180d263950d305bd84a9e7272555ad0bc855d6d84c740e873d2b0ece85d76f3","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-5305f5faf016402634ea","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v05\n- v05 --supports--> v02\n- v07 --supports--> v03\n- v02 --refutes--> v04\n- v07 --refutes--> v06\n- v03 --supports--> v01\n- v05 --supports--> v00\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fe90f11f02b3a05927caae6cccd30a2a4fc676811bec2fcd5f09b12f1448d5a3","schema":"livegraph.boolean-propagation.v1","spec_hash":"5305f5faf016402634ea426eed5fc6ee0a9374d1edf4fd4e89040e529d9b8ab9","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-2a6563d9955db4fb4890","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v02\n- v05 --refutes--> v07\n- v02 --supports--> v04\n- v01 --supports--> v03\n- v01 --refutes--> v00\n- v04 --supports--> v05\n- v03 --supports--> v06\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"08988821cabb1b9545c20d788439e394cdf0535286158e78b5f1b90f4088cbba","schema":"livegraph.boolean-propagation.v1","spec_hash":"2a6563d9955db4fb489065833e0e5f35db59bf81951762152fac18f398a63efe","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":5,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-4382829ce02f70c6054a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v02\n- v04 --refutes--> v01\n- v06 --refutes--> v07\n- v07 --supports--> v03\n- v00 --refutes--> v06\n- v07 --refutes--> v05\n- v04 --supports--> v00\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6f2342a7036c510662703903c7556412e1163508bc34cc8eab3d99d337c36690","schema":"livegraph.boolean-propagation.v1","spec_hash":"4382829ce02f70c6054a5586ea0dc2e8fa495dcd78a5532feff53ddede0f5e8a","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-9bf507e6978b8b5cd0a8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v06\n- v06 --refutes--> v07\n- v00 --supports--> v02\n- v00 --refutes--> v03\n- v05 --refutes--> v04\n- v03 --refutes--> v01\n- v07 --refutes--> v05\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"896fc57ef60c2f96e2e66686a4a7ad9df2e67a6cc0229b21a97ed310899896d9","schema":"livegraph.boolean-propagation.v1","spec_hash":"9bf507e6978b8b5cd0a8504cdcf0569bbf3d91dd5298b10c9d6af3a5c9bc0d76","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-742a4bd13f808e20ca17","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v06\n- v07 --refutes--> v03\n- v04 --supports--> v01\n- v04 --supports--> v00\n- v00 --refutes--> v05\n- v02 --refutes--> v04\n- v06 --refutes--> v07\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bca4f768d1533922815d4660c8d519e9d7f511f94a8ed3319a799a2c2a4c3ffe","schema":"livegraph.boolean-propagation.v1","spec_hash":"742a4bd13f808e20ca175b7a471cd4b57910a0b1ef4c0c3bdee053ca78ebd6c4","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-d3600b003b3b241416a2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v03\n- v02 --refutes--> v07\n- v03 --refutes--> v06\n- v00 --supports--> v05\n- v00 --supports--> v01\n- v06 --supports--> v04\n- v00 --refutes--> v02\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b768f953dfe596d38531d3b5fd5da3376d81c2a0689190460843349c6da3b7fb","schema":"livegraph.boolean-propagation.v1","spec_hash":"d3600b003b3b241416a2a9fed561f3b06b4579558ee748bb4ab21bd16dc20c79","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-d83ae7a0c5b1f5ea0701","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v06\n- v06 --supports--> v05\n- v07 --supports--> v03\n- v01 --supports--> v04\n- v07 --supports--> v01\n- v00 --refutes--> v07\n- v00 --supports--> v02\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"abfa4008b601c05c9517523f0b8ec4b0f0553c35b90aacc1bb2b429a52960978","schema":"livegraph.boolean-propagation.v1","spec_hash":"d83ae7a0c5b1f5ea070130a0713d8c770c9be5b26e5376dc6bfca728d1cb3bb4","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-483cdca7d02c2372aac0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v05\n- v00 --refutes--> v02\n- v05 --supports--> v04\n- v04 --refutes--> v01\n- v04 --supports--> v03\n- v03 --supports--> v06\n- v03 --supports--> v07\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"efd5014146f3e77c1114c1f7666ac3778ded9193ef50a442fce93f38c906785f","schema":"livegraph.boolean-propagation.v1","spec_hash":"483cdca7d02c2372aac0ef6219b919e1e0ba14693ee1260ebf8817af50718c4a","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-77afe8815efe29aae020","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v02\n- v05 --refutes--> v06\n- v01 --refutes--> v05\n- v07 --refutes--> v01\n- v05 --refutes--> v00\n- v06 --supports--> v04\n- v04 --supports--> v03\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ac9e35a1116399da4ac234d41158e710709d6b98305a7c1c13fea5c40038ea58","schema":"livegraph.boolean-propagation.v1","spec_hash":"77afe8815efe29aae020cfef36fef302dac1c651f5d42db29cf1a2faa7b14792","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-349e43432240eebc9c67","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v03\n- v05 --supports--> v07\n- v07 --refutes--> v04\n- v00 --supports--> v06\n- v05 --supports--> v01\n- v04 --supports--> v00\n- v06 --refutes--> v02\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e484936efdec18634f527c5a3044b02651271b7c2bc4a2345c7c9579e92a3926","schema":"livegraph.boolean-propagation.v1","spec_hash":"349e43432240eebc9c6732c4571df6e88609af8fd688d8a7c0886d9ee918cff5","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-894c80c707ea4d910dc4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v00\n- v00 --refutes--> v06\n- v01 --refutes--> v05\n- v06 --refutes--> v02\n- v01 --supports--> v04\n- v03 --refutes--> v01\n- v05 --refutes--> v07\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9e575afa3ffb96e9a653be62661963d7a36fdb6fefbcb089d5dab0c240bad86b","schema":"livegraph.boolean-propagation.v1","spec_hash":"894c80c707ea4d910dc42f939dd418307fb0a92352fa316e6802fe3a08d52a2b","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-65780e760174d0e50d5c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v03\n- v07 --supports--> v02\n- v05 --supports--> v01\n- v00 --refutes--> v05\n- v04 --supports--> v06\n- v01 --refutes--> v04\n- v07 --refutes--> v00\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b7ab66b828a4acf4b56cc850742532c48834389e28af9037408491cc8f7c8500","schema":"livegraph.boolean-propagation.v1","spec_hash":"65780e760174d0e50d5c91cca44d0a7fa48757d5acc89a99fe997095b79c6a74","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-07348c67a1b148efdebf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v04\n- v01 --refutes--> v03\n- v05 --supports--> v01\n- v02 --supports--> v07\n- v01 --refutes--> v00\n- v05 --supports--> v06\n- v04 --refutes--> v05\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7c55e1235f8674b1acaf4feb7a9fa82d8ad861988b4af31d15b761a894fa9dea","schema":"livegraph.boolean-propagation.v1","spec_hash":"07348c67a1b148efdebf6867ce5f530f0d5772533fe2e4d8c96e728560b415e9","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-cb2d27d93f71953ec2c0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v01\n- v02 --supports--> v04\n- v06 --supports--> v03\n- v03 --refutes--> v05\n- v02 --supports--> v06\n- v00 --supports--> v02\n- v07 --refutes--> v00\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c586770ff4bf6aba881df247effd9f4c05200bd7fc6be7baa7c82d0a4f58dba0","schema":"livegraph.boolean-propagation.v1","spec_hash":"cb2d27d93f71953ec2c04e40d0bb64c228cedbde4caa521962d92e3dbae24709","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-ccb7c235b58e3806c232","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v06\n- v04 --refutes--> v00\n- v04 --supports--> v05\n- v02 --supports--> v03\n- v06 --supports--> v07\n- v01 --refutes--> v02\n- v05 --refutes--> v01\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f1652ccc82d099e4560418c5adaabbf8886d53f5cdb62488ac74267a61ae87be","schema":"livegraph.boolean-propagation.v1","spec_hash":"ccb7c235b58e3806c232cbe931dc6daf2803b35d859e2d5f774eb63e91c95b6d","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-114eaab815410b377bc9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v03\n- v00 --refutes--> v05\n- v06 --refutes--> v02\n- v04 --supports--> v07\n- v05 --refutes--> v06\n- v05 --refutes--> v01\n- v07 --refutes--> v00\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"de9f8e7cc7c768440031754837ff7ce894f6c6228ba06ab71e1da4988aa7d6b1","schema":"livegraph.boolean-propagation.v1","spec_hash":"114eaab815410b377bc9c3f1cfdaeb68eca9a472186398d8c06d04d605e226da","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-e0eca18f6b0ea79d4fb6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v01\n- v07 --refutes--> v00\n- v04 --supports--> v06\n- v04 --refutes--> v02\n- v05 --refutes--> v07\n- v03 --refutes--> v04\n- v01 --supports--> v03\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e6170a84469663e9596b2bfccaf9626d9937855089a9ae196b7cd476a88bdc69","schema":"livegraph.boolean-propagation.v1","spec_hash":"e0eca18f6b0ea79d4fb687a7f032fa09ce8c9fa51655b8c1b8e08b089308d555","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-10b50fd8c95ae3960c22","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v07\n- v03 --supports--> v00\n- v01 --refutes--> v02\n- v05 --supports--> v06\n- v02 --refutes--> v03\n- v06 --refutes--> v04\n- v03 --supports--> v05\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"690609f4b47e6bd38a3f0d3e18912352a8852c4f01171e2651e444bd1d7c0cf1","schema":"livegraph.boolean-propagation.v1","spec_hash":"10b50fd8c95ae3960c226980ebc624a7c80e206e34fd8824e9f7a1ab0bc61c0d","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-200fd01bf54f5c77f730","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v06\n- v02 --supports--> v05\n- v00 --supports--> v04\n- v00 --supports--> v07\n- v04 --supports--> v01\n- v01 --supports--> v03\n- v03 --refutes--> v02\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"569e2d10a539bcba5502ae1798b0a1b4c1eb3adecf12ac3b13af30677b60e423","schema":"livegraph.boolean-propagation.v1","spec_hash":"200fd01bf54f5c77f730e26f2b3ddd7912e3b6beafee37637ffa145af43e9a83","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-e11b1d2623e0f6ade555","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v06\n- v05 --refutes--> v03\n- v07 --supports--> v05\n- v00 --refutes--> v07\n- v05 --refutes--> v02\n- v02 --refutes--> v04\n- v07 --supports--> v01\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8d9e838aa42dffdc280496a242b58d61d94870e4fb684e2e186ad820539751b5","schema":"livegraph.boolean-propagation.v1","spec_hash":"e11b1d2623e0f6ade55588a9d682ac7a152da3dd62b2eaf9d0b0e0a0ce23c0a7","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-f94417b585c19ff38df3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v06\n- v03 --refutes--> v05\n- v07 --refutes--> v00\n- v05 --refutes--> v02\n- v02 --refutes--> v01\n- v07 --refutes--> v04\n- v01 --supports--> v07\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bf8e717111d2d6a753071df8e41fce43a73ef8def79ded1815ae5ebeddd7d920","schema":"livegraph.boolean-propagation.v1","spec_hash":"f94417b585c19ff38df324379cdcebb27db88f717083eb296b6e962c10d0b432","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-4c933d4f3a75ea200398","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v07\n- v06 --refutes--> v01\n- v02 --refutes--> v06\n- v04 --supports--> v03\n- v02 --supports--> v00\n- v06 --refutes--> v04\n- v07 --supports--> v02\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f33c0bd822a87921a57ef3f1b536447a7cbd4b7a22534d0ed7b312575f5304a2","schema":"livegraph.boolean-propagation.v1","spec_hash":"4c933d4f3a75ea2003989917226e315a8de8fe174197fe6c80e86ab0981a552e","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-bbc9c17aa840980cd943","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v00\n- v07 --supports--> v03\n- v05 --supports--> v01\n- v00 --refutes--> v04\n- v04 --refutes--> v02\n- v07 --refutes--> v05\n- v05 --refutes--> v06\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1e9c5ea63b80ce7c34708f7253a875e40cf535f68dabe65f86eb848f8f5ade71","schema":"livegraph.boolean-propagation.v1","spec_hash":"bbc9c17aa840980cd943fd5c568eed4908c6f8b0258a649dacee3fdbf6145faa","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-57e53d44228a11155694","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v07\n- v05 --refutes--> v06\n- v02 --supports--> v00\n- v04 --supports--> v03\n- v00 --supports--> v05\n- v07 --refutes--> v02\n- v05 --supports--> v01\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3b46d5784972f9835a4b2cbbcd514da2537758ebd59e43f785d4bc83963d6b8f","schema":"livegraph.boolean-propagation.v1","spec_hash":"57e53d44228a1115569495b424d4cd682a0b4ead5234303f86b443f11eab9117","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-d481cb73d53feed7f185","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v04\n- v03 --refutes--> v01\n- v04 --supports--> v03\n- v03 --supports--> v05\n- v02 --refutes--> v06\n- v02 --refutes--> v07\n- v01 --refutes--> v00\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"695043bf28e92e9ff82d58ebac4b676ce55d4038cec3aa7b12edaf648d2e847c","schema":"livegraph.boolean-propagation.v1","spec_hash":"d481cb73d53feed7f18522ec43c55739a57394535ed123fcfec7bc817de66e20","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-4d61186dc227ae5cbb71","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v04\n- v00 --supports--> v07\n- v03 --refutes--> v05\n- v03 --refutes--> v01\n- v07 --refutes--> v02\n- v04 --supports--> v06\n- v06 --refutes--> v00\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"394a4b39aea808970eeb29624940cf4bbe13f540ab602cf1caac633be3be0b5f","schema":"livegraph.boolean-propagation.v1","spec_hash":"4d61186dc227ae5cbb71dc4d8303aa2aadb2e2adf132f7391445212d7a5d9b91","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-42d79c24653a3a10f312","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v02\n- v01 --supports--> v04\n- v07 --refutes--> v05\n- v07 --supports--> v03\n- v02 --refutes--> v07\n- v06 --refutes--> v00\n- v07 --refutes--> v06\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"735f3260020aebf3c1c9aa712c834157adfeeb28a7f66180fa7734356e88c133","schema":"livegraph.boolean-propagation.v1","spec_hash":"42d79c24653a3a10f312863ed9e42dd21fdc6def4d1ea95ef4eae2e8e0a9bdf7","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-9d78b303c74fef27e3ff","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v06\n- v02 --supports--> v01\n- v06 --supports--> v02\n- v05 --refutes--> v04\n- v01 --refutes--> v03\n- v06 --refutes--> v00\n- v04 --supports--> v07\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4805c9561cdb02e51f8efe81e76505d86178ab01a2cd867d4ebf6eb44483c3c7","schema":"livegraph.boolean-propagation.v1","spec_hash":"9d78b303c74fef27e3ff7c85082d3d27a53fc37ebe63ffc6d9950f1c6d6a9849","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-5967a3b51ce95677d511","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v06\n- v01 --supports--> v05\n- v07 --supports--> v03\n- v01 --supports--> v00\n- v06 --refutes--> v01\n- v05 --supports--> v04\n- v05 --refutes--> v07\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"85e0ed6ff36222274a01b2ea486bb7108e5cfb17511de50c67ade9a73122cc26","schema":"livegraph.boolean-propagation.v1","spec_hash":"5967a3b51ce95677d5117484f1d5f6348a62d5d06f38d0a012ba19ee4b8a1074","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-a78eb64d320ab7d6a6c6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v05\n- v07 --refutes--> v03\n- v02 --refutes--> v07\n- v05 --supports--> v02\n- v03 --refutes--> v01\n- v02 --refutes--> v04\n- v00 --supports--> v06\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bbff68cc67f73dfb74ff2fe9754c4c4db11369571227e78c29663d451fe87e80","schema":"livegraph.boolean-propagation.v1","spec_hash":"a78eb64d320ab7d6a6c65e739429ffcfef965384738879ee48b435e98e239ed1","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-17ba59666017de68fcfd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v02\n- v03 --refutes--> v01\n- v07 --refutes--> v03\n- v00 --refutes--> v05\n- v07 --supports--> v04\n- v02 --supports--> v00\n- v05 --supports--> v06\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"142515d7b1e843011d20e2062a3eb1269d5414dbe8b95874277fd4cccfb8915d","schema":"livegraph.boolean-propagation.v1","spec_hash":"17ba59666017de68fcfdd23a3cea492628788d4d6918b68ab22a43f78961380a","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-b282e9bbdc0426921c12","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v01\n- v07 --supports--> v00\n- v00 --supports--> v05\n- v05 --supports--> v04\n- v03 --supports--> v07\n- v05 --refutes--> v06\n- v06 --supports--> v02\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ad66cba3e0337fe2dded2ee16cd066bf40e1fe1a648445b50afe3808ea00120c","schema":"livegraph.boolean-propagation.v1","spec_hash":"b282e9bbdc0426921c12c54ef4911d716b352f38d966ea48fde22135aaae120d","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-c4bb5408c3fa0f344254","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v04\n- v04 --supports--> v07\n- v06 --refutes--> v00\n- v05 --supports--> v01\n- v04 --refutes--> v05\n- v03 --refutes--> v06\n- v03 --supports--> v02\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e2f9b78f745f30a61fcc52319f0f751ada0b462c7efea6ca97e3d3b287ce4a25","schema":"livegraph.boolean-propagation.v1","spec_hash":"c4bb5408c3fa0f344254608fbd32a06596bf34a0612def85e96735c7be89a025","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-2e81c979b46f78aeaf5e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v05\n- v05 --supports--> v06\n- v05 --refutes--> v03\n- v01 --supports--> v02\n- v07 --refutes--> v00\n- v01 --supports--> v07\n- v06 --refutes--> v04\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5e1891f71863bdca828947ecff333c9dec806eaa76909947208a963b68b5ff48","schema":"livegraph.boolean-propagation.v1","spec_hash":"2e81c979b46f78aeaf5efa3b52ffd1c5a7feb9efdf8be141a3331ad380fc4bb1","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-e4dfd4287e8cd01fb066","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v00\n- v04 --refutes--> v03\n- v07 --refutes--> v01\n- v02 --refutes--> v04\n- v05 --supports--> v07\n- v07 --refutes--> v06\n- v01 --refutes--> v02\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5070ac07761930891452919edbf4219bfa64fc034ded97a589db55c62a3159bc","schema":"livegraph.boolean-propagation.v1","spec_hash":"e4dfd4287e8cd01fb0660ec8c01f041bc0c12b23d15413cb23cf0198090b9e83","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-904756bb333630e0421d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v06\n- v02 --supports--> v07\n- v04 --supports--> v05\n- v03 --refutes--> v01\n- v01 --refutes--> v04\n- v07 --refutes--> v03\n- v03 --refutes--> v00\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ffca98601ef5a309e8666ba221ad22e693e98f32aa451e80345e1007f9908ffc","schema":"livegraph.boolean-propagation.v1","spec_hash":"904756bb333630e0421da710d35a7b6badd7b13e8ccb9b343df0179d6143adff","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-629df3a4cb823e718188","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v04\n- v00 --supports--> v07\n- v05 --refutes--> v02\n- v07 --supports--> v05\n- v06 --refutes--> v03\n- v01 --supports--> v00\n- v06 --refutes--> v01\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"77321a2f8663b30358373d6e349f14ceac0efd2ab44fddd671bbdacd04bd72cd","schema":"livegraph.boolean-propagation.v1","spec_hash":"629df3a4cb823e7181885657076b6f9faa98056eb53450aa52d9076342ef6306","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-a4b27762548804142537","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v00\n- v03 --refutes--> v05\n- v05 --supports--> v04\n- v07 --refutes--> v06\n- v06 --supports--> v02\n- v03 --refutes--> v01\n- v04 --refutes--> v07\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"121d0aaddb7ec0ea1996b7d0656f3067841dbf276fc40b7453e465be75417417","schema":"livegraph.boolean-propagation.v1","spec_hash":"a4b2776254880414253782eb7b116981e199d6acf51ff627ffa1d6413f54e632","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-c2343e63918842d4c052","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v00\n- v00 --supports--> v06\n- v00 --refutes--> v02\n- v04 --supports--> v05\n- v01 --supports--> v07\n- v07 --refutes--> v04\n- v07 --supports--> v03\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b1722a487ce060dd901c01b1938080e23baab17df33c0f8a4e95ad075fb46207","schema":"livegraph.boolean-propagation.v1","spec_hash":"c2343e63918842d4c052f7ec3be12d783d07fc97b465f7e4334445e92c461c8d","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-9c0ee87f9e149ae8a50f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v01\n- v06 --refutes--> v02\n- v01 --refutes--> v03\n- v05 --refutes--> v07\n- v07 --refutes--> v04\n- v01 --refutes--> v06\n- v01 --refutes--> v00\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8ade3cad9abfe4e4cc8ab1effe69f4800758ebcfb8d84595dbba83305d9083f5","schema":"livegraph.boolean-propagation.v1","spec_hash":"9c0ee87f9e149ae8a50f14b682993c0b9fcc27167aa582e21e4b496d99910c96","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-f320763a727f4c462bd2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v05\n- v07 --refutes--> v01\n- v01 --supports--> v00\n- v04 --supports--> v03\n- v04 --refutes--> v02\n- v02 --supports--> v07\n- v02 --supports--> v06\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1c099ed61fa0e4dc1708102c28ca152bf758a8e390719566d3b5c1521d42b489","schema":"livegraph.boolean-propagation.v1","spec_hash":"f320763a727f4c462bd270f53b60d64f070af33408632f1ea5a06574c7317b0a","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-47428fbc397ad34dd164","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v04\n- v02 --supports--> v01\n- v00 --supports--> v07\n- v04 --refutes--> v00\n- v02 --supports--> v05\n- v06 --refutes--> v02\n- v00 --refutes--> v03\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"031a760e86b1d4deb0891d0dec7c98312bb87a424d2e735805e94e1de01c1147","schema":"livegraph.boolean-propagation.v1","spec_hash":"47428fbc397ad34dd164df47118473a3a696ef2b43f227ba0692fe1cf90dacf0","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-75a9a3b398831198207e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v04\n- v06 --supports--> v07\n- v05 --supports--> v03\n- v07 --refutes--> v05\n- v00 --refutes--> v02\n- v03 --refutes--> v00\n- v00 --supports--> v01\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"62955261d648da0abec19a4b42011552cf604f100bc0cb7e0d00ec79c6dc28e3","schema":"livegraph.boolean-propagation.v1","spec_hash":"75a9a3b398831198207e684216e1a18d36b85061683fa290da56b779c504f51c","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-a908445257dab5d9f812","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v01\n- v06 --refutes--> v00\n- v04 --refutes--> v07\n- v06 --supports--> v02\n- v04 --supports--> v03\n- v05 --refutes--> v04\n- v01 --refutes--> v05\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a5b23846dc2030c918846624d5b240c70e703dc730d2f22e5b8d9764d0ed0047","schema":"livegraph.boolean-propagation.v1","spec_hash":"a908445257dab5d9f812952a3956b7bfcc5021b504d671a1cf8d7b1f53c0cd23","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-767d7edc0f3c64b9e7fd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v05\n- v05 --supports--> v04\n- v04 --supports--> v07\n- v01 --supports--> v02\n- v07 --supports--> v01\n- v04 --refutes--> v00\n- v04 --refutes--> v06\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1b15ac395cde114cc624bd52afa7a98ef5883426658a3bd452f8806ab1305d92","schema":"livegraph.boolean-propagation.v1","spec_hash":"767d7edc0f3c64b9e7fdad5a68f65ad22e456a671dc6c4941872c1b1324b4a36","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-105916a60bd86779d803","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v05\n- v05 --refutes--> v02\n- v02 --refutes--> v07\n- v00 --supports--> v01\n- v00 --refutes--> v06\n- v03 --supports--> v04\n- v02 --supports--> v00\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1f55d236e559999cb31afe47466124c8b90620b2e4f4e76fa1a74d5b50dd5502","schema":"livegraph.boolean-propagation.v1","spec_hash":"105916a60bd86779d803dbd1ff4e59f3774decf08463817e629fdd66294f45bb","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-e9b4847118126ead5f27","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v05\n- v04 --refutes--> v06\n- v01 --supports--> v03\n- v04 --supports--> v02\n- v02 --supports--> v00\n- v07 --supports--> v04\n- v03 --refutes--> v07\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1f2b6feb35d5e3af36736d3be6635b486a15f8a0560ad5d1a69f92e251ed8211","schema":"livegraph.boolean-propagation.v1","spec_hash":"e9b4847118126ead5f2709efc61808b616403706af89fefb10c7f487c531d1a5","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-4d4e5ccaadc0556488b7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v00\n- v07 --supports--> v02\n- v03 --supports--> v04\n- v07 --refutes--> v01\n- v05 --supports--> v07\n- v03 --refutes--> v06\n- v01 --refutes--> v03\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2393f7c8c878972da88b5e9dab6c92a51b01b8be625243937c8dd5dcb1e44271","schema":"livegraph.boolean-propagation.v1","spec_hash":"4d4e5ccaadc0556488b700df4bc51ae9e9b9cfd43a0f0258a56eb7026f948788","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-15ae5bf7e77f27f39b84","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v02\n- v06 --supports--> v03\n- v02 --refutes--> v00\n- v07 --supports--> v01\n- v06 --refutes--> v07\n- v07 --supports--> v05\n- v00 --supports--> v06\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"999e9b822dbfacb1e3a2ae77790e60bd9d9e6ff4875eeef6f785ebb3f631dae2","schema":"livegraph.boolean-propagation.v1","spec_hash":"15ae5bf7e77f27f39b84c981783a18c8f708740b1602c2f425cc20dc70b5f901","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-7c8c23c33f9a3d24020c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v06\n- v04 --refutes--> v03\n- v04 --supports--> v00\n- v00 --refutes--> v01\n- v01 --supports--> v05\n- v05 --refutes--> v07\n- v04 --supports--> v02\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1d72d232eb164e95fc6f8d50c80fa5022408bae6efe6c1a30365874fc4310636","schema":"livegraph.boolean-propagation.v1","spec_hash":"7c8c23c33f9a3d24020c877e2c180bd760a8a87a9a8d57af891c6e7de10ea938","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-90d1a264c3383c63b840","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v07\n- v03 --refutes--> v00\n- v05 --supports--> v04\n- v02 --supports--> v01\n- v01 --supports--> v03\n- v00 --supports--> v05\n- v03 --supports--> v06\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3b541325729f22393dda526cf5ed9326945787f920cfd8b655ddfe04467f4e91","schema":"livegraph.boolean-propagation.v1","spec_hash":"90d1a264c3383c63b8400f0ac3e87be4c9638698a67dec0047fbb9ea12fba505","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-6a1290fc6fa8e0449b99","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v00\n- v01 --supports--> v05\n- v02 --supports--> v01\n- v00 --supports--> v06\n- v06 --refutes--> v02\n- v02 --refutes--> v04\n- v02 --supports--> v03\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bb7dba98cae853c1ca67270fde8b7a9d44907f860e734c13981e27dbf9e3c393","schema":"livegraph.boolean-propagation.v1","spec_hash":"6a1290fc6fa8e0449b99fea6226f6f397e58f364e567c27956537da5ef8f8ad5","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-fae38559f65e6371ff4c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v02\n- v04 --supports--> v03\n- v06 --refutes--> v05\n- v02 --supports--> v06\n- v05 --supports--> v07\n- v05 --supports--> v01\n- v04 --supports--> v00\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7e3a736543922b17381da823697d3f881f0d5dd647500965903ca33f1663df29","schema":"livegraph.boolean-propagation.v1","spec_hash":"fae38559f65e6371ff4c1bdf959a1a87a2ce709ac7a14cc5b842cb9fbfe0931e","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-b9cdd74b7f7d41b78410","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v07\n- v06 --refutes--> v00\n- v02 --supports--> v04\n- v00 --refutes--> v05\n- v04 --refutes--> v03\n- v00 --supports--> v02\n- v01 --supports--> v06\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2096b70d34013b4e704a6749b6fa7b9da7086736c8dbfd16779f3779c58dc881","schema":"livegraph.boolean-propagation.v1","spec_hash":"b9cdd74b7f7d41b78410bc8c0eeda055065aff3fbe09e67147afb06efffd9eb7","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-07c6628e8587d50adb93","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v04\n- v05 --refutes--> v06\n- v01 --refutes--> v07\n- v04 --supports--> v02\n- v00 --supports--> v01\n- v02 --refutes--> v00\n- v02 --supports--> v03\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c1831213a05bc6201b1bb88d258f754bf450b93e872e74e4f4e375fc4d8e58c6","schema":"livegraph.boolean-propagation.v1","spec_hash":"07c6628e8587d50adb93c4372e20b058f9c1e26eccf4b498651f058aae3a813a","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-68055662a870122227a4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v04\n- v05 --refutes--> v00\n- v01 --supports--> v05\n- v02 --supports--> v03\n- v05 --supports--> v07\n- v02 --refutes--> v06\n- v06 --refutes--> v01\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"47e22e6da4190a0fdd6dd0b974a696ce56dc6f87c2b77f873dd574a094270f7e","schema":"livegraph.boolean-propagation.v1","spec_hash":"68055662a870122227a4a266c84709748e16a0742bde86dd5679b1e05b26e1d3","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-8b1ded57b653c6399a53","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v07\n- v05 --refutes--> v00\n- v01 --refutes--> v06\n- v06 --supports--> v05\n- v01 --supports--> v02\n- v03 --refutes--> v01\n- v07 --supports--> v04\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cf9ddd7cec2096e3e734716e6f10f523c41339ef8ca6c7543e24655e7830d29f","schema":"livegraph.boolean-propagation.v1","spec_hash":"8b1ded57b653c6399a53016c6197aea7c7db00751afa7e989d3de866ad994e51","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-8ceb0b18fff2b5834ffd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v01\n- v05 --refutes--> v03\n- v04 --supports--> v06\n- v00 --refutes--> v05\n- v07 --supports--> v00\n- v00 --supports--> v02\n- v06 --supports--> v07\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"95e4955760ee5475e278d32030c1127d702b081787383eaed8f5733428ca23c3","schema":"livegraph.boolean-propagation.v1","spec_hash":"8ceb0b18fff2b5834ffd0e9e9775022bc121a15717df9d70636bd80939ab86e5","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-3fbee15510808f78ebc6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v06\n- v00 --refutes--> v01\n- v04 --refutes--> v02\n- v01 --refutes--> v05\n- v02 --refutes--> v00\n- v05 --refutes--> v03\n- v05 --supports--> v07\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7958e15cb50e2604b7ebc542f9bd680f76c24dbe772ea3f18de72c3bd6a66aae","schema":"livegraph.boolean-propagation.v1","spec_hash":"3fbee15510808f78ebc6d71dd1ec5770619fd6bba8dc61c16a150d2a973a45d3","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":5,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-7ee098f83acfbe53580d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v01\n- v06 --supports--> v07\n- v07 --supports--> v00\n- v07 --supports--> v02\n- v01 --supports--> v04\n- v04 --refutes--> v03\n- v05 --supports--> v06\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"09ccba98032cd97e02f6b88977edfc3038cd3a15c3d7e274ca5d5b049281a5e7","schema":"livegraph.boolean-propagation.v1","spec_hash":"7ee098f83acfbe53580d673e8e7be4b33eb9f0d35f107a448b927326b8a4fb71","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-74aafd69c751f44b73f9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v04\n- v04 --refutes--> v02\n- v07 --supports--> v06\n- v04 --refutes--> v03\n- v00 --supports--> v07\n- v05 --supports--> v01\n- v01 --supports--> v00\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"131fbd2d3b0c6cac4aff2ee80f0c8df30fb5a761a7a56b171dcf7c28c564bb8e","schema":"livegraph.boolean-propagation.v1","spec_hash":"74aafd69c751f44b73f992469dea68efd8421c8f12e4f51a33c8d500880e6108","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-2201cc8691e10cf6d99d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v00\n- v01 --refutes--> v04\n- v03 --refutes--> v07\n- v02 --supports--> v05\n- v03 --refutes--> v06\n- v04 --refutes--> v03\n- v06 --refutes--> v02\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1ad2f497c75b783502afa1e23a96c9b76017cd727631b49d011ae373055ea85a","schema":"livegraph.boolean-propagation.v1","spec_hash":"2201cc8691e10cf6d99d4ac4ab9df6fbc0e0625ce8973f72edc4b0c7b6499ea3","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-49f6f1a2e94e06359a74","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v01\n- v06 --refutes--> v02\n- v01 --refutes--> v03\n- v04 --supports--> v00\n- v04 --supports--> v05\n- v03 --supports--> v07\n- v06 --supports--> v04\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b3a455f3c989826efea3d433a748ec38db6312e578820d2e45e469a55fb162db","schema":"livegraph.boolean-propagation.v1","spec_hash":"49f6f1a2e94e06359a7491797cd0fb709f708b6602c682934935e5f51428f5da","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-5cdf718780af89315a46","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v04\n- v05 --refutes--> v00\n- v07 --supports--> v02\n- v00 --refutes--> v01\n- v06 --refutes--> v05\n- v00 --supports--> v03\n- v02 --refutes--> v06\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3947f131f551818cbacd00c95eb40b1b4b515186476f95ee95ccb3504eacdf86","schema":"livegraph.boolean-propagation.v1","spec_hash":"5cdf718780af89315a4652b120631a1908a1bca7f147bf31e562d1c59a85a0f3","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-2bce8b95372a2b607080","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v01\n- v03 --refutes--> v00\n- v06 --refutes--> v07\n- v01 --supports--> v05\n- v00 --supports--> v02\n- v04 --refutes--> v03\n- v06 --refutes--> v04\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1b519df5109ad58270335ca61a259c327224007c477fa6c865f1876fd0f5f90d","schema":"livegraph.boolean-propagation.v1","spec_hash":"2bce8b95372a2b607080752a184cccb90114e5e25c260b586b5e275732c55469","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-d931b03dfca752db753a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v04\n- v02 --supports--> v07\n- v05 --supports--> v06\n- v06 --supports--> v01\n- v00 --supports--> v05\n- v03 --supports--> v02\n- v02 --supports--> v00\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5281404ca8400115acda4908cb235b9fa7dd571398e8ae1d93324eff6c94ead6","schema":"livegraph.boolean-propagation.v1","spec_hash":"d931b03dfca752db753a7284fc3cc60e880c37bd487c0094f72a1e7ffd0a7b57","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-06fc3f23c01c692812d2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v03\n- v05 --refutes--> v00\n- v02 --refutes--> v06\n- v06 --refutes--> v07\n- v01 --refutes--> v05\n- v06 --refutes--> v04\n- v00 --refutes--> v02\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"283ae3dc17ada4fc4940aae2cb75c85d3334779569563f98ac803d9fab30762c","schema":"livegraph.boolean-propagation.v1","spec_hash":"06fc3f23c01c692812d2e9f381e83a6321ddc15b7792cf694d223cf9c6f37a92","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-201ae5f06e178ca6f1f9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v02\n- v00 --supports--> v03\n- v05 --supports--> v07\n- v00 --supports--> v06\n- v01 --refutes--> v05\n- v04 --refutes--> v01\n- v07 --refutes--> v00\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8207c2edf633d25ba7907f17abfa0699f5b98d70d7e0266be42f1f9525c47b65","schema":"livegraph.boolean-propagation.v1","spec_hash":"201ae5f06e178ca6f1f95d104ba840fe453738eb6c92437a0a0e9ba36ec4eefc","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-ba68de6f00006c769d4e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v06\n- v03 --supports--> v02\n- v01 --refutes--> v03\n- v03 --supports--> v07\n- v05 --refutes--> v04\n- v00 --supports--> v01\n- v04 --supports--> v00\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6216861011682e135563c8efbf47dec9a431a6d8ae12da8f4f8992e1533b6307","schema":"livegraph.boolean-propagation.v1","spec_hash":"ba68de6f00006c769d4e38dc4801d09a7bc001a7823542d700341215292dbd6e","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-a68815775fbd0d933240","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v04\n- v00 --supports--> v03\n- v06 --refutes--> v05\n- v02 --refutes--> v06\n- v03 --supports--> v02\n- v06 --supports--> v01\n- v05 --refutes--> v07\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2e6b183f09f3befc2e850eca0f2af2f3f841bae1c2d3daddf7d25b62a2b33c53","schema":"livegraph.boolean-propagation.v1","spec_hash":"a68815775fbd0d933240f21abe25107372977750c3dddcf03ffdd8facc2146d0","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-4e6c4473050c92d3a8e9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v00\n- v03 --supports--> v04\n- v02 --refutes--> v05\n- v05 --supports--> v06\n- v03 --supports--> v07\n- v04 --supports--> v01\n- v01 --refutes--> v02\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cec542032650e2de1b30f0877c10a56e57b08fea431b10b5fd583304903795d9","schema":"livegraph.boolean-propagation.v1","spec_hash":"4e6c4473050c92d3a8e95606688dce2f41ce9a63d6a8658b332aafbd8b4d9717","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-f1173d70247f321dff43","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v03\n- v03 --refutes--> v06\n- v05 --refutes--> v00\n- v07 --supports--> v05\n- v04 --supports--> v07\n- v03 --supports--> v02\n- v01 --supports--> v04\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8b734a2b2d9c2cd7ee813e62a45419cc7111a4af1794309ed850108cf6db9382","schema":"livegraph.boolean-propagation.v1","spec_hash":"f1173d70247f321dff4310aea9a4f8a61f24c3c6f85b170caba3e807b46c153c","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-727a5989e4aa24767d25","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v05\n- v06 --refutes--> v01\n- v04 --supports--> v03\n- v03 --supports--> v07\n- v07 --refutes--> v06\n- v07 --supports--> v00\n- v02 --supports--> v04\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6c31651042f81ea73176cf090d4655d4cdd641fbd32475f5a95b79a71e176b70","schema":"livegraph.boolean-propagation.v1","spec_hash":"727a5989e4aa24767d25f4f83ceab3343a85882bdc6a9e34c6fa97eea9ca5a30","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-910139bdb8703b477662","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v03\n- v03 --supports--> v07\n- v04 --supports--> v05\n- v06 --supports--> v00\n- v01 --supports--> v06\n- v05 --refutes--> v02\n- v06 --refutes--> v04\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6e99b80169ec4612b4c4747f8681e5ba1754b4f0f54ee8fa1aeac14aea54dc6a","schema":"livegraph.boolean-propagation.v1","spec_hash":"910139bdb8703b4776623d5a89695581e54240b61dd917adf2641cbf086dd4c3","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-3af0c1ab8fc5a5a55a96","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v06\n- v07 --supports--> v02\n- v07 --refutes--> v01\n- v01 --supports--> v03\n- v04 --refutes--> v05\n- v06 --refutes--> v07\n- v04 --supports--> v00\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5f9e09bc80e7fc76d5cfe44d7c9bdcbb5bf990e087dad4c83c83e5221c7e8c88","schema":"livegraph.boolean-propagation.v1","spec_hash":"3af0c1ab8fc5a5a55a96ba07e8d87dd5903922e2f725d0e7493fd3c1250402b5","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-8dafab8c6d3ee95801ce","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v00\n- v00 --supports--> v05\n- v00 --supports--> v02\n- v04 --supports--> v01\n- v03 --refutes--> v06\n- v03 --refutes--> v07\n- v05 --supports--> v04\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"78fc3dcde1bfa06d0ded161bd04284d3fe085bc3855f5afb7b5d99d9ecc752e8","schema":"livegraph.boolean-propagation.v1","spec_hash":"8dafab8c6d3ee95801ce7dc65318b656531d7d10829d56b75a9613005fd9770d","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-1e6c8fdba4a47bd073ca","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v07\n- v07 --supports--> v04\n- v03 --supports--> v05\n- v03 --supports--> v02\n- v03 --refutes--> v00\n- v04 --supports--> v01\n- v01 --supports--> v03\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7804ce1ca26cabdc8e0a3b87adea03cf99271584efc723b58f027ea15d95d0ec","schema":"livegraph.boolean-propagation.v1","spec_hash":"1e6c8fdba4a47bd073cae1c94f2d85cfdc7a0e4a786f76f57c8031272caaaab1","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-1d14123527363e3aaa12","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v00\n- v02 --refutes--> v06\n- v06 --supports--> v07\n- v06 --refutes--> v05\n- v04 --refutes--> v03\n- v00 --refutes--> v02\n- v04 --supports--> v01\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0198511bd35d2ca5b5cd0677f2d5033b69fa0a6456fe5aeedad653c00acca833","schema":"livegraph.boolean-propagation.v1","spec_hash":"1d14123527363e3aaa120dbf8f0fde6da01a56a120aedc8dee6ea6363d63eada","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-64846df5e5e580674ce2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v02\n- v01 --supports--> v04\n- v06 --refutes--> v07\n- v02 --supports--> v01\n- v03 --supports--> v05\n- v04 --supports--> v00\n- v06 --supports--> v03\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"293ecbe36fd98e30f6ced07dc138ca02755942166e908455b29f5a7bf528fc69","schema":"livegraph.boolean-propagation.v1","spec_hash":"64846df5e5e580674ce265407f8ffb2f64c3ec5e091f74a286058dcb331aa4fd","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-61b5e4807b308c095b92","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v04\n- v03 --refutes--> v05\n- v03 --refutes--> v06\n- v02 --refutes--> v07\n- v00 --supports--> v03\n- v05 --supports--> v01\n- v07 --refutes--> v00\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6807b5f2cd3c293fdad43a422ddcbb060d32d21ff4776a6577155f7a2ba6022f","schema":"livegraph.boolean-propagation.v1","spec_hash":"61b5e4807b308c095b92f11e550aa898071c76f2434b3356fdfc0e6724196c4f","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-fada9b3621fcbd40cc40","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v03\n- v05 --supports--> v04\n- v07 --refutes--> v05\n- v05 --supports--> v06\n- v02 --refutes--> v00\n- v01 --supports--> v07\n- v06 --refutes--> v02\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4aaedb3c1fb2101c45640d4bcf08b109145f12a60e1aee7c07025931e1ccf082","schema":"livegraph.boolean-propagation.v1","spec_hash":"fada9b3621fcbd40cc40238352cddbdbd6071137a020a8e1235656261d6eaa32","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-7910eda20f3de1a14072","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v00\n- v07 --supports--> v04\n- v02 --supports--> v01\n- v04 --supports--> v06\n- v06 --supports--> v03\n- v02 --refutes--> v07\n- v06 --supports--> v05\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8422fd3b7f7351da8acde045d84edc0b76e0d5bf86089d8dc46748a829d59910","schema":"livegraph.boolean-propagation.v1","spec_hash":"7910eda20f3de1a14072409f8df7dae4e01f26aa3f892ab75235c5e55d890272","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-e6c784ebc962bde5a9df","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v01\n- v01 --refutes--> v07\n- v07 --refutes--> v00\n- v02 --refutes--> v06\n- v00 --supports--> v04\n- v01 --supports--> v03\n- v02 --refutes--> v05\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"12a3e9b1d002fdeae455a0e0d34a34109d56acd6ed296f699e6385b98e0c612f","schema":"livegraph.boolean-propagation.v1","spec_hash":"e6c784ebc962bde5a9df08603a655b51f3001d53ad6b4c82ab1b4ac6d389165d","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-7fd887fb35a2c9efd16f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v03\n- v07 --refutes--> v04\n- v03 --refutes--> v06\n- v01 --supports--> v05\n- v00 --refutes--> v07\n- v07 --supports--> v02\n- v01 --supports--> v00\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"53ec3931130dd2dbfb376275ea1d926204578372c5d22e4b0df219aa45e2053b","schema":"livegraph.boolean-propagation.v1","spec_hash":"7fd887fb35a2c9efd16f75beea3e46f3430c66cf3e89679d33c54203c9745b73","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-fe54606f8f42d8765b76","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v02\n- v07 --refutes--> v04\n- v05 --refutes--> v06\n- v01 --refutes--> v05\n- v04 --supports--> v01\n- v05 --supports--> v03\n- v05 --supports--> v00\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"53f2a4c1d7a6b7a65caef9b7759b72296b978bfb7699f81e225e3beeaeb5ce44","schema":"livegraph.boolean-propagation.v1","spec_hash":"fe54606f8f42d8765b76388143bb11e0ee528a47d2fb0651097d9c1d58cb3568","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-bead570ec128c2afbde7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v00\n- v04 --refutes--> v06\n- v05 --refutes--> v04\n- v05 --refutes--> v01\n- v06 --refutes--> v07\n- v07 --refutes--> v02\n- v02 --supports--> v03\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"78a04304bf6c692d233db30fc4ef3990efb446e05b19756ca366630be3cfb9f0","schema":"livegraph.boolean-propagation.v1","spec_hash":"bead570ec128c2afbde79f1a5c83ab2a169ac695230776c8a1b5f99c57277a63","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-57fa28d4eeee620e416f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v00\n- v03 --supports--> v01\n- v01 --supports--> v07\n- v00 --supports--> v03\n- v01 --supports--> v04\n- v07 --supports--> v05\n- v07 --supports--> v06\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a27f2ed8adf03ba5e31f5d71316809e418727d66a16ee924aee6f6819735af5e","schema":"livegraph.boolean-propagation.v1","spec_hash":"57fa28d4eeee620e416f907176d5fd7c227405e38abe245ceed3cd01821bf309","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-2067179f2e145d2e9fa2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v07\n- v01 --refutes--> v02\n- v01 --supports--> v03\n- v03 --refutes--> v05\n- v05 --supports--> v04\n- v00 --supports--> v06\n- v06 --supports--> v01\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a2e01ba16148e8f92419f827185a28dab3ae9b675e0a5d10aa0e7cb3a682f91f","schema":"livegraph.boolean-propagation.v1","spec_hash":"2067179f2e145d2e9fa27326900ade51e70a5fea89374e8f2398c72ce4153d4e","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-c90a59e50af5b2ee2cf2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v01\n- v02 --refutes--> v03\n- v03 --supports--> v00\n- v05 --refutes--> v07\n- v00 --supports--> v06\n- v04 --supports--> v05\n- v00 --supports--> v04\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"49aa9cd9e8cc2b0ebfa7f8f8e52b6828d2cbcc62b8c021a1d4c45157731abe2f","schema":"livegraph.boolean-propagation.v1","spec_hash":"c90a59e50af5b2ee2cf23c12cae5ad4068498e83b0e0a98691dcb306cbe26601","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-b6720b597a1fe79c5cec","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v02\n- v01 --refutes--> v07\n- v03 --refutes--> v00\n- v05 --supports--> v06\n- v02 --supports--> v03\n- v03 --supports--> v05\n- v01 --supports--> v04\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6ad5518c75af9173479bc952e4a9c8e4854d1fb3aa7534b02801453f5bd7938e","schema":"livegraph.boolean-propagation.v1","spec_hash":"b6720b597a1fe79c5cec720375c373d615cc2622f03666b6859dc359fb2414f6","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":5,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-1c648be595b279523cc1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v04\n- v04 --supports--> v01\n- v00 --refutes--> v06\n- v05 --supports--> v00\n- v02 --supports--> v03\n- v02 --supports--> v07\n- v04 --supports--> v02\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f76c914f9ee9f6781fb8a9c3c7c9a543a928a4ea5642658f472cc1700725e7ad","schema":"livegraph.boolean-propagation.v1","spec_hash":"1c648be595b279523cc10691559ff6a5de50b92b5323460f330e447353ce7dc8","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-3d9b9be9f4e136b4436a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v05\n- v07 --supports--> v06\n- v07 --supports--> v04\n- v00 --refutes--> v01\n- v03 --supports--> v00\n- v06 --refutes--> v03\n- v01 --supports--> v02\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5b8a9f8702a2ed264e7d245b47f5e9cbf0889fb3767265684a0d26148a69e2bd","schema":"livegraph.boolean-propagation.v1","spec_hash":"3d9b9be9f4e136b4436a1edd89f70af9aab16ed69b67d6aa1274eef172154ce2","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-ee95b07b504526da5472","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v01\n- v06 --supports--> v00\n- v01 --supports--> v03\n- v06 --supports--> v07\n- v01 --supports--> v05\n- v04 --supports--> v06\n- v02 --refutes--> v04\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"34507b2d8428769dd8cb09ee65d9be9ffb000c375e190945c4c826bdb7c30e95","schema":"livegraph.boolean-propagation.v1","spec_hash":"ee95b07b504526da547299e5b390dd43dc615e0845011b830f54299642a47af0","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-16ca496819b4b3400892","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v05\n- v01 --refutes--> v04\n- v01 --refutes--> v00\n- v01 --supports--> v07\n- v02 --supports--> v01\n- v06 --supports--> v03\n- v03 --supports--> v02\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"90a269edb2c4851e272f885c14e4a69f0fc961822d7dc9a0ad51b23b956bb023","schema":"livegraph.boolean-propagation.v1","spec_hash":"16ca496819b4b3400892a2ac01a9704839305e573b3ca7dcbdd9fe1c34ec6913","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-4d9bff1a976d8458dc52","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v06\n- v05 --refutes--> v04\n- v06 --supports--> v07\n- v06 --refutes--> v00\n- v01 --refutes--> v05\n- v04 --supports--> v02\n- v02 --supports--> v03\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9f5129fc54ba8271138a34be0c6e93e4cf794b45e703113059db53fa3d272636","schema":"livegraph.boolean-propagation.v1","spec_hash":"4d9bff1a976d8458dc52e66788ce6b586c277ef16a3245311228250d4ea86fd3","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-90c2e063a572910798ec","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v07\n- v07 --supports--> v02\n- v06 --refutes--> v03\n- v01 --supports--> v04\n- v01 --supports--> v06\n- v07 --supports--> v00\n- v00 --supports--> v01\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8a59522dcb86352f36f45ba2f0ef7fcefd1c9120b8f52f39deb2776f17b47d0a","schema":"livegraph.boolean-propagation.v1","spec_hash":"90c2e063a572910798ec2bd596a9e59d47679e134e3c5c7c0861ba4c59977fdd","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-c666d1d735b3fad72129","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v06\n- v02 --supports--> v07\n- v01 --refutes--> v03\n- v01 --supports--> v02\n- v05 --refutes--> v04\n- v07 --refutes--> v00\n- v00 --refutes--> v05\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f4fa5baa68adb3afe9abc7df547e16aea9c8dbc58901ccd6e1cb9065d27f5991","schema":"livegraph.boolean-propagation.v1","spec_hash":"c666d1d735b3fad72129fe77b016e342a4172d396bc0c4ec1b8c33f3f0b60bc1","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-c3b5cbb7212cf018f7fe","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v03\n- v05 --supports--> v06\n- v04 --supports--> v00\n- v02 --supports--> v07\n- v05 --supports--> v01\n- v00 --supports--> v05\n- v01 --supports--> v02\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"68a5843da1a505f8303e52c2eb4b9ce5502fb77dfa64ca081b711dad37ce5ea0","schema":"livegraph.boolean-propagation.v1","spec_hash":"c3b5cbb7212cf018f7fe5e40d588beefb38ebb3acaf2f92ca7378ad0a09a5c17","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":5,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-3253069fa68462c1623f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v06\n- v03 --supports--> v05\n- v04 --refutes--> v00\n- v04 --supports--> v07\n- v06 --supports--> v04\n- v02 --supports--> v03\n- v05 --supports--> v01\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ff5784ffa0287ecc2c90c3efc16ed523bd03e3d01fe9c16b9a7da4f12a7c1b9f","schema":"livegraph.boolean-propagation.v1","spec_hash":"3253069fa68462c1623f605bbe9e28b38363f46b3d9a11a8ee258c0390974e76","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-ed7540a10534724c2298","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v00\n- v02 --supports--> v04\n- v01 --supports--> v03\n- v03 --supports--> v02\n- v05 --supports--> v01\n- v05 --refutes--> v06\n- v03 --refutes--> v07\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6ecd6ad256984d8b807e0c0d86b53302d7d6460dc2800629dc4fee190a6a241d","schema":"livegraph.boolean-propagation.v1","spec_hash":"ed7540a10534724c2298b3a2f17d6acb48d5fec4b32634884355ead7a71448ee","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-67b5dc46629a0871132a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v06\n- v05 --refutes--> v07\n- v05 --refutes--> v02\n- v06 --refutes--> v05\n- v03 --supports--> v04\n- v06 --refutes--> v01\n- v07 --supports--> v00\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8ac056c33c04663e5e372ed2876781e292e5e708190ae69d31cac61f34c1f746","schema":"livegraph.boolean-propagation.v1","spec_hash":"67b5dc46629a0871132a977e7c54578044ac7d7b93d50cec407a3111d6b1503a","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-053611d1686c8a748971","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v06\n- v04 --supports--> v05\n- v07 --refutes--> v02\n- v07 --refutes--> v01\n- v04 --refutes--> v00\n- v01 --refutes--> v04\n- v05 --supports--> v03\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"17fb0335e1b30a16a1b26d802c388fda260cc85524bf1aa86a3f4c0206ad0c6c","schema":"livegraph.boolean-propagation.v1","spec_hash":"053611d1686c8a7489719b52cdefda6864cb256dcd580028a28d9b13dc4c498a","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-409fac70c7b6fefcf49e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v05\n- v06 --refutes--> v03\n- v00 --supports--> v06\n- v04 --supports--> v00\n- v04 --refutes--> v01\n- v02 --supports--> v04\n- v02 --refutes--> v07\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"65a766b2871310073a4a02d7f0ad7224f98d1fe0be0c7bcc6d572de214683785","schema":"livegraph.boolean-propagation.v1","spec_hash":"409fac70c7b6fefcf49e618a00138e4b89dab7dfa041dc0b9fbe667cf5ada5ac","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-9602e70c68a913aa3c49","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v04\n- v07 --refutes--> v03\n- v02 --refutes--> v00\n- v06 --refutes--> v02\n- v07 --supports--> v05\n- v04 --supports--> v06\n- v04 --refutes--> v01\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2e4a9d74cf8348a3cac96b4ea93c39836cbb2118aba90ef17944d8988578ca6c","schema":"livegraph.boolean-propagation.v1","spec_hash":"9602e70c68a913aa3c49a5ee82b713d529921a9bcf1173818cc435f298b605dc","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-7d67dcd8b4aff6bca696","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v01\n- v00 --supports--> v04\n- v02 --refutes--> v07\n- v04 --supports--> v02\n- v03 --refutes--> v00\n- v04 --refutes--> v05\n- v06 --refutes--> v03\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"955b1701ccbf6a1c460617c516cf016ba18f3c4b2a4d7c29ab6584765841a78b","schema":"livegraph.boolean-propagation.v1","spec_hash":"7d67dcd8b4aff6bca696b3859077a6edec7338c80f122cb3a3e3aa4efc9f971f","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-9779a81c29c79908a2db","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v02\n- v06 --refutes--> v05\n- v06 --refutes--> v00\n- v02 --supports--> v07\n- v00 --refutes--> v03\n- v04 --refutes--> v06\n- v04 --refutes--> v01\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"396d18ac9a27e2b13638aa7ebe553d5ec5ac155bc1878505dfab326bd8b54204","schema":"livegraph.boolean-propagation.v1","spec_hash":"9779a81c29c79908a2db777d625fe0320665b8b0fba95d271f4680035fdbbf5c","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-953a99a2f60184da5826","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v05\n- v05 --supports--> v06\n- v06 --refutes--> v03\n- v04 --refutes--> v00\n- v00 --supports--> v02\n- v04 --refutes--> v07\n- v05 --refutes--> v01\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e8f4650b76a9391e271b65d39556a1c37b3f25b7ea7a26a6564cd7c6aa232d64","schema":"livegraph.boolean-propagation.v1","spec_hash":"953a99a2f60184da5826327b72c390898371c3574031e9638f55304ed7dcb627","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-5b3aefbd8f57c30db81b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v02\n- v00 --supports--> v06\n- v02 --refutes--> v04\n- v04 --supports--> v05\n- v07 --refutes--> v03\n- v04 --refutes--> v01\n- v04 --refutes--> v00\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"800f8af449f98c0c44c0565444871e64a2c771f379a888955d25b7bad0e7b88d","schema":"livegraph.boolean-propagation.v1","spec_hash":"5b3aefbd8f57c30db81b77c675716bd98114995295b8c8c2b5c09099fd2a2e40","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-630cd3282e0f26c52c12","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v01\n- v00 --supports--> v06\n- v05 --refutes--> v02\n- v01 --supports--> v04\n- v00 --refutes--> v07\n- v02 --refutes--> v03\n- v04 --refutes--> v05\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"806ee5fb0dfd7c3598904a7e3aecc03ceae35fe1fc511433c77ab04d3e49e9ea","schema":"livegraph.boolean-propagation.v1","spec_hash":"630cd3282e0f26c52c12a89e2526e5ee14f72a4b4e9bf095362123b1be70e18f","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-6cbf7824d1505f018200","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v07\n- v07 --refutes--> v05\n- v01 --refutes--> v04\n- v07 --refutes--> v01\n- v07 --refutes--> v02\n- v00 --supports--> v06\n- v04 --refutes--> v00\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"820c7686fd8b8d5da125044e6e9dd6d028c722f289e7180a444cefa8a0513fcd","schema":"livegraph.boolean-propagation.v1","spec_hash":"6cbf7824d1505f01820090caa3330f1159607417949edfdd8e71d51f5e0a77a9","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-c9ee928fd0fc4b38d8a6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v05\n- v07 --refutes--> v03\n- v02 --supports--> v04\n- v06 --supports--> v02\n- v07 --refutes--> v00\n- v00 --supports--> v01\n- v04 --supports--> v07\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"67d8b2d44f1c467d6389563ba839c514b146498aff4f5aedc9ae5ee306480093","schema":"livegraph.boolean-propagation.v1","spec_hash":"c9ee928fd0fc4b38d8a67597dc764ec7fcea7680a0381799ab0a8002789680a8","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-82d501ebb6f4e6999283","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v04\n- v07 --supports--> v03\n- v00 --refutes--> v01\n- v04 --refutes--> v02\n- v07 --refutes--> v00\n- v01 --supports--> v05\n- v04 --supports--> v07\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"931b9315d695cd8c9801289639b4430b9371a9550df5f4d62aaf32502f4f24f5","schema":"livegraph.boolean-propagation.v1","spec_hash":"82d501ebb6f4e699928330305698274986ff67a0ae0a5fa0856a0e2189187ceb","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-2e9895d4e5e3864c9518","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v02\n- v05 --refutes--> v01\n- v02 --supports--> v07\n- v00 --supports--> v04\n- v00 --supports--> v03\n- v05 --supports--> v06\n- v07 --supports--> v05\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"261ad441b170c1e53b57726513ac64fac62cd55c82fafbf58024dabd173f18bb","schema":"livegraph.boolean-propagation.v1","spec_hash":"2e9895d4e5e3864c95183495e543ee6d6ba20cc5b8114f0784edfcd45d919fe6","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-b2b0d820fcaa42b13888","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v00\n- v01 --refutes--> v04\n- v04 --refutes--> v05\n- v05 --refutes--> v07\n- v05 --refutes--> v02\n- v03 --supports--> v06\n- v06 --refutes--> v01\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c84bf78c753f9f3abb5a7bd866683bfeb7e1c369fa3f0f9db425f97e629984fd","schema":"livegraph.boolean-propagation.v1","spec_hash":"b2b0d820fcaa42b138883db90bb13635c0a046fc695ae06e344c403611620158","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-8665468a4f97c8a3c100","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v06\n- v03 --refutes--> v02\n- v05 --supports--> v00\n- v04 --supports--> v03\n- v07 --supports--> v04\n- v03 --refutes--> v01\n- v00 --supports--> v07\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1fbce22d887ea903d9bbed04838a528c358eed4fa246db8ca207ea5ad2af7a26","schema":"livegraph.boolean-propagation.v1","spec_hash":"8665468a4f97c8a3c100c90350acb962e5093bab07b5b0364caddc3f8d0817ea","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-4e056b375944cdd8699c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v05\n- v01 --supports--> v04\n- v01 --supports--> v00\n- v04 --supports--> v03\n- v05 --refutes--> v01\n- v04 --refutes--> v06\n- v03 --refutes--> v07\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5162f5d9030c06decf8cee4087454e5637ab4a664dd34dc9ceaa20a1cc12a219","schema":"livegraph.boolean-propagation.v1","spec_hash":"4e056b375944cdd8699c496fa7ffce560fc7f47b92663b30a438c40c847ced4d","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-1948b6b119f774028949","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v01\n- v01 --supports--> v06\n- v07 --refutes--> v00\n- v00 --refutes--> v03\n- v06 --supports--> v07\n- v01 --supports--> v02\n- v00 --supports--> v04\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c762cd6878fbb1e6b61011816491fd6e04fa53d2f6c223029100cb24d58ecef7","schema":"livegraph.boolean-propagation.v1","spec_hash":"1948b6b119f774028949a54c33586c9adab35f8341af29833e6a79a36ad0d8b6","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-40e36a965562faa27f0f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v00\n- v01 --supports--> v03\n- v01 --refutes--> v07\n- v02 --refutes--> v01\n- v06 --refutes--> v02\n- v03 --refutes--> v04\n- v01 --supports--> v05\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1bcdb4ad294f8f48efdbee874baeaef8363d5d78e0cf73a2ac3e97eabe2168e1","schema":"livegraph.boolean-propagation.v1","spec_hash":"40e36a965562faa27f0f29ff75d3ee1b71b5b77601cc1f01fd540b2518a47eaa","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-bd0a084e47323135fc32","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v03\n- v04 --supports--> v00\n- v06 --supports--> v07\n- v06 --supports--> v01\n- v05 --supports--> v02\n- v03 --supports--> v06\n- v04 --refutes--> v05\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4090bd69b412ad0482001978a74019f2fd6dc918e2b638e0d44a9dab4d7e97a3","schema":"livegraph.boolean-propagation.v1","spec_hash":"bd0a084e47323135fc32f219b2d6412bc8a02dca740315bc2c8ee32e30febda6","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-11a9b467f85158fe07d3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v00\n- v01 --refutes--> v05\n- v05 --supports--> v07\n- v00 --supports--> v02\n- v02 --refutes--> v03\n- v05 --refutes--> v06\n- v02 --refutes--> v04\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"49a988ac2ddd0448d181e4a7c488fd8532c1d51057fb0e3daa5b13d59ee546d8","schema":"livegraph.boolean-propagation.v1","spec_hash":"11a9b467f85158fe07d3685ad31b18ceb717fb853704725e6561a5c67c259e19","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-ecbfd80cf51471a2c5b1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v02\n- v03 --refutes--> v05\n- v02 --supports--> v07\n- v04 --supports--> v03\n- v01 --refutes--> v00\n- v07 --supports--> v04\n- v03 --refutes--> v06\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"50f7fb4d3b6fc705887ff830e91087fc16a3ac1248a981c3a665117f59e11a4a","schema":"livegraph.boolean-propagation.v1","spec_hash":"ecbfd80cf51471a2c5b17db544d75bcfe7759d13a8bde6937045f3799ab00a36","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-ee12c774bf46db32f58a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v05\n- v07 --supports--> v04\n- v01 --refutes--> v07\n- v06 --supports--> v00\n- v02 --refutes--> v01\n- v07 --supports--> v03\n- v00 --refutes--> v02\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"710e5f0abb964cc701671c608e702bc457efdaee95c612506cf0c9f4b6d7b27a","schema":"livegraph.boolean-propagation.v1","spec_hash":"ee12c774bf46db32f58a2f2b1b7b8e5f5b887308d5ecf514a17c88be371787a4","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-0cc43310d88f228731cd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v04\n- v02 --refutes--> v07\n- v03 --refutes--> v02\n- v02 --supports--> v00\n- v00 --supports--> v01\n- v05 --refutes--> v03\n- v01 --supports--> v06\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"642963ce531a889b0eaa4fbc985cea367b0bbb374dab71310771219eba95d328","schema":"livegraph.boolean-propagation.v1","spec_hash":"0cc43310d88f228731cddd2bccb35523869a2ef2613c8e151572c15bea8899ee","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-fa87efaf45659606d5fc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v02\n- v06 --supports--> v03\n- v07 --refutes--> v06\n- v05 --refutes--> v01\n- v01 --supports--> v00\n- v05 --supports--> v04\n- v03 --refutes--> v05\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b42045ea8bd2ac936a035e398daaef5ce65a5095b3a5eff04a8a483f5596afac","schema":"livegraph.boolean-propagation.v1","spec_hash":"fa87efaf45659606d5fce85b0a78418bb9646899e019bcf751d6e24dcee6e44e","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-199f0cebdd090e7426e6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v00\n- v05 --refutes--> v06\n- v00 --refutes--> v01\n- v03 --refutes--> v02\n- v01 --refutes--> v03\n- v00 --refutes--> v07\n- v02 --refutes--> v04\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9f48ea002a02c682025571c5268b7d955b44fc4f39e027273f8569f817b3f530","schema":"livegraph.boolean-propagation.v1","spec_hash":"199f0cebdd090e7426e6130fc17c4edd8f404606e6d3c59af8ba280bf647ce78","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-56cc9d9af74aed4fef5d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v07\n- v02 --refutes--> v00\n- v00 --supports--> v05\n- v05 --refutes--> v03\n- v03 --refutes--> v04\n- v00 --supports--> v06\n- v04 --supports--> v01\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a19e045f28437e6cbb963134ec225d9a4a0928c39c938cdf26bfe7655dd43dcd","schema":"livegraph.boolean-propagation.v1","spec_hash":"56cc9d9af74aed4fef5d6cf5c4d842796668cd184595b4494be39b1616593131","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-9f447213ac59165f1a5d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v06\n- v02 --refutes--> v05\n- v07 --refutes--> v03\n- v06 --refutes--> v00\n- v06 --supports--> v04\n- v07 --supports--> v01\n- v04 --supports--> v07\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4509c110e59a5b8cd459dbf20acc886114dc6a9be54ddcb1d3e11dd265777ca7","schema":"livegraph.boolean-propagation.v1","spec_hash":"9f447213ac59165f1a5d01f00f9d61e59af4b10b118f858837693de7d3c45c59","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-5922f37337caaf6da86b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v04\n- v05 --supports--> v02\n- v04 --refutes--> v06\n- v07 --supports--> v01\n- v05 --supports--> v00\n- v03 --supports--> v05\n- v02 --supports--> v07\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5a94c051fabfff8df139d47033fb5355f8386da4acf689be81b9285dae419e2c","schema":"livegraph.boolean-propagation.v1","spec_hash":"5922f37337caaf6da86b81678dbb43b7761d2056694b0da7621dad06346cb788","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-c06ad7feaf816ecf5fe4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v01\n- v01 --supports--> v00\n- v07 --refutes--> v05\n- v03 --refutes--> v02\n- v06 --supports--> v03\n- v07 --refutes--> v06\n- v01 --supports--> v07\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"83e16e4fbb467c3bfac0b674cf83c5238f5ee3c2d4cadab818cee3c00c8670e5","schema":"livegraph.boolean-propagation.v1","spec_hash":"c06ad7feaf816ecf5fe4118372fc815a9f3de9994f052dda68aa9e27697f819b","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-d85ec9e29f2f675c20fc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v01\n- v02 --supports--> v06\n- v04 --supports--> v05\n- v02 --refutes--> v00\n- v00 --refutes--> v03\n- v05 --supports--> v07\n- v00 --refutes--> v04\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"20d83b1804e9b90446333e90596ac78aa4188006551ab7f96ad07d711ddf6858","schema":"livegraph.boolean-propagation.v1","spec_hash":"d85ec9e29f2f675c20fc4516a293602d8d304ca2bb3d082d26c0f00d303ad82f","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-c7af97aca4bb5f050c30","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v06\n- v05 --refutes--> v04\n- v01 --refutes--> v00\n- v00 --supports--> v03\n- v04 --supports--> v01\n- v03 --supports--> v02\n- v00 --refutes--> v07\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ebc72146a12830662fb635c04f3e3bbbe29dd20a415baa47d7ad589f8d854785","schema":"livegraph.boolean-propagation.v1","spec_hash":"c7af97aca4bb5f050c302b09199acaffa83812ed5b549303b787f406a110b3ce","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-69dd69eb23a9fe552bff","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v06\n- v07 --refutes--> v04\n- v06 --refutes--> v03\n- v01 --supports--> v00\n- v07 --refutes--> v05\n- v00 --refutes--> v02\n- v06 --supports--> v01\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6b43a98095a0379b6dc293ff5af3dfb058474037eab01868c869bdae2c006c3c","schema":"livegraph.boolean-propagation.v1","spec_hash":"69dd69eb23a9fe552bff57349e5c9ccc8d7ee6f37a7b4c7195902d70121e31be","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-38d384ceb3b5de816ce2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v03\n- v02 --refutes--> v06\n- v01 --supports--> v00\n- v02 --supports--> v05\n- v01 --supports--> v02\n- v03 --supports--> v01\n- v07 --supports--> v04\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"290ee16c5fa7baf0ca1df05a73c1da55d712b4debd70866a7707b3f5dddb2091","schema":"livegraph.boolean-propagation.v1","spec_hash":"38d384ceb3b5de816ce2a55e4823d27057af4db9a111697520ac40cd0ce3c713","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-768b538626c3e44f72b8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v06\n- v03 --refutes--> v07\n- v07 --refutes--> v05\n- v02 --supports--> v04\n- v07 --refutes--> v01\n- v04 --refutes--> v00\n- v06 --refutes--> v02\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3d8f6a7b7b8836e1035338168f38dc2c941a7b408785332602cec09e249c7912","schema":"livegraph.boolean-propagation.v1","spec_hash":"768b538626c3e44f72b82b202dec1e576eaa788c6ad5b85957c0bfa7ebe8d7ec","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-392fa77289a54971fb6d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v00\n- v04 --refutes--> v01\n- v05 --supports--> v04\n- v07 --supports--> v05\n- v03 --refutes--> v06\n- v04 --supports--> v02\n- v06 --refutes--> v07\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"33c8d77a6f3e0a77b032ba66e85a1eb174b19b729f71529edb1332f3afc38169","schema":"livegraph.boolean-propagation.v1","spec_hash":"392fa77289a54971fb6d8f1b9e4c872d83c3b1685dccfcf2409b3f90b5f09b95","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-5a41682aa325dd19ee42","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v01\n- v07 --supports--> v03\n- v05 --refutes--> v02\n- v00 --refutes--> v06\n- v06 --supports--> v05\n- v05 --supports--> v04\n- v04 --supports--> v07\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b9f51f62483f7a8c1ff0cf4ab093eae9306518a097f2bef749fc7cb856e1853e","schema":"livegraph.boolean-propagation.v1","spec_hash":"5a41682aa325dd19ee42e7cea86a446da06b4191162044a344f506e3925ce7a9","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-2c0c1de5cf47062bd1a1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v06\n- v00 --refutes--> v03\n- v05 --supports--> v00\n- v03 --refutes--> v02\n- v01 --refutes--> v05\n- v02 --supports--> v04\n- v02 --supports--> v07\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3117dfa2342fed0aae427e02e38226d9cf815bc0a4ccf6a017c46dcae5204409","schema":"livegraph.boolean-propagation.v1","spec_hash":"2c0c1de5cf47062bd1a1d71604f2971c9e7690bc854f780bb3d5aef813655185","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":5,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-d5fc37932e125bcccec8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v01\n- v00 --refutes--> v04\n- v01 --refutes--> v02\n- v02 --refutes--> v07\n- v01 --supports--> v05\n- v04 --refutes--> v06\n- v06 --supports--> v03\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"47632bbaa03abafb2dde40f5c7cf06eeb182eb7f7ddf3dedd4e369bbc9f5c19f","schema":"livegraph.boolean-propagation.v1","spec_hash":"d5fc37932e125bcccec8e430118f7b248d44c7cb2dc4c449ff1b6c90e70b30c4","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-e8c2a33caa4c24aff738","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v02\n- v03 --supports--> v01\n- v03 --supports--> v07\n- v04 --refutes--> v05\n- v07 --refutes--> v00\n- v05 --supports--> v06\n- v05 --refutes--> v03\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e144f635985aa2aa40bc960801916a14141a00232c2842194ffaaadbe977fa27","schema":"livegraph.boolean-propagation.v1","spec_hash":"e8c2a33caa4c24aff73875f875b3e7a12a9ab3502f28e71b60217906964ee13c","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-0f738630d916a1d65c9a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v03\n- v04 --supports--> v00\n- v00 --supports--> v05\n- v03 --supports--> v04\n- v02 --refutes--> v01\n- v00 --supports--> v07\n- v02 --supports--> v06\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e3c0f1caafc13f67ea3075ed81543c68e6ee6b569629abb6e901fa0754f49ab5","schema":"livegraph.boolean-propagation.v1","spec_hash":"0f738630d916a1d65c9a6b183dd904f377c1349f77badb9198128b2d6be5a828","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-e4a1934ed7c043fe92f8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v01\n- v04 --supports--> v07\n- v05 --refutes--> v00\n- v05 --supports--> v04\n- v07 --refutes--> v02\n- v02 --supports--> v03\n- v03 --refutes--> v06\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e2eb5ed993c16f0bd343c7c022fd722dc412a11a1e16e3d273b46d1bba933e8b","schema":"livegraph.boolean-propagation.v1","spec_hash":"e4a1934ed7c043fe92f83a2752cc20fd43872b33f9959d0d519b2ed4e1c75a9a","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-0541b781a025401e1362","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v05\n- v06 --supports--> v00\n- v07 --supports--> v04\n- v02 --supports--> v01\n- v03 --supports--> v07\n- v01 --supports--> v06\n- v07 --supports--> v02\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"be483bee274bc77fa6a1479a1e0c3c8ce69eb783e2183cf8e8f3d57f82ec96b7","schema":"livegraph.boolean-propagation.v1","spec_hash":"0541b781a025401e13624f8f104837ec84e5435a92e86dadd15824b565f24198","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-4fcf74d766fce7eca095","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v05\n- v01 --supports--> v02\n- v00 --supports--> v03\n- v01 --supports--> v06\n- v03 --supports--> v01\n- v06 --supports--> v07\n- v07 --refutes--> v04\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"35aa84d626b7dcc470bd4c0614135d26af20a43f4bbff089ab27771abb2e144d","schema":"livegraph.boolean-propagation.v1","spec_hash":"4fcf74d766fce7eca0959bc9ded39a0a39377568f4182b46fe8b8d513f4eff73","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-bd5761c0d4040836eca4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v03\n- v01 --refutes--> v00\n- v05 --refutes--> v01\n- v04 --supports--> v05\n- v06 --refutes--> v07\n- v06 --supports--> v02\n- v06 --supports--> v04\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d2dbe258efce0f2ab2b3a0c8b329b075bd65fbe304f809915bb6a9b8e78b2fb0","schema":"livegraph.boolean-propagation.v1","spec_hash":"bd5761c0d4040836eca402648f0bffee86bbc36bade367f319066ac04cf17319","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-e9781fb3619e144c6747","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v04\n- v02 --refutes--> v00\n- v04 --refutes--> v07\n- v00 --supports--> v01\n- v04 --refutes--> v03\n- v02 --supports--> v06\n- v07 --refutes--> v02\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8e38858a3569a933fcf1c091f61b98256c099d31f738075b6c9f056aaaa42dfb","schema":"livegraph.boolean-propagation.v1","spec_hash":"e9781fb3619e144c674715a4438b3d3c6589012f20d6889d022addee24bcf478","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-64b62336e618aa9fa4d3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v04\n- v01 --supports--> v07\n- v04 --supports--> v06\n- v03 --refutes--> v02\n- v03 --supports--> v00\n- v07 --supports--> v03\n- v00 --refutes--> v05\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"38608e0a21eb8f35af50d94b46b24fd91e67cee52b9592e740f8c48d4fc1bebc","schema":"livegraph.boolean-propagation.v1","spec_hash":"64b62336e618aa9fa4d31fe29a02383ec39a7d81ba4252193232f6c9ecf89d6d","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-d4a275ce2fb14ae631fb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v03\n- v05 --refutes--> v04\n- v04 --refutes--> v01\n- v00 --refutes--> v05\n- v03 --supports--> v02\n- v03 --supports--> v07\n- v05 --supports--> v06\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5a4f8b0fe5add92bf107aa29734abf448c27f40a6f19a68527bf3d1ed4c5c09b","schema":"livegraph.boolean-propagation.v1","spec_hash":"d4a275ce2fb14ae631fbfb06ba5fac1c04a5c606b44bce7ae02d6f1bfb7322c3","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-40c546233e774faa1e18","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v05\n- v00 --supports--> v02\n- v01 --refutes--> v00\n- v06 --supports--> v03\n- v07 --supports--> v01\n- v04 --refutes--> v07\n- v06 --supports--> v04\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e696fe03fae4a3eeed53e7382a5143b15defe27be879eb5fb82b51b43ea2fecb","schema":"livegraph.boolean-propagation.v1","spec_hash":"40c546233e774faa1e18b24d574e39f5fd2975edb7b4f0a37aa7cf1a7076c34b","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-b79a5658cc2cffec06dc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v05\n- v00 --refutes--> v07\n- v05 --supports--> v02\n- v01 --supports--> v04\n- v06 --supports--> v03\n- v06 --refutes--> v00\n- v00 --refutes--> v01\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2940824a745eb65ede0ac2f387017903bf912d145bf9f425ee35a812ade530bd","schema":"livegraph.boolean-propagation.v1","spec_hash":"b79a5658cc2cffec06dc636eb13b10f8526288b16e96cf321a366d8cea07453f","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-7cf9094bfb3a1cbed5dd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v03\n- v00 --refutes--> v04\n- v02 --supports--> v05\n- v05 --supports--> v00\n- v01 --supports--> v06\n- v03 --supports--> v07\n- v03 --refutes--> v01\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3a6ee70dc929cadb0ceccce39238b8c28c5de64183046efe330fc65eed196410","schema":"livegraph.boolean-propagation.v1","spec_hash":"7cf9094bfb3a1cbed5dd1b905eca785d66c9fb711e6cf77fb9ff0d34a93d2df5","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-e7a558961c167c209432","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v02\n- v03 --refutes--> v07\n- v07 --refutes--> v06\n- v00 --supports--> v04\n- v04 --supports--> v03\n- v02 --supports--> v01\n- v07 --supports--> v05\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"03500da65c15e5f5409f3a7318565d16efe977426004d3e5e8643e4480749536","schema":"livegraph.boolean-propagation.v1","spec_hash":"e7a558961c167c20943216b06620288a1cf916d139edc0d4056020a3a4243824","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-fdbafc396d75302f7fcb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v03\n- v07 --refutes--> v05\n- v03 --refutes--> v02\n- v02 --supports--> v00\n- v02 --refutes--> v07\n- v04 --refutes--> v06\n- v02 --refutes--> v01\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"89000578c0b86db2fe17e82f9f1a88a4a2dab690d13a0f74d3aef70d4350c4f4","schema":"livegraph.boolean-propagation.v1","spec_hash":"fdbafc396d75302f7fcb8e7a6f0a4d7084f8aa0621f053449b3e5a33f4b0019d","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-debaffff3d8cf5f76016","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v04\n- v02 --refutes--> v05\n- v02 --refutes--> v03\n- v04 --supports--> v02\n- v07 --supports--> v06\n- v03 --refutes--> v01\n- v01 --supports--> v00\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4ed394cd5d6d16c513e085083941e8a62ff99accc4e9efa5ed1a2f0e4e1562c0","schema":"livegraph.boolean-propagation.v1","spec_hash":"debaffff3d8cf5f760165023b852035b0d0195d987be86a9c26bb2a804584744","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-787e2a8cc3935cd74aa7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v00\n- v01 --supports--> v02\n- v05 --supports--> v06\n- v02 --supports--> v07\n- v01 --refutes--> v04\n- v06 --refutes--> v01\n- v03 --refutes--> v05\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"db0fd9b0e2ee8d9e74683792ef39e52dd4d23eaf41061982361aa63c328b2cc2","schema":"livegraph.boolean-propagation.v1","spec_hash":"787e2a8cc3935cd74aa7eaaa5bb58f3d370d7ae6b7ca3fa3bd22b1eb5a2f6815","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-33fb954fff61af31369f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v02\n- v00 --supports--> v07\n- v06 --supports--> v00\n- v01 --supports--> v03\n- v04 --supports--> v01\n- v02 --refutes--> v04\n- v06 --refutes--> v05\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"40a31ad8708148e6e1a0bc47b613c7099b5139509f35af28e3f3dcf54321dcb9","schema":"livegraph.boolean-propagation.v1","spec_hash":"33fb954fff61af31369ff072a92741fa97a070d90d08b12d4ed5d7a4ac8f1a1e","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-d8ecdb7956136c6130b2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v06\n- v06 --refutes--> v02\n- v03 --refutes--> v04\n- v02 --refutes--> v00\n- v01 --refutes--> v07\n- v03 --supports--> v01\n- v01 --refutes--> v05\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a5f8afdc42577d7922c602ef6fd204272504ea5bff26b56db27d00966308b032","schema":"livegraph.boolean-propagation.v1","spec_hash":"d8ecdb7956136c6130b29fee3caf2e93b94f8b5f47b9ce7ff6ee2b793835c02d","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-a6925237139fe51aca0b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v02\n- v06 --refutes--> v07\n- v04 --supports--> v06\n- v03 --refutes--> v04\n- v00 --refutes--> v03\n- v04 --refutes--> v01\n- v05 --supports--> v00\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e50ad78daf65dfd070725eeaec1f43445ed0064c3372157b9df5efb0fab93b74","schema":"livegraph.boolean-propagation.v1","spec_hash":"a6925237139fe51aca0b2d99af7c3ad35a03c01c9354a6f5981920b13b036579","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-359fa1004cf312bab99d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v02\n- v06 --supports--> v05\n- v04 --supports--> v01\n- v00 --supports--> v06\n- v02 --supports--> v04\n- v00 --supports--> v07\n- v01 --refutes--> v03\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4d82c55a360a5a7d3101fb073d199aa012804953496e0ec585d852debc6a4aa4","schema":"livegraph.boolean-propagation.v1","spec_hash":"359fa1004cf312bab99d9b0474565a4eb3040e3d3d35daeee78c416d500eaa7e","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-8a44cffa86834c681681","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v06\n- v04 --supports--> v05\n- v00 --supports--> v07\n- v03 --supports--> v02\n- v00 --supports--> v01\n- v07 --refutes--> v03\n- v02 --refutes--> v04\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e6577c0dd02fd748eb0ae37000e6cc4100cb067aa0bd4d11f59ff753e35236cc","schema":"livegraph.boolean-propagation.v1","spec_hash":"8a44cffa86834c681681df79d062de01dd94666610f475df4e35f9e78439f121","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-0e7f698a44b260c08824","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v05\n- v07 --supports--> v02\n- v02 --refutes--> v01\n- v06 --supports--> v00\n- v01 --refutes--> v03\n- v04 --supports--> v07\n- v06 --supports--> v04\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3fc2ebf5c886521c588c60410ff09b4b7ae24c479521393a6222b1bc150f88e0","schema":"livegraph.boolean-propagation.v1","spec_hash":"0e7f698a44b260c088246d8d7e562ebe452e9a297d9f5e51da4730802e08a985","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-e75cdb4098ebeee219f7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v07\n- v05 --refutes--> v02\n- v02 --refutes--> v01\n- v06 --refutes--> v03\n- v02 --supports--> v06\n- v04 --supports--> v05\n- v00 --supports--> v04\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"08f4cba4de63b0ef250cfb1b8ed063734b9e9c072c62327c54593325dda08d85","schema":"livegraph.boolean-propagation.v1","spec_hash":"e75cdb4098ebeee219f798db2e3724bca58492a5ef3b29079744ab48b2d51499","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-c24ac452445f20364267","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v01\n- v05 --supports--> v06\n- v02 --refutes--> v03\n- v02 --refutes--> v05\n- v05 --supports--> v00\n- v04 --refutes--> v07\n- v07 --refutes--> v02\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6ae67dfdf88c408ed9d66fc76d1d3d123773ffb117b9110f88b7373ffed2fe1a","schema":"livegraph.boolean-propagation.v1","spec_hash":"c24ac452445f20364267133cd11b740ee2fd9e229d0ac56111e1e4ee9368fe33","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-fd1daefec50d396b1c13","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v01\n- v03 --supports--> v04\n- v00 --refutes--> v02\n- v07 --refutes--> v06\n- v02 --refutes--> v07\n- v03 --supports--> v05\n- v04 --refutes--> v00\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"570e0a56d222d958458ed27a054e160bb39ac07a46e8443124a891607644eebd","schema":"livegraph.boolean-propagation.v1","spec_hash":"fd1daefec50d396b1c132e9bf087e18a6cbc8d0b53f743c91abf8b54e9aa8409","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-b26f0bb8c110e8449b28","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v00\n- v07 --refutes--> v06\n- v00 --refutes--> v03\n- v06 --supports--> v05\n- v06 --supports--> v02\n- v05 --refutes--> v04\n- v05 --supports--> v01\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"451d6a0107e0d8a136f4afda1ef6f204494a941cf235718e719a1863f7379ef2","schema":"livegraph.boolean-propagation.v1","spec_hash":"b26f0bb8c110e8449b280e7a907e65da1d2690c76877eae5e28033bcb95441cb","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-edaa9129b372c0f9731e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v05\n- v04 --refutes--> v07\n- v05 --refutes--> v02\n- v06 --refutes--> v03\n- v04 --refutes--> v00\n- v05 --supports--> v04\n- v00 --refutes--> v06\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ea8398eff55cc58b207b2c119e083c2619c2941912daae67f77bf68a715e7a7c","schema":"livegraph.boolean-propagation.v1","spec_hash":"edaa9129b372c0f9731e52519e6863d8f51859e9e55a8e14fc0503ea30a24b73","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-75bba4a1702619b1fcce","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v00\n- v01 --refutes--> v05\n- v04 --refutes--> v03\n- v06 --refutes--> v04\n- v05 --refutes--> v07\n- v03 --refutes--> v01\n- v06 --refutes--> v02\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2755174af1f535ea3e9a86c740f9f0da8375395731903fddc6da151dec7061d4","schema":"livegraph.boolean-propagation.v1","spec_hash":"75bba4a1702619b1fcceaaa8d2d541fe5a28bdd3f466a96d639b2dcf2564c25c","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-cdfe9cf5f394534c19e4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v03\n- v04 --refutes--> v01\n- v05 --refutes--> v00\n- v02 --supports--> v07\n- v01 --supports--> v05\n- v02 --refutes--> v04\n- v02 --refutes--> v06\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"07813eec4fe5de83e3fce0f97ae8535cf5b63c0dda70dd8f2ce0a21e4a810af4","schema":"livegraph.boolean-propagation.v1","spec_hash":"cdfe9cf5f394534c19e47135ed7b53333299f7d0eed1a5e9e07d0fb0f69ede20","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-8fd18467215a9b6d6ad4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v03\n- v07 --refutes--> v02\n- v05 --refutes--> v06\n- v06 --refutes--> v01\n- v07 --supports--> v00\n- v06 --supports--> v04\n- v01 --supports--> v07\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8e634d6bdd4d9f39bd42489801c952fb8fd5b6672ad45322e3475823ed1058e6","schema":"livegraph.boolean-propagation.v1","spec_hash":"8fd18467215a9b6d6ad4a6680fc421b011ca9e29b150e6f7f3241023c186c2fd","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-024052be02b2f3b1426b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v07\n- v04 --supports--> v03\n- v00 --refutes--> v04\n- v01 --refutes--> v02\n- v05 --refutes--> v06\n- v03 --refutes--> v01\n- v05 --supports--> v00\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"eff2dfd64a7267bf678e248a7f738123ffdb29cdd58b6569f1d000b82f75fc62","schema":"livegraph.boolean-propagation.v1","spec_hash":"024052be02b2f3b1426b15a4000c887efd3901ed099fbc7f633879ba3380d0a4","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-8d8f6e501ea9d540011e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v06\n- v03 --supports--> v04\n- v02 --supports--> v00\n- v01 --refutes--> v02\n- v05 --refutes--> v01\n- v04 --refutes--> v05\n- v05 --refutes--> v07\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"28f609e2c94d3776523544fa7b6a896b1612cb03464f43697c19923b75798cd7","schema":"livegraph.boolean-propagation.v1","spec_hash":"8d8f6e501ea9d540011e33e0a9783a2fb9732b788b5c9682e2d986c26dd3b25a","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-062114564897291f3ccb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v07\n- v01 --refutes--> v02\n- v00 --refutes--> v05\n- v02 --refutes--> v06\n- v03 --supports--> v04\n- v04 --supports--> v00\n- v00 --refutes--> v01\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2496f294b3245bd029d45ed0ca105f30ab7c8ce160309dbb45198466e33902fc","schema":"livegraph.boolean-propagation.v1","spec_hash":"062114564897291f3ccb35b12a24e317092e9f63423da5d5e1b22bc863b2749a","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-7e2c1757df43c9424d71","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v01\n- v03 --refutes--> v06\n- v00 --supports--> v04\n- v06 --refutes--> v07\n- v00 --supports--> v05\n- v07 --supports--> v02\n- v07 --refutes--> v00\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e2f7cddd24438776f0d14e5508ab7f2d348514ec1874ade6be4302894d123070","schema":"livegraph.boolean-propagation.v1","spec_hash":"7e2c1757df43c9424d710f41698111742dd4013c4eccc1575a04d617c516d257","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-c8f3e5833bcc2360f935","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v07\n- v06 --supports--> v04\n- v06 --refutes--> v00\n- v02 --supports--> v05\n- v05 --refutes--> v06\n- v01 --supports--> v03\n- v00 --refutes--> v01\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"27562b5fb9467c2f86ed1f89ec47748178e2b27fc9be724e04262c8da4f0cc21","schema":"livegraph.boolean-propagation.v1","spec_hash":"c8f3e5833bcc2360f935fa17161a15c4e4ef36cd60b5d4f24c5d899b14d257b2","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-eda89874b1f11a068a15","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v01\n- v07 --supports--> v04\n- v05 --refutes--> v07\n- v00 --refutes--> v03\n- v06 --supports--> v02\n- v00 --refutes--> v05\n- v02 --refutes--> v00\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f3254ab65d7f3c40486987e17b9d51a0e140969f693d17799b385ef05d30c183","schema":"livegraph.boolean-propagation.v1","spec_hash":"eda89874b1f11a068a156cbc7daed8590b450be36fa3aa3ea7106b1ed3c14db9","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-8a340554cb4202e5ccc7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v04\n- v04 --supports--> v03\n- v03 --refutes--> v07\n- v02 --supports--> v00\n- v04 --supports--> v06\n- v07 --supports--> v01\n- v03 --refutes--> v05\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"909bcd686c8bfad3b8a0579f6aed307f1c2866c53c11d9edfc07bc2704f73d39","schema":"livegraph.boolean-propagation.v1","spec_hash":"8a340554cb4202e5ccc7cf161d22bd06e3018439061ca28097fc60e1098fc70e","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-5f134eef38efffed7bf0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v05\n- v05 --refutes--> v01\n- v01 --supports--> v06\n- v06 --refutes--> v00\n- v01 --refutes--> v07\n- v01 --refutes--> v04\n- v02 --supports--> v03\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"47b0b6b2f35da82dfa1b8565d4ddd50ce9082ae56335d78bd98ad6e2e9a18858","schema":"livegraph.boolean-propagation.v1","spec_hash":"5f134eef38efffed7bf0503a8f9490e566333c921ff33f798d9ef31baa5dd08d","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-44e5f4a86670c13060a9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v05\n- v02 --supports--> v01\n- v05 --refutes--> v04\n- v02 --supports--> v03\n- v00 --refutes--> v02\n- v04 --supports--> v07\n- v02 --supports--> v06\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b2dd37d756cb840015033975e98e97e33240b2fc928bdc74fb8ff64c182f13fd","schema":"livegraph.boolean-propagation.v1","spec_hash":"44e5f4a86670c13060a940af31c620e149ea5b0e2bdb9933166c6e732398d948","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-74494104629c847630aa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v03\n- v06 --supports--> v02\n- v00 --refutes--> v07\n- v04 --refutes--> v05\n- v06 --refutes--> v04\n- v03 --refutes--> v01\n- v04 --refutes--> v00\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cbd61ac8b0998ee475821be036fdfd21b93b7c970c588e59028f4763e7bb55cb","schema":"livegraph.boolean-propagation.v1","spec_hash":"74494104629c847630aaaa24398cd6d576578e34abafa6308c0b45139b8b0e23","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-f90ca178d8e6becb480d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v00\n- v02 --refutes--> v07\n- v07 --refutes--> v01\n- v01 --supports--> v04\n- v05 --refutes--> v03\n- v03 --supports--> v02\n- v05 --refutes--> v06\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"18ad9a7c06abf2d6c71aeef00838fb0cd178412cd332246d5287d08ecc2610cc","schema":"livegraph.boolean-propagation.v1","spec_hash":"f90ca178d8e6becb480d6faae15752036683557a686115bc33014b408d2fbe8f","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-5a6a8bb13eab0237f935","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v04\n- v06 --supports--> v02\n- v07 --refutes--> v05\n- v03 --refutes--> v01\n- v01 --supports--> v07\n- v02 --refutes--> v00\n- v02 --supports--> v03\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"32299575a86f4b77712697d5f85801c9c74eef69e0f1a1d58075b771a8f5f2dc","schema":"livegraph.boolean-propagation.v1","spec_hash":"5a6a8bb13eab0237f935ba140d5047cfa61095e01bffd2e9ecd9168ffd7eef8a","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-9daf23d92b3c970b939f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v04\n- v01 --supports--> v02\n- v00 --refutes--> v03\n- v03 --supports--> v05\n- v02 --supports--> v00\n- v03 --refutes--> v07\n- v06 --refutes--> v01\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d2efa4e09294be3bca0607a1128b3e886c4bf84b2622d7aa8b75d4a2ad215d5e","schema":"livegraph.boolean-propagation.v1","spec_hash":"9daf23d92b3c970b939f39b24462f63c4f1e1609a6ef7f855e1f36218339655b","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-997ccce1b83733571143","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v03\n- v03 --refutes--> v07\n- v01 --refutes--> v06\n- v07 --supports--> v05\n- v07 --supports--> v00\n- v00 --refutes--> v04\n- v00 --refutes--> v01\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1e98c73b5eb2f847b8974ce4782e9aed67e9ba2e26713fe16c770097f0ca108b","schema":"livegraph.boolean-propagation.v1","spec_hash":"997ccce1b837335711430d6fe4f802aad7e937091816ce809abf1ed7a96e70a1","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-2a5e7f9b877d3ef144e8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v07\n- v04 --refutes--> v02\n- v00 --refutes--> v01\n- v05 --supports--> v03\n- v03 --refutes--> v06\n- v00 --refutes--> v04\n- v02 --supports--> v05\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"10f753b60c0fc3dd45335a451a076990497688afc411b439f6e19695ecb69d34","schema":"livegraph.boolean-propagation.v1","spec_hash":"2a5e7f9b877d3ef144e857c26b743dfdecfca2ba782cba980092cf81d6a08934","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-3124ae2529b97813ada9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v04\n- v06 --supports--> v07\n- v05 --supports--> v06\n- v04 --supports--> v02\n- v01 --supports--> v05\n- v07 --supports--> v03\n- v04 --refutes--> v00\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dc1f69c57c44b91dab44bfa2df61846505bccaa097dbdf0c611baebd29225278","schema":"livegraph.boolean-propagation.v1","spec_hash":"3124ae2529b97813ada997025afdba90230811bbac50073b7db211f8cd98c45f","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-6f13d7ab2c4cf0199a86","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v06\n- v04 --refutes--> v01\n- v04 --refutes--> v05\n- v07 --supports--> v04\n- v00 --supports--> v07\n- v02 --refutes--> v03\n- v06 --supports--> v02\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"24c881116bff8aac5065fe6fc005269b97f0ebf92d4f1daeb304afc130c544e2","schema":"livegraph.boolean-propagation.v1","spec_hash":"6f13d7ab2c4cf0199a8693cd44f56d97e6128383af946292540dd2edebf19f76","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-39d8a6049525d9053f90","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v00\n- v06 --supports--> v02\n- v01 --supports--> v05\n- v02 --refutes--> v03\n- v04 --supports--> v06\n- v01 --supports--> v04\n- v07 --refutes--> v01\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3dd785db0d5e7db64522e7201e7bc04c59d348a58b0479e53d007d908c53095b","schema":"livegraph.boolean-propagation.v1","spec_hash":"39d8a6049525d9053f9059b2ac06d80a32b2fe3196d58243a02478e7bda208df","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-85d407f0ecc5cf1a2477","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v02\n- v03 --refutes--> v00\n- v06 --supports--> v05\n- v06 --supports--> v01\n- v03 --refutes--> v07\n- v07 --refutes--> v06\n- v01 --refutes--> v04\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5cd4357253ecfef9d787be2478589e4b0d716d790cfccde456ed4ef063134867","schema":"livegraph.boolean-propagation.v1","spec_hash":"85d407f0ecc5cf1a24772add54b4a7f29281ee6e5c1226602a2e810a4820d076","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-99c613028b50823308f8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v00\n- v05 --supports--> v03\n- v02 --refutes--> v07\n- v06 --refutes--> v02\n- v04 --supports--> v05\n- v00 --supports--> v06\n- v05 --supports--> v01\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7ea5dfba7a4eb0372bec608d98a7e0bb9f4e5578a94caeb33f9b34f311a18462","schema":"livegraph.boolean-propagation.v1","spec_hash":"99c613028b50823308f843e0bd627067e8066daccae4f941461ce05a39a61f3d","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-921def304dd96aa135c9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v03\n- v05 --refutes--> v07\n- v03 --refutes--> v04\n- v07 --supports--> v02\n- v04 --refutes--> v06\n- v06 --supports--> v01\n- v07 --supports--> v00\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"890c55b5f6eeff26ff5ac9a26518ae65a7149fc5482bcb0b7cd705255782988d","schema":"livegraph.boolean-propagation.v1","spec_hash":"921def304dd96aa135c928dda01aff10dadd42bdf7001c087515480aa59bc947","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-be16c716094875695f33","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v06\n- v04 --refutes--> v03\n- v06 --refutes--> v07\n- v07 --supports--> v01\n- v07 --supports--> v05\n- v06 --supports--> v00\n- v05 --supports--> v02\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0965f2c78863a4e464e970d37e37c5e70d0dc6e753895be9a0f1405fb715f390","schema":"livegraph.boolean-propagation.v1","spec_hash":"be16c716094875695f337efe24393ac1cb955da7c2bb34a1300bb4c729a6d80b","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-c354f68eb1d713dd0aa5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v04\n- v00 --refutes--> v02\n- v04 --supports--> v07\n- v01 --supports--> v03\n- v07 --supports--> v05\n- v01 --supports--> v06\n- v07 --supports--> v01\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ebd1fcefb29dbad12b4745f61efca3f4b545499455a9456d49b4594580071070","schema":"livegraph.boolean-propagation.v1","spec_hash":"c354f68eb1d713dd0aa5438c2bfe7942bd101229cd8696a3305bda61cb8fb911","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-ee8820fb93327a5294ca","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v02\n- v06 --refutes--> v05\n- v05 --supports--> v03\n- v03 --refutes--> v04\n- v04 --refutes--> v01\n- v02 --supports--> v00\n- v02 --supports--> v07\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e42d55a02f4d1f5229d812482372f643a2e1c54ed2d134339d3ba0dce82ca3f6","schema":"livegraph.boolean-propagation.v1","spec_hash":"ee8820fb93327a5294caa5275292bde72498b84af144669dcc7dfc1e18d789ad","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-6e3e927ca7ef28f82f91","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v01\n- v05 --refutes--> v07\n- v00 --supports--> v04\n- v07 --refutes--> v02\n- v03 --supports--> v05\n- v06 --refutes--> v00\n- v07 --refutes--> v06\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9c58424ceef89d8164b7ab6118d8c51e8619d2955f3448abae4d336f0bbcbffa","schema":"livegraph.boolean-propagation.v1","spec_hash":"6e3e927ca7ef28f82f91999a6c18f332c24785960ba3c6155d9e0b70f8d8e3ba","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-07cc998dbd392e7c724f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v03\n- v00 --refutes--> v04\n- v00 --refutes--> v01\n- v04 --supports--> v06\n- v03 --refutes--> v00\n- v04 --refutes--> v07\n- v05 --supports--> v02\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"110bc64d1368f9befe1084d36fd518b898c4c9acf3e8f7b5ff13a6e2348dd4a3","schema":"livegraph.boolean-propagation.v1","spec_hash":"07cc998dbd392e7c724f31bfafd63126fd42c23508cd5c350c6b1fabb9b02f68","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-523327ffab37bf15c9f1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v02\n- v00 --supports--> v03\n- v00 --supports--> v01\n- v05 --supports--> v07\n- v03 --refutes--> v05\n- v02 --supports--> v04\n- v03 --supports--> v06\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3e4f1766709b35a724fb12708f259a9b5a1f0b68dd765e1d89f4371b8bbdf324","schema":"livegraph.boolean-propagation.v1","spec_hash":"523327ffab37bf15c9f1231c9ef70307e46b0b5dc80f2523f63bb6a5531d5e58","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-a5f1c408bfc3e0cd064e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v00\n- v05 --refutes--> v07\n- v04 --supports--> v06\n- v03 --supports--> v02\n- v02 --supports--> v05\n- v01 --refutes--> v04\n- v05 --supports--> v01\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3fce03547bcade94a4254f97e2eae67e8c64e897720059b01fabb3b0977d3d51","schema":"livegraph.boolean-propagation.v1","spec_hash":"a5f1c408bfc3e0cd064e93464c0af1b417a401cbc3beb4c8a069f3ab6ea38576","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-e2ca94b3a654cf21af1a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v00\n- v05 --supports--> v06\n- v04 --refutes--> v01\n- v05 --refutes--> v07\n- v05 --supports--> v02\n- v00 --refutes--> v03\n- v01 --supports--> v05\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8a7775fdb9765e44b3c3b989f32a315770e3445187310ed4d0b7d3ac67150d30","schema":"livegraph.boolean-propagation.v1","spec_hash":"e2ca94b3a654cf21af1ab30138f2c4d416bc6eade7b6638b171fcdd740586223","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-778f513152593a3ab291","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v04\n- v07 --supports--> v05\n- v04 --supports--> v07\n- v05 --refutes--> v06\n- v04 --refutes--> v03\n- v06 --refutes--> v00\n- v05 --refutes--> v01\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"db87c3d8ab20d873792aed0cd40ba50dbab2ab6a9d28daec14e1308d21497468","schema":"livegraph.boolean-propagation.v1","spec_hash":"778f513152593a3ab291ac24cef5e062391d1eaa80ae6545617cf971b002f84e","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-4b6dbfd6e68297fb3a16","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v00\n- v04 --refutes--> v01\n- v04 --refutes--> v03\n- v03 --supports--> v07\n- v02 --refutes--> v05\n- v00 --refutes--> v04\n- v04 --refutes--> v06\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6674a329a90178486fc47ad2ac0ea201fc442958a0fb3150c32542ecde30290e","schema":"livegraph.boolean-propagation.v1","spec_hash":"4b6dbfd6e68297fb3a16393ee3f5b1326ee51823d32998aef16710f3ced7a671","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-ccd64429502ca94e0218","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v05\n- v02 --supports--> v06\n- v07 --supports--> v02\n- v02 --refutes--> v03\n- v06 --refutes--> v01\n- v00 --supports--> v07\n- v01 --supports--> v04\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8f9445f3dc011c9f56cb48d7b4fe8d132a1e2b9fc213161c739dc2637cbbf2ae","schema":"livegraph.boolean-propagation.v1","spec_hash":"ccd64429502ca94e02185acf2aadcaab435028ba619dacdd3f710ba3985387bf","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-155be38b85f03dc8ba48","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v05\n- v06 --refutes--> v07\n- v07 --refutes--> v02\n- v02 --refutes--> v04\n- v03 --refutes--> v06\n- v00 --refutes--> v03\n- v03 --refutes--> v01\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d555fa1b30472dc119f835e5e94476a91a0fec4a42ccf9706d14dad617246ad4","schema":"livegraph.boolean-propagation.v1","spec_hash":"155be38b85f03dc8ba4813f932fefa0271688af247e1ca84a3efa8c2a70e090f","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-b08d403849f1df3e02f7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v06\n- v00 --supports--> v02\n- v05 --supports--> v00\n- v07 --supports--> v03\n- v02 --refutes--> v04\n- v04 --refutes--> v07\n- v05 --refutes--> v01\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"71098847d8c340d648043ea38f5d6a8c0df8c20f3dd157009dafab4bbfd5b3e0","schema":"livegraph.boolean-propagation.v1","spec_hash":"b08d403849f1df3e02f773f92f8927f437ee53339bb9189e9ff9042193ad0cd3","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-ca50b2eebc78d9c69fe7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v06\n- v07 --supports--> v00\n- v00 --refutes--> v05\n- v07 --supports--> v01\n- v07 --supports--> v04\n- v05 --supports--> v03\n- v03 --supports--> v02\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cef6ab3bf261df89b6d08d59598065eda9c40e4cdbde64247a8c52c78232ab6d","schema":"livegraph.boolean-propagation.v1","spec_hash":"ca50b2eebc78d9c69fe79512655609d31a992466efc44d7a754a16624543c245","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-cad12b9def5754686603","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v07\n- v06 --supports--> v04\n- v02 --supports--> v06\n- v06 --refutes--> v03\n- v07 --refutes--> v01\n- v05 --supports--> v02\n- v05 --supports--> v00\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ed0cdb2bd49e2eb08ff518cb86bb8d7c4d8cfbd76c8c89493f5b2e621f14c950","schema":"livegraph.boolean-propagation.v1","spec_hash":"cad12b9def575468660317d2bf7858c321c2fc99bbcd6f98cf3fd3ab8f8214bd","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-e166ffcb52c642c02eea","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v03\n- v04 --refutes--> v05\n- v00 --refutes--> v04\n- v06 --refutes--> v07\n- v00 --refutes--> v02\n- v03 --refutes--> v01\n- v04 --supports--> v06\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"37277cf5bbec8f715a2efc14839ae685fe313a928848230f1e65e46af9aadbec","schema":"livegraph.boolean-propagation.v1","spec_hash":"e166ffcb52c642c02eea92b707dbbce957e3c215b5ff74a6774e46a89d384a2f","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-fd0e35d0a9715c518408","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v04\n- v02 --refutes--> v01\n- v00 --refutes--> v03\n- v05 --refutes--> v00\n- v06 --supports--> v05\n- v05 --refutes--> v07\n- v00 --refutes--> v02\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3650ef684501f6ae588870f64ad7a8b394cc877c07d05eb79b985a8784cc5ec7","schema":"livegraph.boolean-propagation.v1","spec_hash":"fd0e35d0a9715c518408b21deafe0e5afcbfa714e537694a3ae2519102ae5dc5","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-733f2908c671a755e9d6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v00\n- v00 --supports--> v06\n- v07 --refutes--> v03\n- v07 --supports--> v02\n- v06 --supports--> v01\n- v00 --supports--> v05\n- v04 --refutes--> v07\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4736accf2a0aa52ff16450515d74bb0f0851de66103517d58a11a759a5756f33","schema":"livegraph.boolean-propagation.v1","spec_hash":"733f2908c671a755e9d6c58140125d9aa3bf5b1374e1ab4af4ce4ab6b352d4a9","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-cace9c46f5dfa8fcc622","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v06\n- v01 --refutes--> v05\n- v07 --supports--> v03\n- v05 --refutes--> v00\n- v07 --supports--> v01\n- v06 --supports--> v02\n- v06 --supports--> v04\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"419cf876dca506aaa370e0b18d0d69481b4afa8c36b189f1f91c20eb464053e4","schema":"livegraph.boolean-propagation.v1","spec_hash":"cace9c46f5dfa8fcc622e766d7924cb832ce6fa6759997a76811c222e43ac540","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-0e0d0380ac1100bf4bb4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v05\n- v00 --supports--> v04\n- v02 --refutes--> v00\n- v03 --supports--> v07\n- v04 --supports--> v06\n- v06 --refutes--> v03\n- v04 --refutes--> v01\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"16bf38252debeb3a93f0f6a8ba95e8687bb335318b90538e7d943cb3f988c15e","schema":"livegraph.boolean-propagation.v1","spec_hash":"0e0d0380ac1100bf4bb41b3521976bb3978160f98e6bb39cf76406bf991027f9","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-0fd0bf3578a3019d3c30","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v05\n- v03 --refutes--> v07\n- v03 --refutes--> v04\n- v02 --refutes--> v06\n- v00 --supports--> v02\n- v05 --refutes--> v00\n- v05 --supports--> v01\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6eb7e02eedf707fdd0401f850bf0fa635eee08e65e6573f36fbf69e25d7aef6c","schema":"livegraph.boolean-propagation.v1","spec_hash":"0fd0bf3578a3019d3c30609c73f82b53bac1a24e0323ac86e2c4a0d16e55d0da","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-d1bb5536ec04b2645032","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v06\n- v01 --supports--> v05\n- v07 --refutes--> v03\n- v03 --refutes--> v00\n- v05 --refutes--> v02\n- v01 --supports--> v04\n- v00 --refutes--> v01\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1d2bf62ae891476f6526f58483beacf8bfb0f67d1ae42c287ef75960a9c0e18f","schema":"livegraph.boolean-propagation.v1","spec_hash":"d1bb5536ec04b26450327049d7c5d2885eb74fcf95457c53a63c9ec505ba2bda","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-70874ab1f1cf18d40517","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v01\n- v05 --supports--> v02\n- v01 --refutes--> v05\n- v01 --supports--> v06\n- v07 --refutes--> v00\n- v05 --refutes--> v03\n- v03 --supports--> v07\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b3419b4ef3407a066bacad5be4ae585dab0a176fada17d7608440b5f9ffaea77","schema":"livegraph.boolean-propagation.v1","spec_hash":"70874ab1f1cf18d40517cb257a10d533aa87f9a4d514bc35e14b8fa1693e3efa","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-4a758fd97696363d4ae1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v02\n- v03 --refutes--> v05\n- v01 --refutes--> v04\n- v02 --supports--> v03\n- v03 --supports--> v01\n- v06 --supports--> v00\n- v06 --supports--> v07\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"42725ce80fe679ae7b08408afb86a7d383150fa30094716a0419f41a88d4c0c1","schema":"livegraph.boolean-propagation.v1","spec_hash":"4a758fd97696363d4ae13910ed5dab13748977c5f8bfd6ae142952ed9237ab1b","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-5826b82f7a6aef475bec","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v03\n- v02 --refutes--> v04\n- v06 --refutes--> v01\n- v06 --refutes--> v07\n- v01 --refutes--> v02\n- v02 --refutes--> v00\n- v03 --supports--> v06\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"88225af700d7f433796f535f725ae8aa02bf2db6cb0c3fbcb4f98dd703a476b3","schema":"livegraph.boolean-propagation.v1","spec_hash":"5826b82f7a6aef475bec018ba305c85d314bc55b0d6aa90e4b06f03c886e6efe","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-ef788b9c8650b0a8e748","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v06\n- v06 --supports--> v04\n- v04 --refutes--> v07\n- v03 --supports--> v01\n- v02 --refutes--> v03\n- v03 --refutes--> v05\n- v02 --refutes--> v00\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"687800a9a75da8674d5ffba496e0130f1a20e12fe30e4f9e88d19d5383e96a86","schema":"livegraph.boolean-propagation.v1","spec_hash":"ef788b9c8650b0a8e7482bc1c1cf087cc072c6c1390c64a78fc2f4f02a1b7bc3","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-1583023ebb39e25c3b2b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v00\n- v01 --refutes--> v03\n- v03 --supports--> v07\n- v07 --supports--> v06\n- v07 --refutes--> v05\n- v06 --refutes--> v04\n- v02 --supports--> v01\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ee583f49c3d669268162df9ee312134295b8a17f8d67eccc4dd2a844bc687eff","schema":"livegraph.boolean-propagation.v1","spec_hash":"1583023ebb39e25c3b2bbbba82cd1c0cb2ec97a68a0b6d2befd07d1ec9ade263","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-f00622022ed1c4262a02","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v05\n- v01 --supports--> v00\n- v06 --refutes--> v04\n- v07 --refutes--> v03\n- v06 --refutes--> v07\n- v00 --refutes--> v02\n- v07 --supports--> v01\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8dcf7897c425246ad0c014246fd9369deb7f813501595ef39330f795a5c4ce04","schema":"livegraph.boolean-propagation.v1","spec_hash":"f00622022ed1c4262a0224709c892abafe2bc95934d9294a84d3ac7367e7a4a2","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-0f0b8d528dc981be3278","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v02\n- v01 --refutes--> v07\n- v01 --refutes--> v05\n- v00 --supports--> v04\n- v06 --refutes--> v03\n- v04 --supports--> v06\n- v03 --supports--> v01\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7c28d2067ded6b36566360bf85cb716056e5b0c00685f8c505e771ab7b4365d1","schema":"livegraph.boolean-propagation.v1","spec_hash":"0f0b8d528dc981be3278541236536ae5c5b3634d036aa361b3b94abcbfe59bcf","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-8c72bb46f0f48e46be6f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v05\n- v03 --supports--> v01\n- v03 --supports--> v04\n- v05 --refutes--> v03\n- v00 --refutes--> v06\n- v02 --refutes--> v00\n- v07 --refutes--> v02\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"061a47fbe17cd1e3dc3e304db54a05f2153aed62a2ac8942bf56034123b631ee","schema":"livegraph.boolean-propagation.v1","spec_hash":"8c72bb46f0f48e46be6fa4c0a8c79dfbe4b12cdf9c8ee7e1dc2a6009ee289dda","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-327211be3c68a707e098","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v05\n- v01 --supports--> v00\n- v03 --supports--> v02\n- v01 --supports--> v06\n- v07 --refutes--> v03\n- v06 --supports--> v04\n- v02 --supports--> v01\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"971b20e3e28d206329719711e7ed5813c6c04c13d212b0f32c771380dc309561","schema":"livegraph.boolean-propagation.v1","spec_hash":"327211be3c68a707e098c8e46cb7a0bbf124a0753f95554eb3ea5e0af882e437","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-0404b58a18afdafd6c84","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v06\n- v03 --refutes--> v05\n- v05 --supports--> v01\n- v07 --supports--> v03\n- v00 --supports--> v02\n- v02 --supports--> v07\n- v07 --supports--> v04\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b82ed274cab1d47743046c3982e2892162942de09a1f42accd7bcfc7cd81b05d","schema":"livegraph.boolean-propagation.v1","spec_hash":"0404b58a18afdafd6c843c597866a2858f924df4d27c14f19f76ca65267156c8","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-6ff13e68b20aec858de1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v07\n- v06 --refutes--> v03\n- v07 --supports--> v01\n- v06 --supports--> v02\n- v04 --refutes--> v06\n- v05 --refutes--> v00\n- v07 --supports--> v05\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7d3d1b3c5e15c3941ad9c123161504cd0783123e9d824d043f6c54bf321187ba","schema":"livegraph.boolean-propagation.v1","spec_hash":"6ff13e68b20aec858de1b535bb97b11d49ac2db16924e71dd10488dde2cfbf3a","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-ffb6da028ee5ab6d9225","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v04\n- v02 --supports--> v07\n- v02 --refutes--> v01\n- v00 --refutes--> v06\n- v03 --refutes--> v00\n- v07 --supports--> v05\n- v05 --refutes--> v03\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7c287f3b053f111c710b57371b3fafb2b51d0da9cc4f21a71cd1ec99ecf45091","schema":"livegraph.boolean-propagation.v1","spec_hash":"ffb6da028ee5ab6d9225e929b56233b2a97a5c78e391b91640dee9273a2472f7","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-2c421ec54073ebd7d890","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v03\n- v00 --supports--> v02\n- v00 --supports--> v04\n- v07 --supports--> v05\n- v02 --supports--> v07\n- v01 --supports--> v06\n- v06 --refutes--> v00\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"da2eedc563fba1216cb28ee49ced7381e2deec4360c1ddd719d89d14e74ca548","schema":"livegraph.boolean-propagation.v1","spec_hash":"2c421ec54073ebd7d8905404865d0e99f88db40bc72133ff40ef0d98e9f99370","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-98eb3b1b3721da3a4c04","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v01\n- v00 --supports--> v06\n- v07 --refutes--> v05\n- v07 --refutes--> v02\n- v02 --supports--> v00\n- v00 --supports--> v03\n- v01 --refutes--> v07\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"de807964a0bddf02bb6e0a3de3bfc4843eba6f9a1bd3bdcfaf8cbe4d78ef549b","schema":"livegraph.boolean-propagation.v1","spec_hash":"98eb3b1b3721da3a4c04ec277ce537a3798d48e1e1de237c2a12a6e459761bc3","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-7dbee21f15ac16addc15","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v00\n- v04 --supports--> v06\n- v00 --supports--> v02\n- v00 --supports--> v03\n- v07 --supports--> v01\n- v07 --supports--> v05\n- v06 --refutes--> v07\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"84b81e4d11fc4fdfe6ab48a793711cfaea794b14f5b27c4d1a48b2426d754c75","schema":"livegraph.boolean-propagation.v1","spec_hash":"7dbee21f15ac16addc152e86759ff4ea473eceb98cd0d6f9a6d5018931e937a7","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-03de8ae3251c25601110","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v03\n- v03 --refutes--> v05\n- v07 --supports--> v00\n- v03 --supports--> v06\n- v00 --refutes--> v04\n- v02 --refutes--> v01\n- v03 --supports--> v02\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"54c24fc6513463438405801c9394b3be21b7e447f020a33fb96c4c77b8a60e4b","schema":"livegraph.boolean-propagation.v1","spec_hash":"03de8ae3251c256011103e63a84e1800b8412aea55cc5ee0a6157d91eae6a6e9","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-7e70ab013cc13ef899d5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v02\n- v00 --supports--> v05\n- v01 --refutes--> v03\n- v04 --supports--> v01\n- v07 --refutes--> v00\n- v00 --supports--> v04\n- v06 --supports--> v07\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f31891e2d7871018496c1b281d85a4474ec91dbc5dbaf4aa35ad5febcafdda00","schema":"livegraph.boolean-propagation.v1","spec_hash":"7e70ab013cc13ef899d508dfc4516a4a9a91af8cbf2ab8d87fdfb1d00377bf91","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-19fafa3fae613ef747b2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v05\n- v07 --supports--> v06\n- v05 --refutes--> v03\n- v06 --refutes--> v04\n- v05 --supports--> v07\n- v05 --refutes--> v02\n- v04 --refutes--> v01\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"328082653736e3fca9b21cfbfe6e9d5f68e57af90302b2ac90dbc22d6255dcce","schema":"livegraph.boolean-propagation.v1","spec_hash":"19fafa3fae613ef747b2cca28fb383ece9fe4526d2360e55e61632075b796994","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-0a675a5671a3535dafed","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v05\n- v05 --refutes--> v07\n- v00 --supports--> v04\n- v03 --supports--> v06\n- v06 --supports--> v01\n- v04 --refutes--> v03\n- v06 --supports--> v02\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5671d3f28cfcbe602a735edda483e880190ff7194fe9e7c5b1d083eefa4989c4","schema":"livegraph.boolean-propagation.v1","spec_hash":"0a675a5671a3535dafed5249a151a75bba0f700a9b91c60b56ebdf96606a7d20","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-24fc0a45ee5f5931ced4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v02\n- v05 --refutes--> v04\n- v02 --refutes--> v05\n- v05 --supports--> v07\n- v00 --refutes--> v03\n- v05 --supports--> v00\n- v01 --supports--> v06\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7b818421f15ec7bd7aa20eeb7c9c2edd47a6a7872db9a457ed55303fe64b59a8","schema":"livegraph.boolean-propagation.v1","spec_hash":"24fc0a45ee5f5931ced4d17c864b71067e0a093b06da7bf1b3c38b428b63ee50","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-191544075521ae3a2719","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v06\n- v01 --supports--> v05\n- v00 --refutes--> v02\n- v03 --refutes--> v07\n- v07 --supports--> v01\n- v03 --refutes--> v04\n- v02 --refutes--> v03\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"19745c729a0fcf671f231d67c917a731cfeff9f9e8760c5ec8d2ac2da5d2d362","schema":"livegraph.boolean-propagation.v1","spec_hash":"191544075521ae3a2719aa908e9bcc05ad6cd9bc227f848fe012b61813d6dc04","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-8bb9a1334bc61fe17f15","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v02\n- v04 --refutes--> v01\n- v06 --supports--> v04\n- v04 --refutes--> v03\n- v00 --refutes--> v05\n- v06 --refutes--> v07\n- v02 --refutes--> v06\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"42db6f9f376cb97a52e140ba8dfd2aff18505b55c33104047c0373bcec33d239","schema":"livegraph.boolean-propagation.v1","spec_hash":"8bb9a1334bc61fe17f156464251ea5ee909e6c381a3bdf52f97fc450b9d1fd69","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-c9e6b3a80b1dd8fb2c8b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v04\n- v05 --refutes--> v00\n- v04 --supports--> v02\n- v02 --refutes--> v05\n- v02 --refutes--> v03\n- v01 --refutes--> v07\n- v01 --supports--> v06\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6112331b8e95f717abd55cfb806efa987d5d34e234a8b5348c39cf1c80c785b4","schema":"livegraph.boolean-propagation.v1","spec_hash":"c9e6b3a80b1dd8fb2c8b99f30ad14de6c6972531ca03f74245133135c836cb80","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-01d84d4119b2cacf15b1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v03\n- v04 --refutes--> v07\n- v00 --refutes--> v01\n- v03 --supports--> v04\n- v04 --refutes--> v02\n- v06 --supports--> v00\n- v00 --refutes--> v05\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6cf79dd57d26cc2f0a344f501045823f95ed00142c0b3c1cea74e9b379eaefa3","schema":"livegraph.boolean-propagation.v1","spec_hash":"01d84d4119b2cacf15b187d5da44c9169f3972671b59750a6aa010fe56749c19","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-5007a36daea2a309945e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v07\n- v05 --refutes--> v01\n- v07 --supports--> v02\n- v01 --supports--> v06\n- v04 --refutes--> v05\n- v07 --supports--> v03\n- v06 --refutes--> v00\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"750acd720908fe43ae0292554f7a22cac318d6e5daa5b16586cb905d6679b1ee","schema":"livegraph.boolean-propagation.v1","spec_hash":"5007a36daea2a309945e4cc5eaf080eef64f84b8098eb55d12178216f541f6cd","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-0b09f9ead78e6dc6ace9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v03\n- v01 --supports--> v00\n- v07 --refutes--> v04\n- v04 --refutes--> v05\n- v01 --supports--> v06\n- v07 --supports--> v02\n- v03 --supports--> v01\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"abc22e497ce3055fe1500c50b2a6cd64183ee829982c826a2e226dbec98fe55d","schema":"livegraph.boolean-propagation.v1","spec_hash":"0b09f9ead78e6dc6ace97f23710ac0b193efc5f400978556a9c9decaca4f5f26","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-317dc3713bc79a1fd832","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v02\n- v03 --supports--> v06\n- v02 --supports--> v05\n- v03 --supports--> v00\n- v01 --supports--> v03\n- v04 --refutes--> v07\n- v06 --supports--> v04\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"047fdd2182365e11f1f6e6be54a67fd48417070bb708838c235aa4ee0ce8d97d","schema":"livegraph.boolean-propagation.v1","spec_hash":"317dc3713bc79a1fd83297cd8e8188c524e7c3e31dda8bc1d0c8c25e0445d5de","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-db4304680a31a151e93d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v06\n- v05 --supports--> v07\n- v07 --supports--> v02\n- v01 --refutes--> v04\n- v02 --supports--> v03\n- v00 --supports--> v01\n- v02 --refutes--> v00\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"98d860c6a765c66c87260cb48b04c04c8de96d56deb08f8b2347f3209facf725","schema":"livegraph.boolean-propagation.v1","spec_hash":"db4304680a31a151e93d826eded0b92ec9ffc3a8ecaa56b0351941dacd0d6c64","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-17a7413a2735abbc097e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v06\n- v07 --supports--> v05\n- v05 --refutes--> v00\n- v03 --supports--> v07\n- v01 --refutes--> v02\n- v00 --supports--> v01\n- v07 --refutes--> v04\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"61c734ac4f42b3e74d36d87d3e5d88948cd06bc2e797ea19a47a55d7311803da","schema":"livegraph.boolean-propagation.v1","spec_hash":"17a7413a2735abbc097e35e4981d49f95b7be960185eb2f23a66b426d5695b6c","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-a72cb2b34e3af5b3908d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v04\n- v06 --supports--> v01\n- v06 --refutes--> v05\n- v01 --refutes--> v03\n- v04 --refutes--> v02\n- v03 --supports--> v00\n- v04 --refutes--> v07\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"92fcf7a717c504749b9ad8928c87947a4b120e2a6222171be9a2a1ecef750627","schema":"livegraph.boolean-propagation.v1","spec_hash":"a72cb2b34e3af5b3908d8b49c7e8c489bfead30a7f995f9fb333fdfab745d266","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-bfbeb55948972150e156","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v05\n- v06 --refutes--> v07\n- v05 --supports--> v00\n- v00 --supports--> v02\n- v06 --refutes--> v04\n- v00 --refutes--> v01\n- v02 --supports--> v03\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"28b647b6d267c9014f4e9cfcb17f30ed17c2cc0d6d8ac5421cb00c8cf29019d5","schema":"livegraph.boolean-propagation.v1","spec_hash":"bfbeb55948972150e1564e74adc07334122238988b15a4c0f7068379b065a682","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-05a7515b7bc20a34ac0e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v05\n- v02 --supports--> v06\n- v06 --refutes--> v00\n- v01 --refutes--> v02\n- v06 --refutes--> v04\n- v03 --refutes--> v07\n- v07 --supports--> v01\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"47d75592717d73456ad858870c8d8234972be8add03cffaf5c80a36fbacd01f1","schema":"livegraph.boolean-propagation.v1","spec_hash":"05a7515b7bc20a34ac0e118cc8e14e29c327e7060d80b6023b3259e7edea33f7","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-7dd4563fa19fcee7beef","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v07\n- v07 --supports--> v03\n- v02 --refutes--> v04\n- v05 --refutes--> v02\n- v07 --supports--> v00\n- v06 --supports--> v05\n- v02 --refutes--> v01\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"677d4c057c2dc0a33956bc16066466574efb2bcfda32c9cc026da4732ee9cca8","schema":"livegraph.boolean-propagation.v1","spec_hash":"7dd4563fa19fcee7beef39f55597d4f8abe18bbab210b38ff873dbfbd6802039","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-f0cee3f3fcf2ecb8c856","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v00\n- v07 --refutes--> v04\n- v00 --refutes--> v06\n- v02 --supports--> v03\n- v03 --supports--> v07\n- v07 --refutes--> v01\n- v00 --supports--> v02\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9fb38d02d74ba058e3b9c3bff7d087fddcded2b96a2bae975c29cbfd39027e88","schema":"livegraph.boolean-propagation.v1","spec_hash":"f0cee3f3fcf2ecb8c85618fef4ef37fa9b481aa15ffdaeef98ffde6fb0496b55","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-a67babdbc9f96f8ecb43","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v02\n- v06 --supports--> v03\n- v07 --supports--> v01\n- v04 --supports--> v05\n- v06 --supports--> v07\n- v00 --refutes--> v04\n- v07 --refutes--> v00\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"314a8f55167d7fd9fc22a37962f99216f4e564722543e09fcc54ded8fd99e9d0","schema":"livegraph.boolean-propagation.v1","spec_hash":"a67babdbc9f96f8ecb43bba61ba1447bce442bf152b24beaf6eaa572c3b3e036","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-e6a70ad25c713236c4db","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v02\n- v07 --supports--> v01\n- v04 --refutes--> v07\n- v07 --supports--> v00\n- v03 --supports--> v05\n- v02 --refutes--> v04\n- v03 --refutes--> v06\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"24f484469e569fd498c8f118722d1db7a2f7ef51ca580857ea27f3fc1deb02ef","schema":"livegraph.boolean-propagation.v1","spec_hash":"e6a70ad25c713236c4db352695579657d3b26eaf19707bac83d3610e86431b04","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-1f0ea6a9629572d28b27","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v02\n- v03 --supports--> v04\n- v01 --refutes--> v06\n- v07 --supports--> v00\n- v04 --refutes--> v05\n- v01 --supports--> v03\n- v05 --supports--> v07\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fd9443b57afed2aee0e829265216b9f139832d1b293a69c99a25e89882005d6c","schema":"livegraph.boolean-propagation.v1","spec_hash":"1f0ea6a9629572d28b271ddeda006e67da65dfb52b3e90e7b131c3aaabb4ace7","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-5e828fc6c69dcb0de5de","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v03\n- v01 --refutes--> v05\n- v01 --supports--> v06\n- v02 --refutes--> v01\n- v03 --supports--> v04\n- v04 --supports--> v00\n- v02 --supports--> v07\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0af1f4dd4b55d085df5c9d4fe2d3a338c284c4d16c19019a5c14859fce53bccb","schema":"livegraph.boolean-propagation.v1","spec_hash":"5e828fc6c69dcb0de5de6f1a54bf76139f7a4fc8064f7a427aaf5725e0e9b480","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-e71f521d79bd5c56d2fb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v04\n- v05 --supports--> v06\n- v04 --supports--> v01\n- v07 --refutes--> v05\n- v00 --supports--> v02\n- v00 --supports--> v03\n- v05 --refutes--> v00\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7360e2bcfda1cd44a9f941bbf980d5c10e0650fce7dafb15c71d7079d32eb8cc","schema":"livegraph.boolean-propagation.v1","spec_hash":"e71f521d79bd5c56d2fb65d849caf6a7640975d00dff56703e610be22be204bd","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-f8c793f51e994c641fe2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v05\n- v04 --supports--> v02\n- v02 --supports--> v03\n- v04 --supports--> v01\n- v05 --supports--> v00\n- v03 --supports--> v07\n- v05 --supports--> v04\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e6241105d07d4b6333c759a861d386b1ee5277cf75452c57489ad011ac32bd54","schema":"livegraph.boolean-propagation.v1","spec_hash":"f8c793f51e994c641fe263e8015bfe7e89312aa41047cdc3a4c4e562a1c5c897","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-e30b4fd8a6178636fe3a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v06\n- v06 --supports--> v07\n- v06 --supports--> v04\n- v04 --refutes--> v00\n- v00 --refutes--> v03\n- v03 --supports--> v02\n- v06 --refutes--> v01\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"31afdb416e22bdd055894a95ba1d344fb1fcc5a1df76e1ad17fead6ccabe25f7","schema":"livegraph.boolean-propagation.v1","spec_hash":"e30b4fd8a6178636fe3ad14b57e24767231530dd23eca17b50d8cf14ddc6eec0","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-8f9ed94eb03a810bcea9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v01\n- v04 --supports--> v05\n- v04 --refutes--> v06\n- v03 --refutes--> v00\n- v06 --supports--> v02\n- v07 --supports--> v03\n- v02 --refutes--> v07\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5410eed6feeb72c897c3d5f703126b98e5a6a98117fb140a0e4ebd81ad3ef2f8","schema":"livegraph.boolean-propagation.v1","spec_hash":"8f9ed94eb03a810bcea9b7a3809c5d26751a68d6eb437c717fdef944024e14d0","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-5854cdc9480cc80c46ad","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v05\n- v00 --refutes--> v04\n- v03 --refutes--> v02\n- v06 --refutes--> v01\n- v00 --refutes--> v06\n- v02 --refutes--> v07\n- v07 --supports--> v00\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"be2f8f743c95271304f19657f16ec7b7eba1d08752284d95267199923c3fb5cb","schema":"livegraph.boolean-propagation.v1","spec_hash":"5854cdc9480cc80c46ad2be67cb23c16e2be267d0b6164272c1cbb841f00645c","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-4e50552967b349b6c6fe","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v05\n- v02 --refutes--> v03\n- v00 --refutes--> v07\n- v03 --supports--> v04\n- v05 --refutes--> v00\n- v07 --supports--> v01\n- v00 --refutes--> v06\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"18fc0d64bf1ff3b66a5de60327b960beec07b8e831810ce03be235724263e0b8","schema":"livegraph.boolean-propagation.v1","spec_hash":"4e50552967b349b6c6fea85632ace179c39acdc9d938ab1d58c1f69111942d46","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-f11453352d1d3a3186c4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v06\n- v05 --supports--> v03\n- v04 --supports--> v05\n- v01 --supports--> v00\n- v00 --supports--> v07\n- v06 --supports--> v01\n- v05 --supports--> v02\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4c7b5a998204528d6016cd37570d6d76e72df54b74af9a00e93624aa1608974b","schema":"livegraph.boolean-propagation.v1","spec_hash":"f11453352d1d3a3186c48c57d8bf09f0ea51b9c5e9d87b4888595cf995e576a7","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-f9ad5516d7e0ff07f8c6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v03\n- v07 --refutes--> v00\n- v07 --refutes--> v06\n- v09 --supports--> v05\n- v09 --supports--> v04\n- v04 --refutes--> v08\n- v08 --supports--> v07\n- v06 --supports--> v01\n- v03 --refutes--> v02\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"acabc6815bdedc3b94207b6c5eb6203d1f64238c506ccfb121e516489441ca05","schema":"livegraph.boolean-propagation.v1","spec_hash":"f9ad5516d7e0ff07f8c670aeb09c00b287d909edd7bb1d89f9959b12eb33189d","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-4c98320b0ec447380a91","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v06\n- v08 --refutes--> v04\n- v01 --refutes--> v02\n- v03 --refutes--> v00\n- v01 --supports--> v09\n- v05 --supports--> v03\n- v05 --supports--> v07\n- v04 --supports--> v01\n- v00 --refutes--> v08\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"be1985de0077ea1a02b221033e0b3c18ef34e914094e98151c3d4c0350202b26","schema":"livegraph.boolean-propagation.v1","spec_hash":"4c98320b0ec447380a91b4ecf054a47b0ff29456bbccdfc58bad7fd31e358c6a","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-a84acd6cf1b97417b2a5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v01\n- v08 --supports--> v00\n- v00 --supports--> v05\n- v06 --refutes--> v04\n- v03 --supports--> v06\n- v04 --refutes--> v09\n- v03 --supports--> v02\n- v00 --refutes--> v07\n- v09 --refutes--> v08\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7890f8b07bc77fb2043148ecaa37dd01698dc2d0995a4e7a96b50418d5a73836","schema":"livegraph.boolean-propagation.v1","spec_hash":"a84acd6cf1b97417b2a5f2cc9f04def2c70d5a9657ae7398786adeebf0b1214a","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-6a2d8180450cf8148bf2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v08\n- v01 --refutes--> v05\n- v09 --refutes--> v07\n- v09 --supports--> v04\n- v02 --supports--> v03\n- v05 --refutes--> v09\n- v04 --supports--> v02\n- v03 --refutes--> v06\n- v09 --refutes--> v00\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ab51773cbe33d6657d0caf5fcaa29e60f25194c2a8896e4ae39be9a7fab4a852","schema":"livegraph.boolean-propagation.v1","spec_hash":"6a2d8180450cf8148bf2c41cf9c75eb830ef46550190fb71db31bd8fb040db44","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-ef3b80a2076e263ad5e1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v09\n- v09 --refutes--> v01\n- v02 --supports--> v05\n- v00 --supports--> v08\n- v09 --refutes--> v06\n- v04 --supports--> v03\n- v00 --supports--> v02\n- v06 --refutes--> v00\n- v07 --refutes--> v04\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"daea3fb1a683523457753f99ec3cc7fccf35f27589498f8702b8a67dfb210930","schema":"livegraph.boolean-propagation.v1","spec_hash":"ef3b80a2076e263ad5e1d5c4bc3cc7a4933a4c76f540d8bf1d11d0953898fa2a","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-21a7139a4101ceef655d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v07\n- v00 --refutes--> v03\n- v04 --refutes--> v06\n- v05 --refutes--> v09\n- v08 --refutes--> v00\n- v08 --supports--> v01\n- v06 --supports--> v08\n- v03 --refutes--> v05\n- v00 --refutes--> v02\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a123d2a3cea840af94f694dab3e1eed2d2ddbe5441a5bb92b73948afb4ce4026","schema":"livegraph.boolean-propagation.v1","spec_hash":"21a7139a4101ceef655dceec9ef0f868f548e8bf0c0497f8b7af8fc8592c0336","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-49ea8e2b0c041f3201d5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v07\n- v03 --refutes--> v02\n- v02 --refutes--> v09\n- v04 --refutes--> v00\n- v02 --supports--> v08\n- v00 --supports--> v06\n- v00 --supports--> v01\n- v09 --supports--> v04\n- v06 --refutes--> v05\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a5220bc9777746b653df222099810c5fba01ce98c0b43f086d0bfed2cb771cb9","schema":"livegraph.boolean-propagation.v1","spec_hash":"49ea8e2b0c041f3201d5f4b085025f6bf8c9bbca55c71aa6a7b1b4c93b01f120","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-ecdd32e6076f44636b18","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v03\n- v04 --refutes--> v09\n- v06 --supports--> v01\n- v01 --refutes--> v00\n- v04 --supports--> v05\n- v01 --supports--> v02\n- v02 --supports--> v04\n- v02 --refutes--> v07\n- v05 --refutes--> v08\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d8de2bbdcaed7b424b24a12ec745aa448ef8510f30ee0f51d1afb9c3333e96c6","schema":"livegraph.boolean-propagation.v1","spec_hash":"ecdd32e6076f44636b18057bb0f71279058fcc2a52629635c99343194a20211e","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-94668ef165587c46b0f6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v05\n- v04 --supports--> v01\n- v09 --refutes--> v00\n- v05 --supports--> v06\n- v05 --supports--> v02\n- v06 --refutes--> v09\n- v07 --refutes--> v03\n- v07 --refutes--> v08\n- v09 --supports--> v04\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"866118624d6e966277031fe8d71f7e2371f525b6945e56988eeb90e0806a0ea4","schema":"livegraph.boolean-propagation.v1","spec_hash":"94668ef165587c46b0f679c9cbf7d0b8feb64be043faa9db5f8dfeea60bb838a","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-ecd560bdc60bf48cd991","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v09\n- v08 --supports--> v00\n- v06 --refutes--> v02\n- v01 --refutes--> v04\n- v05 --refutes--> v06\n- v06 --refutes--> v08\n- v09 --refutes--> v01\n- v01 --refutes--> v07\n- v08 --supports--> v03\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"60beb19f574949a86738c9c59801f06f80e3b376dff1ae6eb55fe9012f2fcd92","schema":"livegraph.boolean-propagation.v1","spec_hash":"ecd560bdc60bf48cd991b60f1519bfa11ea350e9066d0968a135f434b588b5f6","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-aec19b4ac6dc9607485f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v09\n- v05 --refutes--> v02\n- v04 --refutes--> v06\n- v04 --supports--> v03\n- v07 --supports--> v04\n- v01 --refutes--> v08\n- v06 --refutes--> v00\n- v05 --refutes--> v07\n- v08 --refutes--> v05\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1983ca5a088d1c6013014f3af5e5bcbb8b351cd16d47dd225e7d3541bd4609bb","schema":"livegraph.boolean-propagation.v1","spec_hash":"aec19b4ac6dc9607485fa6746d09de74157e383064884e02e51826d6bae2cb0b","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-dd8cbeaacfe119dfa590","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v09\n- v03 --refutes--> v02\n- v06 --supports--> v00\n- v06 --refutes--> v01\n- v07 --supports--> v04\n- v09 --supports--> v06\n- v02 --refutes--> v05\n- v08 --supports--> v07\n- v07 --refutes--> v03\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"58c5d6cc5d60e0aa9e44b24e9305fd3534ce32f378f3debf78f9a71b5b048170","schema":"livegraph.boolean-propagation.v1","spec_hash":"dd8cbeaacfe119dfa59042ed52f5b86a1e72ac92a268df73ad56c7b0098707a6","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-d13a47e3a33ffa478b89","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v00\n- v09 --refutes--> v05\n- v01 --supports--> v04\n- v00 --supports--> v09\n- v04 --supports--> v06\n- v08 --refutes--> v01\n- v01 --supports--> v02\n- v06 --refutes--> v03\n- v08 --supports--> v07\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"90bd08f7a4ed07f85191bde04395b0f3e244f7a6223de5ba02d170fc3a75a8e8","schema":"livegraph.boolean-propagation.v1","spec_hash":"d13a47e3a33ffa478b890480a1f6cebaec3a05caaacd8ef8673cb2f38f43963e","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-b6759f4c9b6311bbdfd8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v01\n- v07 --supports--> v06\n- v03 --refutes--> v09\n- v00 --refutes--> v03\n- v05 --refutes--> v07\n- v03 --refutes--> v04\n- v07 --supports--> v08\n- v06 --refutes--> v00\n- v05 --supports--> v02\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"31647d70b35b4705e77709dc79d35a2515fac3bee5423b9a3a46d48a1d76540a","schema":"livegraph.boolean-propagation.v1","spec_hash":"b6759f4c9b6311bbdfd84576f299374a41d042b369b8284fea053feb18b8d6c1","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":6,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-15538bb25d2a34ace50f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v05\n- v00 --supports--> v07\n- v00 --supports--> v09\n- v02 --refutes--> v00\n- v05 --supports--> v02\n- v01 --refutes--> v04\n- v03 --supports--> v06\n- v03 --supports--> v08\n- v01 --refutes--> v03\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"81caeda8658f9ae50123b3054b506da7215bd55a465150e70472893cf59971e7","schema":"livegraph.boolean-propagation.v1","spec_hash":"15538bb25d2a34ace50fac04fcb9ed2db3ab1c8bdb4112148c8dd5b655e9afc3","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":6,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-c95fbf328a7ea4729b9c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v02\n- v01 --refutes--> v05\n- v07 --supports--> v03\n- v03 --supports--> v01\n- v02 --supports--> v04\n- v03 --supports--> v08\n- v04 --refutes--> v06\n- v09 --refutes--> v07\n- v09 --supports--> v00\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a78f90ae9bb0cd92cbb2b0d8444e51da90d0eccd0fbd875d39681c784985093c","schema":"livegraph.boolean-propagation.v1","spec_hash":"c95fbf328a7ea4729b9c17f8b20924c15c8047986daa0af9b1be1f1003f62092","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-fd437a8b65b5fc01aa6c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v09\n- v08 --supports--> v00\n- v00 --refutes--> v05\n- v06 --refutes--> v02\n- v05 --refutes--> v01\n- v05 --supports--> v06\n- v06 --supports--> v04\n- v03 --refutes--> v07\n- v02 --refutes--> v03\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"140c3c13a8165dfb329e8926bc8b62f445450b05166d9a66eb8d0f3e83d9e3f1","schema":"livegraph.boolean-propagation.v1","spec_hash":"fd437a8b65b5fc01aa6c0fff158a03ed36c858d7b0f97eb2ac6c1307d322637a","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-489b5be583c41f532a90","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v03\n- v07 --supports--> v08\n- v04 --refutes--> v05\n- v08 --refutes--> v02\n- v01 --supports--> v07\n- v08 --supports--> v00\n- v02 --refutes--> v04\n- v08 --refutes--> v09\n- v04 --supports--> v06\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9a12993fa3090d7d9551fcf242befac6652f809d153249fdcba2d91d09067131","schema":"livegraph.boolean-propagation.v1","spec_hash":"489b5be583c41f532a90f123bb2e1168855272ca31f46462f03007ff5885a73a","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-b1e06d286626000c175b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v04\n- v07 --supports--> v02\n- v00 --supports--> v05\n- v01 --supports--> v00\n- v07 --supports--> v06\n- v02 --supports--> v09\n- v01 --supports--> v03\n- v05 --refutes--> v08\n- v04 --refutes--> v01\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0603a61e1f720989b4afa89d4fa4f9b67d943dd7679986e8ae8b0008f9312415","schema":"livegraph.boolean-propagation.v1","spec_hash":"b1e06d286626000c175b1689a08f5fac8fd370d99dc09426b359863b8da8bd62","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":6,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-12b3a3f23358dbc182b7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v04\n- v04 --refutes--> v02\n- v05 --supports--> v06\n- v09 --refutes--> v00\n- v05 --supports--> v03\n- v06 --supports--> v01\n- v06 --supports--> v07\n- v04 --supports--> v08\n- v00 --supports--> v05\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"59f39bc95bc9d7a265e2d8a30424c73484556754bd7b6f630f30d7569546cb62","schema":"livegraph.boolean-propagation.v1","spec_hash":"12b3a3f23358dbc182b759604f3c082b159eeea6b2dc43d698409c03c60f7b9b","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-961f0b8d85806f843999","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v08\n- v08 --supports--> v00\n- v02 --supports--> v06\n- v06 --supports--> v01\n- v06 --refutes--> v04\n- v01 --refutes--> v09\n- v08 --refutes--> v02\n- v03 --refutes--> v05\n- v08 --refutes--> v07\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1ab2baee9fd1bd407e330c3ad1a3ad7a96679553b56852db7137a0893ae09df1","schema":"livegraph.boolean-propagation.v1","spec_hash":"961f0b8d85806f843999a76fa08bc104dffbaeea2b6459d048fe77e561103e0b","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-3241bbf054dec256d0f6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v02\n- v06 --supports--> v00\n- v07 --supports--> v05\n- v04 --refutes--> v08\n- v03 --refutes--> v04\n- v06 --refutes--> v09\n- v05 --refutes--> v01\n- v01 --refutes--> v06\n- v04 --supports--> v07\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b09b84c069346481ba74254f1a5fe632519165e9bc44be20831e3a389b418186","schema":"livegraph.boolean-propagation.v1","spec_hash":"3241bbf054dec256d0f626559401535f013524bc81b1bef8ec042f28d3b8a9b2","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":6,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-f858a24a1bc69b168604","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v05\n- v02 --supports--> v09\n- v08 --supports--> v07\n- v09 --supports--> v06\n- v03 --refutes--> v00\n- v01 --supports--> v03\n- v04 --refutes--> v08\n- v09 --supports--> v01\n- v08 --supports--> v02\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"29c9308bc590f627d58ec9113537e0d73cbeb1fc4c15f86fe04babb96b0b8765","schema":"livegraph.boolean-propagation.v1","spec_hash":"f858a24a1bc69b168604bd91358466c602a9d3a14592e34eadaf10fd3c952585","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-3e14fe7f1f394b2d2c52","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v03\n- v04 --supports--> v00\n- v08 --supports--> v06\n- v07 --refutes--> v09\n- v09 --supports--> v01\n- v01 --refutes--> v04\n- v05 --refutes--> v07\n- v08 --refutes--> v02\n- v04 --supports--> v08\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c08c11325f830ba81e6efa0bc7af22b963c11657d82c63a19b6011cb246952c1","schema":"livegraph.boolean-propagation.v1","spec_hash":"3e14fe7f1f394b2d2c52ca5708c18da535b92676e99852c4f0db478556d3d32f","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-71ec2c8e315d4cfc5299","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v05\n- v05 --refutes--> v09\n- v07 --supports--> v03\n- v07 --refutes--> v02\n- v07 --refutes--> v08\n- v01 --supports--> v06\n- v06 --refutes--> v07\n- v04 --supports--> v00\n- v09 --supports--> v01\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7029448b311fed285aa9b02fa35c4d0f863b173957033ddb276bfee5702e4a79","schema":"livegraph.boolean-propagation.v1","spec_hash":"71ec2c8e315d4cfc529917599ba01e6b1d4864733b48b11e5babd40dca245dcc","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-ac70576c57e92c57b36d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v06\n- v07 --supports--> v05\n- v02 --supports--> v07\n- v04 --supports--> v09\n- v05 --supports--> v01\n- v01 --supports--> v08\n- v01 --refutes--> v00\n- v08 --refutes--> v04\n- v04 --supports--> v03\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fc747bfe95aac97ac761a14a1f02b5b43139ead32b1bb2a4d83c1ec04293c839","schema":"livegraph.boolean-propagation.v1","spec_hash":"ac70576c57e92c57b36d57f50361fcce80831822f66cd5615131d08c9e0d4fd2","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-1ec34923f4fc8041c643","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v08\n- v03 --supports--> v02\n- v04 --supports--> v00\n- v07 --supports--> v04\n- v08 --refutes--> v03\n- v03 --refutes--> v06\n- v07 --supports--> v09\n- v02 --supports--> v07\n- v04 --supports--> v01\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"46f68f084aceb36db8e57a95f50dbd857362e625eae81c64c4f8d9022bc32dd9","schema":"livegraph.boolean-propagation.v1","spec_hash":"1ec34923f4fc8041c643cbb81607ddf15cd8cb99acf2f86ac26fe4675030590c","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-b7c7b46ed2ad7924bb60","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v05\n- v06 --refutes--> v09\n- v05 --supports--> v03\n- v07 --supports--> v08\n- v04 --supports--> v06\n- v09 --supports--> v01\n- v01 --refutes--> v02\n- v05 --refutes--> v07\n- v07 --supports--> v00\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4fdbe9f90643195eea414642c6c7921d2c8415700fc7c0372d4f3e3c65c007e7","schema":"livegraph.boolean-propagation.v1","spec_hash":"b7c7b46ed2ad7924bb604befa740c82c38bc2dba26d81f4a5eac0299813867c1","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-da45ced821aa38462b9d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v00\n- v07 --supports--> v01\n- v03 --refutes--> v09\n- v09 --refutes--> v02\n- v00 --refutes--> v03\n- v00 --supports--> v05\n- v03 --refutes--> v04\n- v01 --supports--> v08\n- v01 --supports--> v06\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"49b0ea2f71c00d9d7b20dbc799eca5b5a10ef8cfa35b974748d5c866b720c9f5","schema":"livegraph.boolean-propagation.v1","spec_hash":"da45ced821aa38462b9d4a4c2673ba81b5ae6669c2dcba77c068291a69850682","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-6fb0669f5d8b0c47cc96","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v09\n- v09 --refutes--> v07\n- v03 --refutes--> v08\n- v00 --supports--> v03\n- v06 --supports--> v05\n- v02 --supports--> v00\n- v09 --refutes--> v01\n- v07 --supports--> v02\n- v03 --refutes--> v04\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9d3e3c1742fb2eb85bd95df962fc00629faad20297248206c65c5dbc7631faf4","schema":"livegraph.boolean-propagation.v1","spec_hash":"6fb0669f5d8b0c47cc9666841a65ef7529fb663c58ead59853d49155d0c2618a","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-261eab5a697392e3ebed","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v02\n- v05 --supports--> v04\n- v06 --refutes--> v08\n- v07 --supports--> v06\n- v02 --supports--> v05\n- v02 --refutes--> v09\n- v00 --supports--> v03\n- v05 --supports--> v00\n- v08 --supports--> v01\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4ab562d6db79c708cb4378c59ab437520e1a49d33cd0af12d5a698fb76e3a807","schema":"livegraph.boolean-propagation.v1","spec_hash":"261eab5a697392e3ebedb4ac120cc8d6036ab4ae9e4938c78d46e81d5364d977","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-6effb5c7fa388b17003e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v07\n- v06 --refutes--> v00\n- v06 --refutes--> v02\n- v04 --supports--> v08\n- v09 --supports--> v03\n- v00 --refutes--> v04\n- v01 --refutes--> v06\n- v04 --refutes--> v09\n- v05 --supports--> v01\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ae2e16e5e290189a66b17ae5a2d4fe2d1804bc3fba830e05486c5e774f5353c9","schema":"livegraph.boolean-propagation.v1","spec_hash":"6effb5c7fa388b17003eed79f29d46e370f5a0b613ab2c3db4cd013f73c70512","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-0551b313af47ce3707ae","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v08\n- v09 --supports--> v00\n- v08 --supports--> v04\n- v08 --refutes--> v02\n- v06 --refutes--> v03\n- v07 --refutes--> v05\n- v07 --supports--> v06\n- v00 --supports--> v07\n- v06 --supports--> v01\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dedacf04b999011013b4e367a2e26e3bd2cb326fda70c4fea5c5fbf72c3df998","schema":"livegraph.boolean-propagation.v1","spec_hash":"0551b313af47ce3707ae69b9fa83059ca24734c9bc973be4663255a09bb647ba","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-2e4746ba292e386bc3d4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v05\n- v04 --refutes--> v08\n- v04 --supports--> v01\n- v03 --refutes--> v02\n- v00 --refutes--> v09\n- v01 --refutes--> v03\n- v03 --refutes--> v06\n- v05 --refutes--> v04\n- v09 --supports--> v07\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"14c623db4c45db255386e5fc8018b5e316fa081e6a40b5b219db889cb6461f61","schema":"livegraph.boolean-propagation.v1","spec_hash":"2e4746ba292e386bc3d4383424e7506d4e29af40fc9f08e25fb0ae6a63d9675e","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-194230f3d287811a0ce4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v06\n- v08 --refutes--> v01\n- v08 --supports--> v00\n- v01 --refutes--> v03\n- v02 --refutes--> v08\n- v07 --supports--> v02\n- v05 --supports--> v07\n- v03 --supports--> v04\n- v02 --refutes--> v09\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"21638e015d3bd3ba60795763cfdad38286478790e256b5a038f23bdaa71f81cb","schema":"livegraph.boolean-propagation.v1","spec_hash":"194230f3d287811a0ce4ad9d75f2bb0ffa547f1d30956a9ef86d739a24ceacc3","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-61607f6c7ceaccac869b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v07\n- v08 --supports--> v05\n- v04 --refutes--> v02\n- v00 --supports--> v04\n- v05 --refutes--> v00\n- v04 --supports--> v06\n- v06 --refutes--> v03\n- v04 --supports--> v09\n- v03 --supports--> v01\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"463aa84662d41a3f0a4d141aa8bd9b24525fb25c7ceb02ec92b5117204ed9c6f","schema":"livegraph.boolean-propagation.v1","spec_hash":"61607f6c7ceaccac869b0462f02020a9a0c857122983bc891b9bf2b475a01a0c","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-2368e1e2d2ea935c5ce1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v05\n- v07 --refutes--> v03\n- v03 --supports--> v04\n- v09 --supports--> v08\n- v03 --supports--> v06\n- v01 --supports--> v02\n- v06 --refutes--> v09\n- v07 --refutes--> v00\n- v02 --refutes--> v07\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4e191fb1bfbab7c0f2d5edf1b3637201e22b93f52e1c36b2f2f1b16f19f21e43","schema":"livegraph.boolean-propagation.v1","spec_hash":"2368e1e2d2ea935c5ce1eec6dad1830bf7f30cbe0c75b1203b80561aaba6966a","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-ecf15e1a91600911a82f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v02\n- v04 --refutes--> v09\n- v01 --refutes--> v07\n- v02 --refutes--> v03\n- v06 --supports--> v05\n- v05 --supports--> v08\n- v02 --supports--> v00\n- v04 --supports--> v01\n- v02 --refutes--> v04\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ab2f6ef4b57d4cdb9b9242e34d50f8880f1aa08256f99b0a3d8d32e6eb43b5f7","schema":"livegraph.boolean-propagation.v1","spec_hash":"ecf15e1a91600911a82f69b35d3785b255b8ec7490236084ecd229cb7976fd5c","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-8cefae5977c2330ef289","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v03\n- v07 --supports--> v01\n- v09 --supports--> v02\n- v09 --refutes--> v04\n- v07 --supports--> v06\n- v03 --refutes--> v08\n- v02 --supports--> v00\n- v01 --supports--> v05\n- v08 --refutes--> v07\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"597912d932237cbb10ca2563e01a61acda086b875a992e81fbb87c2b851328cd","schema":"livegraph.boolean-propagation.v1","spec_hash":"8cefae5977c2330ef2891275a8786d2f3cd02f2c2a3a4689376be3e9abb57339","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-d2b87c398b1306cddd01","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v03\n- v06 --refutes--> v08\n- v01 --supports--> v04\n- v01 --refutes--> v09\n- v09 --refutes--> v06\n- v02 --refutes--> v07\n- v07 --supports--> v05\n- v01 --refutes--> v00\n- v08 --supports--> v02\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9d72a8680f6a5bfe4dfaa1d1303c3dd8bc70449c1bff28c7a9cd1ec83ac208d9","schema":"livegraph.boolean-propagation.v1","spec_hash":"d2b87c398b1306cddd017014815e5d40d304a836cc7445f99e29a5adc0504d52","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-f350e1f580ad84c3e8d8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v09\n- v04 --refutes--> v08\n- v09 --refutes--> v02\n- v00 --refutes--> v03\n- v09 --supports--> v07\n- v00 --refutes--> v01\n- v06 --refutes--> v00\n- v08 --supports--> v05\n- v02 --refutes--> v06\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cff6d6ba3acadcdb07447f629554775011106030c59d5e4d364700ca9f19ddda","schema":"livegraph.boolean-propagation.v1","spec_hash":"f350e1f580ad84c3e8d856f90469e6d3c903cf34c9c91c310161dd1af00e37f5","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-196649f1abb1c3e707bb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v09\n- v05 --supports--> v00\n- v01 --refutes--> v04\n- v00 --supports--> v02\n- v08 --supports--> v07\n- v06 --supports--> v05\n- v01 --supports--> v06\n- v06 --supports--> v03\n- v07 --supports--> v01\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5e5584784b5f9d8047d25005d15cc91d9522c43b84f746ac2d0213e6c8492740","schema":"livegraph.boolean-propagation.v1","spec_hash":"196649f1abb1c3e707bb5334524bf41f11b5b3cd589a694af1498d79e495c1a1","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-57e2a0f069d14d0ba677","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v09\n- v01 --supports--> v04\n- v08 --refutes--> v02\n- v04 --supports--> v05\n- v02 --refutes--> v07\n- v07 --supports--> v00\n- v06 --supports--> v08\n- v07 --refutes--> v03\n- v04 --refutes--> v06\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1816e375e96b700b6b21292948dbb7f202af358a5b2058683c64e5956201b402","schema":"livegraph.boolean-propagation.v1","spec_hash":"57e2a0f069d14d0ba677802897ef9765907c2966b630204e33120315c9e2a07e","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-451a44b2d1a5b22e3a01","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v09\n- v01 --refutes--> v07\n- v08 --supports--> v00\n- v05 --refutes--> v04\n- v03 --supports--> v08\n- v08 --supports--> v06\n- v04 --refutes--> v02\n- v06 --supports--> v05\n- v01 --supports--> v03\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"eafae3a3b9d7130c2d08c85dc315812f708fb0486194392e2d80a4321f5fa653","schema":"livegraph.boolean-propagation.v1","spec_hash":"451a44b2d1a5b22e3a01bbc187059a835bf46a5f6dc5f957712f9d7ffcef112d","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-8055da579c6bc193bcf0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v03\n- v00 --supports--> v07\n- v00 --supports--> v08\n- v05 --refutes--> v04\n- v06 --supports--> v02\n- v06 --refutes--> v00\n- v03 --supports--> v09\n- v03 --supports--> v01\n- v09 --refutes--> v05\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ebe6fa29ac4571fd61a6816478f52b94d77b55c6850efc3327c13824ecf7f412","schema":"livegraph.boolean-propagation.v1","spec_hash":"8055da579c6bc193bcf0ca0467c917930f114dca7fa9d0994947140b00675c62","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-665fe8c291ff968b9f0b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v08\n- v08 --supports--> v02\n- v03 --refutes--> v07\n- v03 --refutes--> v09\n- v02 --refutes--> v06\n- v03 --supports--> v05\n- v03 --supports--> v01\n- v06 --supports--> v04\n- v07 --refutes--> v00\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d3bd3e7fd9ba978f2df46d5393049e5f01dd579d604ed19c27758dee434be473","schema":"livegraph.boolean-propagation.v1","spec_hash":"665fe8c291ff968b9f0bb39b18729eba12174ebb7ba13526504eb285e9ba5043","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-9554c52703d4305232b9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v00\n- v09 --refutes--> v07\n- v01 --refutes--> v05\n- v07 --supports--> v03\n- v02 --supports--> v01\n- v05 --refutes--> v04\n- v07 --refutes--> v08\n- v02 --refutes--> v06\n- v03 --refutes--> v02\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"de0b1f8f18fdc36aa2e96896226c987ad67e9fb7b4ec2577235615c828e17a07","schema":"livegraph.boolean-propagation.v1","spec_hash":"9554c52703d4305232b9a7114ca1929fc4e44825cc68179c7b5b7c2a608b195c","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-8265c1f46ee4ee138f09","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v08\n- v06 --supports--> v02\n- v03 --refutes--> v01\n- v07 --supports--> v06\n- v00 --supports--> v07\n- v06 --supports--> v04\n- v05 --refutes--> v00\n- v00 --refutes--> v09\n- v03 --supports--> v05\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2329e91d2cee6c81924d99e090359ca0074c286f10e4ce3bfca14dc293a310c2","schema":"livegraph.boolean-propagation.v1","spec_hash":"8265c1f46ee4ee138f0922390e7dc2cd28db2fadf6226ee2b81fab21e277574e","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":6,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-1443278ca3d665a1560b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v05\n- v01 --supports--> v08\n- v07 --supports--> v09\n- v05 --supports--> v03\n- v09 --supports--> v00\n- v04 --supports--> v02\n- v05 --refutes--> v07\n- v09 --refutes--> v04\n- v04 --supports--> v06\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fcf9dc79275c3d952bfd5af39581a6e4ebb405ddecbf1dc331f2bcc0babe16ba","schema":"livegraph.boolean-propagation.v1","spec_hash":"1443278ca3d665a1560b49d5e5141ca9f7df5e7e82adb1903206a56a6923a1d8","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-18537c532accd30ff39d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v03\n- v06 --refutes--> v05\n- v08 --refutes--> v01\n- v00 --supports--> v02\n- v05 --supports--> v07\n- v00 --refutes--> v09\n- v09 --supports--> v06\n- v07 --supports--> v08\n- v05 --refutes--> v04\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"af50ba78433fb7040fc99801ba0196bea41678b9aac454fec701c4939e39b0cb","schema":"livegraph.boolean-propagation.v1","spec_hash":"18537c532accd30ff39d50306986136873113dac2fffac905d4462d2b624cd7c","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-3977b0d1868bfed5ef7b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v06\n- v04 --refutes--> v08\n- v09 --refutes--> v00\n- v09 --refutes--> v05\n- v02 --supports--> v04\n- v07 --refutes--> v09\n- v04 --supports--> v01\n- v03 --refutes--> v07\n- v08 --supports--> v03\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4c22fff66915f3bf9798f4407786baa065aaadf99cdc0ba9df439c882c39afd2","schema":"livegraph.boolean-propagation.v1","spec_hash":"3977b0d1868bfed5ef7b7274255f13bd2f48fdaece8d288209a48df35241b61b","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-d2205e9c7ee2b61f6641","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v05\n- v03 --supports--> v09\n- v08 --refutes--> v03\n- v02 --refutes--> v07\n- v00 --supports--> v04\n- v04 --supports--> v01\n- v04 --refutes--> v02\n- v02 --refutes--> v06\n- v06 --supports--> v08\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b1c8569c3e5fc046184666e95b4cceb3cc52db3f7b98cd45d0ebb0140a6874b2","schema":"livegraph.boolean-propagation.v1","spec_hash":"d2205e9c7ee2b61f6641b4025b6e6accf9a22e67abf4955a4addd5d849c266da","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-2cea4702e5e32461b851","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v04\n- v07 --refutes--> v02\n- v08 --refutes--> v09\n- v03 --refutes--> v06\n- v03 --refutes--> v00\n- v07 --supports--> v08\n- v04 --refutes--> v01\n- v01 --supports--> v03\n- v01 --refutes--> v05\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b6fa1f14a9806f7f7f3ca89b92269627fa6b1246c033b01479218a793e627d58","schema":"livegraph.boolean-propagation.v1","spec_hash":"2cea4702e5e32461b851e53007cd17486c73d5344bbb7bb2067b1dc503509b48","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-87e982da3d7ee5c76162","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v06\n- v03 --refutes--> v05\n- v02 --supports--> v00\n- v07 --supports--> v08\n- v08 --refutes--> v02\n- v07 --supports--> v04\n- v02 --refutes--> v03\n- v09 --supports--> v01\n- v01 --supports--> v07\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2cb4a5f52c5cade8b9be54764fc400847c571af4af254035614615faec067823","schema":"livegraph.boolean-propagation.v1","spec_hash":"87e982da3d7ee5c76162791cfe772a65899886630d27f70466f20a1e27c84d89","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-8b372105bd3c6454c7f3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v06\n- v01 --supports--> v04\n- v03 --refutes--> v07\n- v09 --refutes--> v08\n- v07 --supports--> v05\n- v01 --refutes--> v02\n- v08 --supports--> v03\n- v00 --refutes--> v09\n- v07 --refutes--> v01\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8573bcc98e6db716e848fa8350ec35eaa2561743f66f3867cc1ad6453ea0f052","schema":"livegraph.boolean-propagation.v1","spec_hash":"8b372105bd3c6454c7f36be4a3f19864569912e4accd78d04aab44f5d7a0136a","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-d2505afea377deeca1e6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v09\n- v03 --supports--> v02\n- v00 --supports--> v07\n- v02 --refutes--> v05\n- v09 --supports--> v04\n- v02 --supports--> v01\n- v00 --refutes--> v08\n- v04 --refutes--> v00\n- v00 --refutes--> v03\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"313eabb7c7cd6f75f2f2814ab72f46627e9e989c044065054f56be7af535350b","schema":"livegraph.boolean-propagation.v1","spec_hash":"d2505afea377deeca1e6386122987cf0fae89f2bc88ad629732efa085e744dce","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-29309245a39d1cb7758f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v08\n- v08 --supports--> v06\n- v03 --supports--> v09\n- v06 --supports--> v03\n- v04 --refutes--> v05\n- v03 --refutes--> v04\n- v07 --refutes--> v02\n- v07 --supports--> v01\n- v04 --supports--> v07\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f3f5c2d8cce8889204c0cdf7c4d8255edd2df72b39c461daf5f0ff2155acde1e","schema":"livegraph.boolean-propagation.v1","spec_hash":"29309245a39d1cb7758ffd548bfba181db107a81ec2e2b515a6352a228e9bef5","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-a202c014f8f5e2a3f337","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v03\n- v04 --refutes--> v00\n- v07 --supports--> v08\n- v06 --supports--> v04\n- v08 --refutes--> v05\n- v05 --refutes--> v01\n- v05 --refutes--> v09\n- v01 --supports--> v06\n- v06 --supports--> v02\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b43e926b4b18658b400071512df20873fb5eb4ac02829c6bdefbfebd3c72e496","schema":"livegraph.boolean-propagation.v1","spec_hash":"a202c014f8f5e2a3f337eeb099aa6788fefdfd62dc77adba0c149b7bfeb12739","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-fab2ae5c030a4853ed4c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v08\n- v07 --refutes--> v05\n- v04 --refutes--> v07\n- v07 --refutes--> v06\n- v05 --supports--> v01\n- v00 --supports--> v02\n- v02 --supports--> v04\n- v05 --refutes--> v09\n- v03 --supports--> v00\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"88eb78f880132fa2139d0952d7c1bdec5431f427ba2a185f505885eea3f40473","schema":"livegraph.boolean-propagation.v1","spec_hash":"fab2ae5c030a4853ed4c123d8792ba4f6a0a2c1c7289cd4b988361885c3bdc2b","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":6,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-99753ce12f07ef6ba66a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v05\n- v01 --supports--> v03\n- v07 --supports--> v04\n- v06 --supports--> v00\n- v01 --refutes--> v02\n- v08 --refutes--> v09\n- v05 --refutes--> v06\n- v00 --supports--> v08\n- v07 --refutes--> v01\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7227e9aa4853f52e02ec41dcdadf311363dea50836715c5cd6a760b7f864a69c","schema":"livegraph.boolean-propagation.v1","spec_hash":"99753ce12f07ef6ba66a61cfe2f7564d50cb7e9f3c95c1d81c364b9140aa454e","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-060f7fecd65858b9cb69","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v03\n- v01 --refutes--> v05\n- v05 --refutes--> v08\n- v05 --refutes--> v07\n- v04 --supports--> v01\n- v08 --refutes--> v02\n- v05 --supports--> v06\n- v00 --refutes--> v09\n- v09 --refutes--> v04\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1203a9efaaea2c5ec8178b3421af64686042b466c29b87cf9092bc8e68eff600","schema":"livegraph.boolean-propagation.v1","spec_hash":"060f7fecd65858b9cb69d20edfdea7308e80a679c08300a3c0f27ab0d4f78c62","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-2f8a553328700e4f0a9a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v04\n- v01 --refutes--> v08\n- v05 --supports--> v03\n- v06 --refutes--> v09\n- v05 --refutes--> v07\n- v02 --refutes--> v05\n- v08 --supports--> v02\n- v00 --refutes--> v06\n- v06 --refutes--> v01\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"132e381ec1b5577dd160869d7b40f538ce1b92d18b51b3cc1a32c8e9319525ed","schema":"livegraph.boolean-propagation.v1","spec_hash":"2f8a553328700e4f0a9acde9abd900eb02a5d7b4e129accd432f841e101e0599","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-256668a8ac0aabe494a0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v05\n- v07 --refutes--> v06\n- v06 --supports--> v09\n- v09 --supports--> v04\n- v08 --supports--> v00\n- v09 --refutes--> v03\n- v00 --supports--> v07\n- v01 --supports--> v02\n- v01 --refutes--> v08\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6588417b7075a7a5240ff03d3fdb0dd91b0caa0a911649c96d0a1f7f4d23ecde","schema":"livegraph.boolean-propagation.v1","spec_hash":"256668a8ac0aabe494a0d082ddae0ac86dc9368cb997c6fecbaef054d08cf6db","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-538d50962511290c3e52","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v08\n- v06 --refutes--> v09\n- v04 --refutes--> v01\n- v03 --refutes--> v06\n- v08 --refutes--> v03\n- v06 --refutes--> v02\n- v06 --supports--> v04\n- v01 --supports--> v07\n- v01 --refutes--> v05\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dc585ce6cb7061e3d0be9ea6e1b95e054c5b242586684781b26ae9462a4b9139","schema":"livegraph.boolean-propagation.v1","spec_hash":"538d50962511290c3e5223baff9f883060da4a7e50407c432363fae24d1c4e6a","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-8d3f32d25a2a21626a32","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v05\n- v00 --refutes--> v03\n- v04 --refutes--> v07\n- v08 --refutes--> v00\n- v04 --supports--> v02\n- v00 --supports--> v06\n- v05 --supports--> v09\n- v03 --supports--> v04\n- v02 --supports--> v01\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9b9815ec7e9e2491173d03aa4a30712360410be5785698a662817ca21f9f7f98","schema":"livegraph.boolean-propagation.v1","spec_hash":"8d3f32d25a2a21626a3253297b5f885b326aa6ac5816ec75f54f9dc076a754d5","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-70c346db0de99e55cac0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v06\n- v02 --refutes--> v00\n- v07 --refutes--> v02\n- v01 --refutes--> v05\n- v02 --supports--> v08\n- v01 --refutes--> v07\n- v06 --supports--> v09\n- v04 --supports--> v01\n- v06 --refutes--> v04\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3b8c9af344fe9941aaa1d52e3d63f7337838af49ed06659d3f6cae011cd16b11","schema":"livegraph.boolean-propagation.v1","spec_hash":"70c346db0de99e55cac0a835c1a4b82f9b1cde13863b392580c5e6fae03ea39a","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":6,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-ae9fb95ec8e83b4e4482","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v02\n- v07 --refutes--> v00\n- v02 --refutes--> v06\n- v02 --supports--> v04\n- v01 --refutes--> v03\n- v09 --supports--> v01\n- v06 --refutes--> v09\n- v06 --supports--> v08\n- v06 --refutes--> v05\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d03e123e558fab60d166b8fcbd72cf7b4dcfef14b0acbe2f8068d30537806ac5","schema":"livegraph.boolean-propagation.v1","spec_hash":"ae9fb95ec8e83b4e44826619d7eea3a8d3e1f33a3b04e161435b759a93e3e04e","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-49d5acfb27586b593d5e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v01\n- v04 --refutes--> v09\n- v00 --supports--> v04\n- v05 --supports--> v07\n- v09 --supports--> v05\n- v00 --refutes--> v06\n- v08 --supports--> v02\n- v02 --refutes--> v03\n- v05 --supports--> v08\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2f6f4e620a44d898df1c22b6cdb245f9c467191730a4dcda33bbe6537c7a3146","schema":"livegraph.boolean-propagation.v1","spec_hash":"49d5acfb27586b593d5eb4f4eee9e3c4df0bbf0a28147f23ab89eaee7ba54f42","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-9be02f5e65c7ff522266","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v04\n- v04 --refutes--> v00\n- v07 --refutes--> v06\n- v00 --refutes--> v07\n- v00 --supports--> v02\n- v07 --supports--> v01\n- v08 --refutes--> v05\n- v03 --refutes--> v08\n- v09 --refutes--> v03\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a3ea2488d9baf27defdf9dbb82a455eb8821f30765453f343a70239ec65e4400","schema":"livegraph.boolean-propagation.v1","spec_hash":"9be02f5e65c7ff5222669f0b5051f83f6b16211cc77e837a1aa6d1ecd34de9db","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-c2ae4f2dc35cf2b833e2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v05\n- v03 --supports--> v01\n- v04 --supports--> v00\n- v07 --refutes--> v08\n- v03 --refutes--> v06\n- v05 --supports--> v07\n- v00 --refutes--> v02\n- v00 --refutes--> v03\n- v03 --supports--> v09\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"63cf5d5fa3815aefbbd1cfbb2a8f6021b7c11a1b94aea61ab2a971622a59b7cf","schema":"livegraph.boolean-propagation.v1","spec_hash":"c2ae4f2dc35cf2b833e2ea79632eb944fb5d37d1a843f994a83c0d0d3157fd47","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":6,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-9ad5acf53926261c0902","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v08\n- v05 --supports--> v01\n- v04 --refutes--> v03\n- v05 --refutes--> v09\n- v04 --supports--> v02\n- v06 --supports--> v04\n- v08 --refutes--> v06\n- v00 --refutes--> v07\n- v02 --supports--> v00\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f347746cb0ccd351e6deec8184d72ff087b200386f60a964e2f4563a6567dcdd","schema":"livegraph.boolean-propagation.v1","spec_hash":"9ad5acf53926261c0902484dda8f5f62116bf3aef64d7f03f4d9084050d4d348","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-28fb86f70fdee895e57e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v02\n- v01 --refutes--> v04\n- v09 --supports--> v07\n- v01 --refutes--> v05\n- v00 --supports--> v09\n- v03 --supports--> v08\n- v01 --supports--> v03\n- v02 --supports--> v00\n- v07 --supports--> v06\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f4d4346ccee1ed814d7bf0922283bed25231308fa62476a70e5caad75d0fdf9a","schema":"livegraph.boolean-propagation.v1","spec_hash":"28fb86f70fdee895e57eaa777f76db552adce13e42aadd491dc330ae9b0c2ea1","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-b8a22acc913a1d0b330a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v08\n- v07 --supports--> v05\n- v06 --supports--> v00\n- v04 --refutes--> v02\n- v08 --supports--> v01\n- v06 --refutes--> v04\n- v02 --refutes--> v07\n- v08 --supports--> v09\n- v07 --refutes--> v03\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"946a3f6f7efc7287817ec70f3d0ea83e0a2e961d85e410263086090cc01c5912","schema":"livegraph.boolean-propagation.v1","spec_hash":"b8a22acc913a1d0b330a79ac53ee0a9a9f7914f02394ac6832d6da2e3801fed5","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-25b6e50526627fa572cd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v02\n- v08 --supports--> v01\n- v02 --supports--> v03\n- v02 --refutes--> v04\n- v07 --supports--> v06\n- v03 --refutes--> v05\n- v02 --refutes--> v00\n- v01 --supports--> v07\n- v07 --supports--> v09\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"37bf3c01a35a2a83ab8703bc592db745704535a3c16f51da4556620a7291847a","schema":"livegraph.boolean-propagation.v1","spec_hash":"25b6e50526627fa572cdc3cfd0ac46045f8af7774a8da4e96d6a10698851027b","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-177590f559a8460903bf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v04\n- v06 --supports--> v01\n- v09 --supports--> v00\n- v06 --supports--> v03\n- v05 --supports--> v02\n- v01 --refutes--> v08\n- v01 --supports--> v07\n- v02 --refutes--> v09\n- v00 --supports--> v06\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"02f0397c2c969b55744e6781dc2090435b3e1e0547532998d685d1ea31a78cf7","schema":"livegraph.boolean-propagation.v1","spec_hash":"177590f559a8460903bf0022dbf1992076eaaaa7db26e03d2166792861ffdf50","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-24bb75839b96f81add53","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v02\n- v02 --refutes--> v03\n- v01 --refutes--> v06\n- v07 --supports--> v04\n- v07 --supports--> v09\n- v03 --supports--> v00\n- v03 --supports--> v07\n- v06 --supports--> v08\n- v04 --refutes--> v05\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6aad4c537d5ef3a842b285df174bacab2f51f503b32c0bc7ab4f2fb170e76fcb","schema":"livegraph.boolean-propagation.v1","spec_hash":"24bb75839b96f81add5366276bfdeea02f7eb6fdefcb985ef586e4d25950adf2","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":6,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-692b20eea32e9dfc78ae","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v02\n- v05 --supports--> v06\n- v07 --supports--> v00\n- v08 --supports--> v01\n- v00 --refutes--> v08\n- v04 --refutes--> v07\n- v01 --refutes--> v09\n- v01 --supports--> v05\n- v04 --supports--> v03\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7a453abbf2b703d8135268b72ca5fa9d36ac1ea4da42f881160f3ffd9482da43","schema":"livegraph.boolean-propagation.v1","spec_hash":"692b20eea32e9dfc78aed285e09fe6281cb71d74e06bc2e8ba809d1a0dc9958a","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-c026ece71695913bfa05","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v04\n- v07 --supports--> v06\n- v09 --refutes--> v05\n- v05 --refutes--> v03\n- v00 --supports--> v07\n- v06 --refutes--> v09\n- v02 --refutes--> v01\n- v00 --refutes--> v08\n- v05 --refutes--> v02\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7198e43f48c02ea4e3bd2b07a3837abeeb928a803e855c5af5a0532c1a78d4d1","schema":"livegraph.boolean-propagation.v1","spec_hash":"c026ece71695913bfa05fb4fff1a5698af0b4fcbad51b495013fdf8503e7cb1e","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-45d1c80b869a49754620","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v06\n- v01 --refutes--> v04\n- v00 --refutes--> v08\n- v06 --supports--> v07\n- v06 --refutes--> v05\n- v00 --refutes--> v09\n- v04 --supports--> v03\n- v07 --supports--> v02\n- v02 --refutes--> v00\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a84144c81ee21e1e2fa83475688029e2a533eeee37dcc227c1fd7bcc841e50ca","schema":"livegraph.boolean-propagation.v1","spec_hash":"45d1c80b869a49754620a00c317ec75a41ce34882a2cb933abbaff9eafb41750","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":6,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-a72ec682deee93b15b06","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v01\n- v07 --supports--> v06\n- v05 --refutes--> v03\n- v00 --supports--> v09\n- v00 --supports--> v04\n- v00 --refutes--> v07\n- v01 --refutes--> v00\n- v05 --supports--> v08\n- v06 --supports--> v05\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d7745a176a9f2711dbe98ecbb3d17e01ae95e22db705f2da4ddb7a99c6155874","schema":"livegraph.boolean-propagation.v1","spec_hash":"a72ec682deee93b15b06c3fad3596b925eb5bbb71d877aee47b89bc63fd1e14e","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":6,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-15054c67447774a9d8a5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v05\n- v06 --supports--> v02\n- v04 --supports--> v08\n- v03 --refutes--> v07\n- v09 --refutes--> v00\n- v03 --refutes--> v04\n- v06 --refutes--> v01\n- v04 --refutes--> v06\n- v00 --refutes--> v03\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1eb9a71d31d36fcdb1475bd9314aea1d3e87b8dc7f70e77d17ffde7071f932c3","schema":"livegraph.boolean-propagation.v1","spec_hash":"15054c67447774a9d8a5ec6e10bfbe671dda22e19f1cb32ab117ae0164fc8235","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-ab187789a3726691b0e3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v06\n- v02 --supports--> v00\n- v02 --refutes--> v05\n- v07 --refutes--> v09\n- v08 --supports--> v02\n- v06 --supports--> v04\n- v07 --refutes--> v03\n- v09 --refutes--> v08\n- v05 --supports--> v01\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2405f63c6475c0fc1fde88ff51b1219c07e811348fe8eaa0e0769cd83f64d32f","schema":"livegraph.boolean-propagation.v1","spec_hash":"ab187789a3726691b0e3c67747a95d1a89793248bd1eff142e20b8e647c5b7d6","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-9eeee908b60ae2f3f9d6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v07\n- v03 --supports--> v05\n- v06 --refutes--> v02\n- v00 --supports--> v03\n- v05 --supports--> v01\n- v07 --supports--> v09\n- v07 --refutes--> v08\n- v07 --supports--> v06\n- v04 --supports--> v00\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a297372801b1d10ba2cb94889ea8f0645fc50881b63a0ea4d393ad8c9ec3a90b","schema":"livegraph.boolean-propagation.v1","spec_hash":"9eeee908b60ae2f3f9d68b6189103c8b772eb084a563a7b88c28f244388cde68","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":6,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-bf1c3d91e8f32a725ec6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v05\n- v01 --refutes--> v07\n- v00 --refutes--> v04\n- v09 --supports--> v01\n- v07 --supports--> v02\n- v04 --supports--> v09\n- v03 --supports--> v00\n- v01 --refutes--> v06\n- v07 --refutes--> v08\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ecf690a5b4106f6fefea403654e6174a94e9a372c5fccd930de0c281ac7c2f72","schema":"livegraph.boolean-propagation.v1","spec_hash":"bf1c3d91e8f32a725ec6926ada45fd7c7c7c1266f0ae35e28a52f005de78e9bd","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-ffc286900d7d47c51ba4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v01\n- v04 --refutes--> v06\n- v09 --supports--> v07\n- v04 --refutes--> v05\n- v09 --supports--> v03\n- v00 --supports--> v04\n- v03 --refutes--> v08\n- v05 --refutes--> v09\n- v08 --refutes--> v02\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5c73abde728a2b432d7a8a07826e32c30eae761a52f4f6fc46122b8289880ea7","schema":"livegraph.boolean-propagation.v1","spec_hash":"ffc286900d7d47c51ba454570cd6537c4eed4b26e03165e902e208453a9ed098","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-91a64a7446b5f0ea7fb7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v01\n- v02 --supports--> v03\n- v03 --supports--> v07\n- v05 --refutes--> v00\n- v01 --supports--> v08\n- v07 --refutes--> v05\n- v01 --refutes--> v02\n- v05 --supports--> v04\n- v09 --supports--> v06\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"51c9183354556fccdf5289ec92aed44d147844fde25b4073024229ddde539bae","schema":"livegraph.boolean-propagation.v1","spec_hash":"91a64a7446b5f0ea7fb7ee57d321b2fc9630aefe0b70a3e1efcfb205794e86ed","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-2ae690b684c2c565db70","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v09\n- v09 --refutes--> v08\n- v01 --refutes--> v02\n- v06 --refutes--> v04\n- v00 --refutes--> v06\n- v09 --supports--> v01\n- v05 --refutes--> v00\n- v01 --refutes--> v05\n- v06 --supports--> v07\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bd2169d0dd119c581155b21b755498b4050b9c3d8c1678fe04f896395b828fe6","schema":"livegraph.boolean-propagation.v1","spec_hash":"2ae690b684c2c565db708cdbd809556e7efb7a8f64f8b2241fb932365b198500","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-f4f6dab6caecf3482f49","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v06\n- v05 --supports--> v03\n- v08 --supports--> v05\n- v06 --refutes--> v08\n- v01 --supports--> v00\n- v08 --supports--> v04\n- v03 --supports--> v07\n- v05 --refutes--> v09\n- v06 --supports--> v02\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"edb2c4f8e38d3cbf8de0b4c1d1e81dc7d5debf349ba3d46275bbd66ce2f7e8d5","schema":"livegraph.boolean-propagation.v1","spec_hash":"f4f6dab6caecf3482f494782377cbebca48de6be735c2de752f9f633d5a4329d","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-afa9be368183c52a1e10","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v04\n- v05 --supports--> v07\n- v01 --supports--> v03\n- v03 --refutes--> v08\n- v02 --refutes--> v05\n- v03 --supports--> v06\n- v03 --supports--> v00\n- v09 --supports--> v02\n- v07 --supports--> v01\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"615b86d3c02b2dafb4e019742237a4f7dc95a7b31166dca5b9d673ef0684ccc3","schema":"livegraph.boolean-propagation.v1","spec_hash":"afa9be368183c52a1e10b580980cf444f7b9861b70028b9398bfc809862828b5","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-655752b762e618bced9e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v01\n- v05 --supports--> v06\n- v08 --supports--> v00\n- v02 --supports--> v08\n- v07 --refutes--> v04\n- v07 --supports--> v02\n- v03 --refutes--> v05\n- v06 --refutes--> v07\n- v05 --supports--> v09\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"81f66f66d1aa4b825f040ae48c87c0772de11d4c5d1ea761dbe26025a77c9d04","schema":"livegraph.boolean-propagation.v1","spec_hash":"655752b762e618bced9ed936ba1e54c55efe9fbef7d23033699f1ed043f341cc","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-eda96dfce1ef502d8eb9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v04\n- v06 --supports--> v09\n- v04 --supports--> v05\n- v01 --supports--> v08\n- v01 --refutes--> v02\n- v03 --refutes--> v01\n- v08 --supports--> v00\n- v09 --refutes--> v03\n- v03 --refutes--> v07\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6929bcfd134e4c04a54b9eaced0ffd3c5fb1e823a27454f95e3d8ff56362f690","schema":"livegraph.boolean-propagation.v1","spec_hash":"eda96dfce1ef502d8eb910731c1d4e4724cef9c365cbf05ac0598c03900d47f4","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-1f7a4fcdaebb365cfd37","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v01\n- v05 --refutes--> v04\n- v07 --refutes--> v06\n- v09 --supports--> v08\n- v06 --refutes--> v09\n- v04 --refutes--> v00\n- v07 --supports--> v03\n- v09 --refutes--> v02\n- v01 --refutes--> v05\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"26b9952fbfb139eaff4fff5670aaed36904bc16f5ff41471c56719b9a5071d9e","schema":"livegraph.boolean-propagation.v1","spec_hash":"1f7a4fcdaebb365cfd37c1fb024c166a06b13ae1f6d8b078f6b709a90f3ae1de","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-558f0688a155dd65b32d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v05\n- v09 --refutes--> v06\n- v02 --supports--> v04\n- v00 --supports--> v03\n- v05 --supports--> v07\n- v00 --refutes--> v09\n- v09 --refutes--> v08\n- v04 --refutes--> v00\n- v09 --supports--> v01\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"261d34a24800933710b0894c4147a731edb43a28c89f20c051b2253cd86be519","schema":"livegraph.boolean-propagation.v1","spec_hash":"558f0688a155dd65b32dbf3eb24e591aa1d8906d4932f2088465ed1768f722eb","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-d0cdfd4a589a2b695432","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v04\n- v00 --supports--> v03\n- v04 --supports--> v02\n- v04 --supports--> v09\n- v04 --refutes--> v08\n- v05 --refutes--> v01\n- v00 --supports--> v07\n- v03 --supports--> v05\n- v01 --supports--> v06\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3e052566c3fdf94c12d655a5ab789d36221fae64579a61b449e6c097228efbe4","schema":"livegraph.boolean-propagation.v1","spec_hash":"d0cdfd4a589a2b69543283da9fa7007dc9ff350671cbae442ecb566d857be30e","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":6,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-035a9cce0ddd8e2e838e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v07\n- v05 --refutes--> v03\n- v08 --refutes--> v02\n- v02 --refutes--> v00\n- v04 --supports--> v05\n- v01 --supports--> v09\n- v02 --refutes--> v06\n- v09 --supports--> v08\n- v05 --refutes--> v01\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1f7c807af3ca7ff069c6089963d2b497f8b2d4f40ea3fa9f7ba07940560747f5","schema":"livegraph.boolean-propagation.v1","spec_hash":"035a9cce0ddd8e2e838e06a878659895248b3569abc64e196df20450d44db680","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-9e7b33c529c76b3eae69","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v04\n- v08 --refutes--> v06\n- v05 --refutes--> v01\n- v01 --refutes--> v09\n- v02 --refutes--> v00\n- v08 --refutes--> v07\n- v00 --supports--> v05\n- v00 --supports--> v03\n- v09 --refutes--> v08\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9e5941db51609ed5dfd9a3187eb5a17aeef432aad1ca9e9cf66c54a9ca17c8bf","schema":"livegraph.boolean-propagation.v1","spec_hash":"9e7b33c529c76b3eae69c8d2f326b38b5a59f9946a475b858220785e1f985262","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-8c9ad70ef38145a8c8e8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v04\n- v07 --supports--> v00\n- v09 --supports--> v03\n- v00 --refutes--> v06\n- v09 --supports--> v08\n- v06 --refutes--> v01\n- v09 --refutes--> v07\n- v04 --refutes--> v09\n- v09 --refutes--> v05\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9c867c5ba4e4850a955c1d741f9c6765600c4001ead994bdd10b05f979b7bb9d","schema":"livegraph.boolean-propagation.v1","spec_hash":"8c9ad70ef38145a8c8e8b618997ca3b209a3616184bb54b967492f5e2637f7fd","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-1f5d3f00e76214d27386","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v03\n- v03 --refutes--> v06\n- v06 --refutes--> v04\n- v09 --supports--> v02\n- v06 --supports--> v09\n- v00 --refutes--> v07\n- v02 --supports--> v00\n- v01 --supports--> v05\n- v00 --supports--> v08\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"18f72aeb900976025e7ce2a37d834a945884c37b772c801579c254cb4cbf4030","schema":"livegraph.boolean-propagation.v1","spec_hash":"1f5d3f00e76214d27386e393640c2bf5a47cbc1abcde49850920567be5cdca83","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-8e4f5c15433a53e6e8a4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v04\n- v07 --refutes--> v06\n- v09 --supports--> v08\n- v05 --supports--> v03\n- v04 --refutes--> v07\n- v03 --supports--> v00\n- v03 --supports--> v02\n- v07 --refutes--> v01\n- v03 --supports--> v09\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e0523185e9373236417f02c2ab364568ea8fd76bfa491ba48a1d4c554d8a5948","schema":"livegraph.boolean-propagation.v1","spec_hash":"8e4f5c15433a53e6e8a49d0f5d722e29ca3967bdd5728e07d4e6ddafab0cb7f0","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-49d71133bfe584ddaf6b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v06\n- v04 --refutes--> v08\n- v05 --supports--> v09\n- v02 --refutes--> v04\n- v09 --supports--> v01\n- v04 --supports--> v03\n- v01 --refutes--> v02\n- v08 --refutes--> v00\n- v02 --supports--> v07\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"87fab0a29e95852d969ef73ed2282c281ddb7c35dfa829e0ef012a6bfa34b6fc","schema":"livegraph.boolean-propagation.v1","spec_hash":"49d71133bfe584ddaf6b06e587abe7945c634ef8f9e02cb1ef35cad5dbc6fcf8","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-e17d0a8ac2b4544b3616","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v03\n- v07 --refutes--> v06\n- v09 --supports--> v05\n- v04 --supports--> v01\n- v04 --refutes--> v07\n- v08 --refutes--> v09\n- v02 --refutes--> v00\n- v06 --refutes--> v02\n- v02 --supports--> v08\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"be75614107f727830fd5ac733e57861b5079ec0d353a7909f85cfe41e817b7ee","schema":"livegraph.boolean-propagation.v1","spec_hash":"e17d0a8ac2b4544b361659638b2a3434347d03b0e213fc5b7acc9ff3262a8e71","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-f3a39725bb1cc89fb282","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v04\n- v06 --supports--> v01\n- v05 --supports--> v08\n- v06 --supports--> v09\n- v05 --refutes--> v06\n- v00 --supports--> v03\n- v09 --supports--> v00\n- v03 --refutes--> v07\n- v02 --refutes--> v05\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0b0dce2ff1437502ed2d7f0dc536955985064af71734e2ce164e6c342d9c49f5","schema":"livegraph.boolean-propagation.v1","spec_hash":"f3a39725bb1cc89fb28248df2079719a42fee149cc414939e623fa7a2f9ff61e","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-8f516fce6b82f584d2f4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v00\n- v07 --supports--> v08\n- v09 --supports--> v02\n- v07 --refutes--> v04\n- v09 --supports--> v03\n- v06 --refutes--> v09\n- v04 --supports--> v05\n- v05 --refutes--> v01\n- v01 --supports--> v06\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8e7de9936b374814604792fb3c59f60d458ab8efb66201b358fd5c63fcde2c88","schema":"livegraph.boolean-propagation.v1","spec_hash":"8f516fce6b82f584d2f4e171ab53e06aec8b395f07bf7bf04b83bae50d6fb59f","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-23c9320bb751857266b1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v00\n- v09 --refutes--> v06\n- v04 --refutes--> v09\n- v08 --refutes--> v05\n- v03 --supports--> v01\n- v07 --supports--> v08\n- v04 --refutes--> v02\n- v08 --supports--> v03\n- v01 --refutes--> v04\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"18a6ef44dfa3c309ea8f09509495ed6fa4a4d0c4b1040429cbf6fe0a6a85a706","schema":"livegraph.boolean-propagation.v1","spec_hash":"23c9320bb751857266b1d04709d66a75c1bb9aa70068218fbce7580c368af84a","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-75391acafd8e4e614333","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v00\n- v05 --refutes--> v07\n- v07 --refutes--> v08\n- v02 --supports--> v03\n- v01 --refutes--> v04\n- v07 --supports--> v09\n- v03 --refutes--> v06\n- v07 --supports--> v02\n- v00 --supports--> v01\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0653bc54e63e330045ea9c695a41bee7f7ec9a85abe05c8a96aa97b72c9c7257","schema":"livegraph.boolean-propagation.v1","spec_hash":"75391acafd8e4e614333ba0517c1acb89561ea90f6800c16252280b45462e913","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":6,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-fe04578613d5ed837dd4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v00\n- v07 --supports--> v02\n- v09 --refutes--> v01\n- v09 --refutes--> v08\n- v03 --supports--> v06\n- v05 --supports--> v07\n- v01 --supports--> v03\n- v02 --refutes--> v09\n- v03 --refutes--> v04\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"31e578a2a49fcc956c084716f43b48b0b1e13bacd1025e35dfba298bede53509","schema":"livegraph.boolean-propagation.v1","spec_hash":"fe04578613d5ed837dd4acadbb27f1cee1759ccd1cc5f2939717abb29b999a27","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-d1cb0f9f1c88b23f29ca","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v07\n- v01 --supports--> v03\n- v01 --supports--> v06\n- v04 --refutes--> v09\n- v02 --refutes--> v05\n- v00 --refutes--> v04\n- v08 --supports--> v02\n- v09 --supports--> v08\n- v02 --refutes--> v01\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ba7ff79470f1300251b3f0c3d9ee1a8c6f9716a94b942540837f8950153a9bb8","schema":"livegraph.boolean-propagation.v1","spec_hash":"d1cb0f9f1c88b23f29caa920647a6eb22c614cd4213ee8c2a4859ce019ade691","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-1a45b7b9ac3ab251840d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v05\n- v07 --refutes--> v01\n- v04 --supports--> v07\n- v06 --refutes--> v04\n- v05 --refutes--> v00\n- v06 --supports--> v09\n- v01 --supports--> v03\n- v06 --refutes--> v02\n- v01 --supports--> v08\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6e1e93699c177a8717191c3abd1e4fb8986ad67fd690db796531b041dcb9cc73","schema":"livegraph.boolean-propagation.v1","spec_hash":"1a45b7b9ac3ab251840d6a02f117f79ecf0fd18a79dc3b5a6b845052b554f64e","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":6,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-91ad48829a426b7f6d94","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v03\n- v06 --refutes--> v09\n- v02 --supports--> v05\n- v08 --refutes--> v06\n- v01 --refutes--> v02\n- v00 --refutes--> v01\n- v09 --refutes--> v00\n- v01 --refutes--> v04\n- v00 --supports--> v07\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"eca85e590cf9eb15a5ecced7ff79f9b937c8475ed701e510214d0291830a5ee6","schema":"livegraph.boolean-propagation.v1","spec_hash":"91ad48829a426b7f6d94cf2e53636874d0bbbdc516e309d406b0c677af1e8e48","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-ea4dee6aef0e45418fc3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v03\n- v00 --supports--> v04\n- v09 --refutes--> v07\n- v09 --refutes--> v06\n- v02 --supports--> v00\n- v03 --refutes--> v05\n- v03 --supports--> v01\n- v02 --refutes--> v08\n- v05 --refutes--> v02\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"16d98fb32858de3d1565b968f805deba37a8f862795aad5a015201fc1f0b041a","schema":"livegraph.boolean-propagation.v1","spec_hash":"ea4dee6aef0e45418fc36a2da273c2a7e35347472347225eb22781e4077d7d95","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-21fb7db0fd1937e8f774","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v05\n- v00 --refutes--> v02\n- v06 --refutes--> v08\n- v07 --supports--> v06\n- v02 --refutes--> v03\n- v03 --supports--> v07\n- v08 --supports--> v04\n- v08 --refutes--> v09\n- v02 --supports--> v01\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d717d35eb8592db91921171ae8d37c59b5268443e21910d62540f00b4949baef","schema":"livegraph.boolean-propagation.v1","spec_hash":"21fb7db0fd1937e8f7749bb3696f769f263ee320a24cc0b7b52cea7cf0dab0dd","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-b687a2d0a576ecf1d5e1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v07\n- v00 --supports--> v05\n- v08 --refutes--> v04\n- v04 --refutes--> v00\n- v08 --refutes--> v02\n- v03 --supports--> v06\n- v06 --supports--> v09\n- v00 --supports--> v01\n- v06 --refutes--> v08\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"02d3f6117b00ec51687c376477b5f9ae5d16d831839d68f99857d7c981ae792f","schema":"livegraph.boolean-propagation.v1","spec_hash":"b687a2d0a576ecf1d5e1f4a0a53accc83f30fc5d0c9e749e989d7ed746238287","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-c6cdfb7020f2d53fc943","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v06\n- v06 --supports--> v00\n- v01 --refutes--> v07\n- v01 --refutes--> v02\n- v05 --supports--> v03\n- v07 --refutes--> v05\n- v06 --supports--> v04\n- v09 --refutes--> v08\n- v09 --refutes--> v01\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"18715cdd6c9a8bc8bd1b7025bd90122fc00136e66c290d9cabf63d2590ca5c52","schema":"livegraph.boolean-propagation.v1","spec_hash":"c6cdfb7020f2d53fc94360c03160c57e047fcf8a316b6f8704bbec49259904c9","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-56a86904bcd291352ea9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v08\n- v01 --refutes--> v00\n- v01 --supports--> v07\n- v04 --supports--> v05\n- v09 --refutes--> v06\n- v05 --refutes--> v09\n- v00 --supports--> v03\n- v00 --refutes--> v04\n- v09 --supports--> v02\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6e12256eaf7ecb821681fcc4b5bf3db27d0fa3416e2366d0b2b7885dd1710f5c","schema":"livegraph.boolean-propagation.v1","spec_hash":"56a86904bcd291352ea9b48151a9269e772063ce5d6459048e7d6d3a7c0d0080","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-f54e7115edc194cb1779","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v03\n- v05 --supports--> v02\n- v08 --refutes--> v01\n- v00 --supports--> v06\n- v08 --refutes--> v07\n- v03 --refutes--> v00\n- v01 --refutes--> v05\n- v00 --refutes--> v08\n- v08 --supports--> v04\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e2a557b48786ead3bcb2630afc5122a468b3fce26b6da0ec25b1aff48ebe26f3","schema":"livegraph.boolean-propagation.v1","spec_hash":"f54e7115edc194cb1779cbb54f9371f38f99d51f3ea3ebdec77edf7c0647149e","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-e26c3ac3c665d21dc84f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v00\n- v02 --refutes--> v01\n- v00 --refutes--> v09\n- v01 --supports--> v08\n- v01 --refutes--> v05\n- v00 --supports--> v04\n- v09 --refutes--> v06\n- v01 --refutes--> v03\n- v06 --refutes--> v07\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"63d01bfb5421fc01d1cc578c29a92a01f42915412cd489ba381e8bd94abad0d5","schema":"livegraph.boolean-propagation.v1","spec_hash":"e26c3ac3c665d21dc84f8f921dfd87b2a0cb49d7b7c2a15001602b32f9602933","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-12874a2daeba5a24fc1c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v08\n- v03 --supports--> v04\n- v04 --refutes--> v00\n- v02 --supports--> v05\n- v00 --refutes--> v09\n- v02 --supports--> v07\n- v08 --refutes--> v01\n- v01 --refutes--> v02\n- v02 --supports--> v06\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e308051ce0c9c1e6c79f6f0e2ae7f414788a5087da8aff9ef1f49fcba91cf44b","schema":"livegraph.boolean-propagation.v1","spec_hash":"12874a2daeba5a24fc1cdd147aaaaa120c8e224a6a303a5ec7e82d3f3bf2fb40","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-0b614bb9541c7f3424a1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v07\n- v00 --refutes--> v02\n- v05 --supports--> v03\n- v03 --supports--> v06\n- v01 --supports--> v09\n- v09 --supports--> v08\n- v03 --supports--> v01\n- v08 --supports--> v00\n- v05 --supports--> v04\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9c93ab71edb8c364751e7b9ba70fea6f29aac567342a76903fe1cc5cbab5e377","schema":"livegraph.boolean-propagation.v1","spec_hash":"0b614bb9541c7f3424a1c09a59a718c7b6b126370d6199deb24c42b5fdd555a7","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-b1f5abe76d209e2a3e67","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v06\n- v08 --refutes--> v09\n- v06 --supports--> v03\n- v05 --refutes--> v04\n- v03 --refutes--> v08\n- v07 --supports--> v02\n- v08 --refutes--> v00\n- v02 --refutes--> v01\n- v02 --supports--> v05\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d4b12fa3ccffe4e693b79703209d916cfc1af938c7cf6388d92cce6d850b3a5e","schema":"livegraph.boolean-propagation.v1","spec_hash":"b1f5abe76d209e2a3e6709c52c6efc5c03c3360486a951a7d5683546090c7b53","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-4e63e354a4bca014704c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v00\n- v02 --refutes--> v08\n- v07 --refutes--> v09\n- v01 --refutes--> v04\n- v00 --refutes--> v07\n- v06 --refutes--> v02\n- v07 --refutes--> v03\n- v04 --refutes--> v05\n- v07 --refutes--> v01\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a58425eca1d1311e514a466344135d6921eb2b5be769b592506b6673f17a6388","schema":"livegraph.boolean-propagation.v1","spec_hash":"4e63e354a4bca014704ce2aacb646c784a6284161a96e2e8f7ad55b67c74d36b","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":6,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-932849ea2153280958da","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v03\n- v04 --refutes--> v08\n- v05 --refutes--> v07\n- v00 --supports--> v05\n- v03 --refutes--> v09\n- v03 --supports--> v02\n- v06 --supports--> v00\n- v05 --refutes--> v01\n- v03 --refutes--> v04\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e51b792250aea1798a9dd79f46883c03dc182e4b6970b7a034f7f9d7f24174c3","schema":"livegraph.boolean-propagation.v1","spec_hash":"932849ea2153280958dae6c0fd3dc4dbc6ac95afcbb8f902ea156ebd8893f510","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":6,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-42540d433e8c9b5bed07","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v06\n- v07 --supports--> v08\n- v06 --supports--> v03\n- v05 --supports--> v00\n- v01 --supports--> v07\n- v06 --supports--> v09\n- v05 --supports--> v01\n- v04 --supports--> v05\n- v06 --supports--> v02\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"415889ac9efbefd163589af6ee8d8da23442f2928b936570917c36e7103e05be","schema":"livegraph.boolean-propagation.v1","spec_hash":"42540d433e8c9b5bed070e352cc8073a47d903e7c0a05a9c5a446232c28c97c1","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-f13f1ed828902dbd3a13","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v01\n- v09 --supports--> v06\n- v05 --supports--> v08\n- v04 --supports--> v00\n- v09 --refutes--> v02\n- v05 --refutes--> v07\n- v07 --supports--> v09\n- v04 --refutes--> v03\n- v06 --refutes--> v04\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dfeeccefab5dc84505247951e7eb663976d0ef37d086a23cd621f87e2c593882","schema":"livegraph.boolean-propagation.v1","spec_hash":"f13f1ed828902dbd3a134566bbec9285e9b080112ffed406ee16770686640652","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-0995cf7763d3181040f9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v02\n- v02 --supports--> v08\n- v06 --supports--> v01\n- v09 --refutes--> v03\n- v08 --refutes--> v06\n- v06 --refutes--> v05\n- v09 --supports--> v04\n- v05 --supports--> v07\n- v03 --supports--> v00\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2a4294a7ad5ef6c2017358470c990a85d0036b13b0b63a6414a94c4f35527ff5","schema":"livegraph.boolean-propagation.v1","spec_hash":"0995cf7763d3181040f961508826a62a48532d1c5ba7753f4132a76138cf6aec","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-77f05b99a2bd05909480","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v00\n- v06 --supports--> v05\n- v06 --supports--> v08\n- v01 --supports--> v06\n- v09 --supports--> v02\n- v02 --refutes--> v01\n- v04 --supports--> v03\n- v02 --supports--> v07\n- v00 --supports--> v09\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8577dd97a61c8ba1f2340f2e75bb3ad9755005035f090ae4c7b2a266f1a8849d","schema":"livegraph.boolean-propagation.v1","spec_hash":"77f05b99a2bd05909480b3e701d4f5df0e5b364de986969df75a9085b6edb735","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-6341ef557b5482ea1abe","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v06\n- v07 --refutes--> v09\n- v02 --refutes--> v05\n- v09 --refutes--> v08\n- v03 --supports--> v01\n- v05 --refutes--> v00\n- v09 --refutes--> v02\n- v05 --supports--> v04\n- v03 --refutes--> v07\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"749413e95f6bd1773810a41638ff56635cf71182286ba360adc913c6c1a9fcd9","schema":"livegraph.boolean-propagation.v1","spec_hash":"6341ef557b5482ea1abe431bc32acf5a64cc375107a4bb57c094f8716ac6ff9f","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-05f54622dddd3ad914c8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v02\n- v07 --refutes--> v09\n- v00 --supports--> v03\n- v07 --refutes--> v06\n- v01 --refutes--> v00\n- v09 --supports--> v08\n- v08 --refutes--> v01\n- v00 --refutes--> v04\n- v07 --supports--> v05\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9a96bb1bf0d2a63f1898b8730907909595d737081109d902d77a54e5f65604e8","schema":"livegraph.boolean-propagation.v1","spec_hash":"05f54622dddd3ad914c8d3f7355ced778e91a04f2bc935bb60d0e9e9ec0877ba","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-92644e4c006a4508e913","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v03\n- v01 --refutes--> v02\n- v04 --supports--> v00\n- v02 --supports--> v06\n- v09 --supports--> v04\n- v03 --supports--> v05\n- v02 --supports--> v08\n- v02 --supports--> v09\n- v03 --supports--> v07\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"27538412f9b2362e4952b68528d611e3652e65485d169e71b7df9acd0cfa7123","schema":"livegraph.boolean-propagation.v1","spec_hash":"92644e4c006a4508e91326507cdf0ba88e08f3b6d1c56bfeeed5b9658c396716","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-b61adf4954da111cd7c9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v02\n- v05 --supports--> v06\n- v08 --supports--> v03\n- v03 --refutes--> v01\n- v00 --supports--> v05\n- v02 --supports--> v00\n- v06 --supports--> v07\n- v08 --refutes--> v09\n- v06 --refutes--> v04\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e7ed905af4829b9dba796d5c4824d0cf6612cb0b889b0e099dca305e429635e9","schema":"livegraph.boolean-propagation.v1","spec_hash":"b61adf4954da111cd7c9e2db1392a9a0511a2a3071204cc6cdca9300c9d46ee7","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-77d871042ff136ffdbc9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v05\n- v08 --refutes--> v03\n- v08 --refutes--> v07\n- v06 --refutes--> v08\n- v03 --refutes--> v00\n- v09 --supports--> v06\n- v01 --supports--> v02\n- v02 --supports--> v04\n- v03 --supports--> v01\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0969f04bc93c2df502d02c4ffd87bcac809c804333d4688481893937e7370376","schema":"livegraph.boolean-propagation.v1","spec_hash":"77d871042ff136ffdbc9b69714e23371680197c9d05119aca9abd3d689e3e5d9","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-e0ef00e5ecbfed8632bc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v05\n- v06 --refutes--> v03\n- v00 --supports--> v04\n- v05 --supports--> v09\n- v01 --refutes--> v02\n- v09 --supports--> v06\n- v04 --refutes--> v08\n- v00 --supports--> v07\n- v06 --refutes--> v00\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d5b59c2664d65dae890da6a29eddda676d18765dae119659409cc8d8323c7477","schema":"livegraph.boolean-propagation.v1","spec_hash":"e0ef00e5ecbfed8632bc0cd039879b8594e7ce10aa1068f03d771f34c0ec3408","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-1a0f781106e0b9c674cd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v08\n- v06 --supports--> v00\n- v05 --supports--> v03\n- v03 --refutes--> v02\n- v03 --supports--> v07\n- v08 --refutes--> v04\n- v07 --refutes--> v01\n- v03 --supports--> v09\n- v04 --supports--> v06\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"27489e9cb8f7e9fdea372d5b938b193a31f584e9cd888162ada31cc2d17d7638","schema":"livegraph.boolean-propagation.v1","spec_hash":"1a0f781106e0b9c674cd550b43d30ec18edd3477ccfdda0758c2078fa92a5fe6","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-11b3ca5cf1a9797a8d37","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v05\n- v07 --refutes--> v08\n- v08 --supports--> v04\n- v06 --refutes--> v02\n- v06 --refutes--> v09\n- v00 --supports--> v07\n- v03 --refutes--> v01\n- v03 --refutes--> v00\n- v04 --supports--> v06\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0611e9348004b6c6075886d22d355ee778d7270bd934310c4221850414704fd3","schema":"livegraph.boolean-propagation.v1","spec_hash":"11b3ca5cf1a9797a8d3725b9413e227b2777c03646a69c9ad465162d7a9bf336","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-e5d19537893bd85f4aa1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v02\n- v02 --refutes--> v03\n- v04 --supports--> v06\n- v09 --refutes--> v00\n- v00 --supports--> v05\n- v08 --supports--> v07\n- v03 --refutes--> v09\n- v04 --supports--> v01\n- v00 --refutes--> v04\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6245d106c9ed0b816191be830ad73ee089283e66db1f473e03ed711f87fde9f4","schema":"livegraph.boolean-propagation.v1","spec_hash":"e5d19537893bd85f4aa156413ee6b0ff09ffa2839e68edfbb456072e9cfba9ba","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-1ee22fb3c3453e6e0ccf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v04\n- v00 --supports--> v09\n- v03 --supports--> v02\n- v05 --supports--> v03\n- v08 --refutes--> v05\n- v04 --supports--> v01\n- v01 --refutes--> v00\n- v00 --supports--> v07\n- v08 --refutes--> v06\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1bb5babcaaef430b8fd289e3b0fbfc1da9b30d575765fe30a0ced5f021e464bd","schema":"livegraph.boolean-propagation.v1","spec_hash":"1ee22fb3c3453e6e0ccf1d8d4312a226cc05a218d5fa3d86c68025005fe90950","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-1c08437fa4a3664df70c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v01\n- v02 --refutes--> v08\n- v02 --refutes--> v04\n- v01 --supports--> v09\n- v02 --refutes--> v07\n- v04 --refutes--> v05\n- v09 --supports--> v00\n- v00 --refutes--> v06\n- v06 --refutes--> v03\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a03f5442b26deaa6ecbe25787e6ee71285e1bf1e0e7f2e77a0c41cee2553520b","schema":"livegraph.boolean-propagation.v1","spec_hash":"1c08437fa4a3664df70c30f4e9a4e22c7d875bef7119b273217de13985a67d49","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-e5a404bc41c6df9656f5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v05\n- v08 --refutes--> v06\n- v03 --supports--> v01\n- v00 --refutes--> v03\n- v08 --refutes--> v07\n- v08 --refutes--> v09\n- v02 --refutes--> v08\n- v01 --supports--> v02\n- v09 --refutes--> v04\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0bd1db4b39dd1819e0c01a7891f85607217dc5ad14569d95ed0bff313c3635f3","schema":"livegraph.boolean-propagation.v1","spec_hash":"e5a404bc41c6df9656f53aff15d9259a915d2b2df1e368283670df7c3a9632a9","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-640ed080c96f89b89a20","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v00\n- v05 --refutes--> v08\n- v01 --supports--> v03\n- v04 --refutes--> v02\n- v03 --supports--> v04\n- v06 --supports--> v05\n- v07 --supports--> v06\n- v05 --supports--> v01\n- v04 --supports--> v09\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0ef7d4c9b729e7491d05835bb28f1920b16039a7cbc0d665993aab1b83bac47d","schema":"livegraph.boolean-propagation.v1","spec_hash":"640ed080c96f89b89a20435bbac11ced7b825eadf243635d40e4eb1c7131fc4c","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":6,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-c5864564216dd96c049b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v08\n- v08 --supports--> v02\n- v07 --refutes--> v03\n- v05 --refutes--> v01\n- v01 --refutes--> v07\n- v02 --refutes--> v00\n- v00 --supports--> v09\n- v00 --supports--> v04\n- v07 --refutes--> v06\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"597389fd552dbe8b4a05473f6c41a3d5d28499100c3ca8442ef3c1941e4f08c7","schema":"livegraph.boolean-propagation.v1","spec_hash":"c5864564216dd96c049bcf07d85f9f9e49f2dd94a2bd86b67494e054a7a7c9a7","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-d828e0f741c380130240","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v09\n- v01 --supports--> v00\n- v03 --supports--> v04\n- v00 --refutes--> v03\n- v03 --supports--> v05\n- v02 --supports--> v01\n- v07 --supports--> v08\n- v03 --refutes--> v07\n- v06 --supports--> v02\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f1cce944a5884d24b09f6789767399176e149b9babb4d38f513802d691b91a20","schema":"livegraph.boolean-propagation.v1","spec_hash":"d828e0f741c3801302407351badf2ff4b8d947a5ad58f28e74900969796e9568","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":6,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-0a6d5af26cc4358d38a0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v08\n- v06 --supports--> v03\n- v09 --supports--> v01\n- v00 --supports--> v04\n- v00 --refutes--> v05\n- v07 --refutes--> v00\n- v02 --refutes--> v07\n- v06 --supports--> v09\n- v01 --refutes--> v02\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bbb84fd6ba0512cad085560c1a8ab70c87d13308c3310e8b59fdee095cf9e887","schema":"livegraph.boolean-propagation.v1","spec_hash":"0a6d5af26cc4358d38a0e20bfbb0f0c541d8b10fae9aa9b0691e297bcf0aac50","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-8d9a001339e70c54a2d3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v04\n- v05 --supports--> v09\n- v00 --supports--> v07\n- v08 --refutes--> v03\n- v06 --refutes--> v08\n- v09 --supports--> v02\n- v09 --supports--> v06\n- v05 --refutes--> v01\n- v07 --supports--> v05\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e5770bef760b18a9bbb923e23d2e42c98f3e4e6fcc6c777b3a160736cb2f7224","schema":"livegraph.boolean-propagation.v1","spec_hash":"8d9a001339e70c54a2d3649195a913364addcaa05444367620f02cbabb652365","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-50f9381a8bafc1bc78ad","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v01\n- v01 --supports--> v03\n- v05 --supports--> v02\n- v02 --supports--> v08\n- v04 --supports--> v00\n- v04 --refutes--> v06\n- v03 --refutes--> v05\n- v07 --refutes--> v04\n- v04 --supports--> v09\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a4760a0f27e04c92fabcfa0430d1d717d80d97ac78661f9e9b9f269e8d864e28","schema":"livegraph.boolean-propagation.v1","spec_hash":"50f9381a8bafc1bc78ada28edecb40257dcddafe6f40450285e2cd44ad616d0c","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":6,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-ed9b8cb11837fe7afef1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v07\n- v06 --refutes--> v02\n- v03 --refutes--> v08\n- v07 --refutes--> v04\n- v05 --refutes--> v01\n- v08 --refutes--> v05\n- v07 --supports--> v03\n- v03 --refutes--> v09\n- v08 --supports--> v00\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ebc0cd21646a577165ae5fb3ea05e8461f2f184184d88844706f226f65607fdc","schema":"livegraph.boolean-propagation.v1","spec_hash":"ed9b8cb11837fe7afef1e0d73befee5b60df9366a3e554c2a8b06e695b7c349e","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-6c4d8a32b55c4663742d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v02\n- v02 --refutes--> v09\n- v04 --refutes--> v06\n- v04 --supports--> v07\n- v03 --supports--> v01\n- v03 --refutes--> v00\n- v05 --refutes--> v03\n- v00 --refutes--> v08\n- v04 --refutes--> v05\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9d3b070f26c036e7e967a942835177e8db3f63a9395fcf97cd768557168b7021","schema":"livegraph.boolean-propagation.v1","spec_hash":"6c4d8a32b55c4663742de5448d7b0ec95aab5ca04bceb53e1a62fdaff7f0d9d9","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-de71bf245853ce07486f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v03\n- v00 --supports--> v01\n- v05 --supports--> v02\n- v06 --refutes--> v08\n- v06 --refutes--> v05\n- v07 --supports--> v04\n- v07 --supports--> v06\n- v09 --refutes--> v00\n- v02 --supports--> v09\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"458281acaeb6df8094b5d1aa602a72b99d5740a05762df2286abb88b6c952d5e","schema":"livegraph.boolean-propagation.v1","spec_hash":"de71bf245853ce07486f6fe06214da8c1818092de8cbf4df3f9bf2d611136664","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-510d39d8403570e6d31f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v07\n- v07 --refutes--> v01\n- v01 --refutes--> v00\n- v06 --refutes--> v03\n- v08 --refutes--> v09\n- v06 --supports--> v05\n- v04 --refutes--> v06\n- v06 --supports--> v08\n- v07 --supports--> v02\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"310fb2be1dc72d23a396d1caeb5805ca49bc3da4e4558d197d1d47d7ab316f03","schema":"livegraph.boolean-propagation.v1","spec_hash":"510d39d8403570e6d31f5c8ffcad6d38a5b51820f3446e2b896657826ad10ddb","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-49c3750bb5af116aa6f3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v05\n- v07 --supports--> v09\n- v03 --refutes--> v08\n- v08 --refutes--> v07\n- v02 --refutes--> v01\n- v00 --refutes--> v02\n- v06 --refutes--> v00\n- v02 --refutes--> v04\n- v02 --refutes--> v03\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cb3b8d1852351d35751d0c179141446ebc45d7410eae0070e5a70173241b7425","schema":"livegraph.boolean-propagation.v1","spec_hash":"49c3750bb5af116aa6f3ab5967d31584351281d3b50dea3b1d433d972183f76b","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-249246effe26557e03a6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v07\n- v06 --supports--> v00\n- v07 --refutes--> v01\n- v00 --supports--> v03\n- v06 --refutes--> v05\n- v06 --refutes--> v09\n- v01 --supports--> v06\n- v00 --refutes--> v08\n- v04 --supports--> v02\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6cda4d849bcb92a125fca8cbf5d3d9fce23887a6e09bd245b4e56a21fd32bd52","schema":"livegraph.boolean-propagation.v1","spec_hash":"249246effe26557e03a6ff1fe52c04745922247b2bc9f05377cc4ba059df3042","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":6,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-0cc9673c5fbb630025a0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v05\n- v09 --refutes--> v04\n- v06 --refutes--> v01\n- v04 --supports--> v08\n- v03 --supports--> v00\n- v00 --refutes--> v07\n- v00 --supports--> v06\n- v02 --refutes--> v03\n- v04 --refutes--> v02\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f00720343bb09efff7ce8fa262d538699f654216a65dd0b1400633c419110d5c","schema":"livegraph.boolean-propagation.v1","spec_hash":"0cc9673c5fbb630025a08b0924c99241643eff0c1aa9ddd366c9114b4f8904a0","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-9e5e3cc90840b927923f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v04\n- v01 --supports--> v07\n- v08 --supports--> v09\n- v04 --refutes--> v01\n- v06 --supports--> v05\n- v00 --supports--> v06\n- v09 --supports--> v00\n- v04 --supports--> v02\n- v09 --supports--> v03\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fbd58e00908acf002096611a7eeb74a1de5da482320ea711bb0ff7821e60f248","schema":"livegraph.boolean-propagation.v1","spec_hash":"9e5e3cc90840b927923f9e836301a895b8a92feae5bdd91d176001d0cdf7f5dc","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-d7360aed840f797b7095","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v01\n- v07 --supports--> v00\n- v03 --refutes--> v07\n- v02 --refutes--> v09\n- v07 --refutes--> v05\n- v09 --supports--> v08\n- v08 --supports--> v03\n- v02 --refutes--> v04\n- v02 --supports--> v06\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5b4ae428add6d2866bcafc3b1371604ff91e62175d2d0f121a4d2e6069e2985f","schema":"livegraph.boolean-propagation.v1","spec_hash":"d7360aed840f797b709582450c69d9b3731cdf1a0a81687392f1b46dd24d265d","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-a43b75fd55d0b17f58ed","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v06\n- v01 --refutes--> v07\n- v03 --supports--> v01\n- v02 --supports--> v03\n- v06 --refutes--> v09\n- v02 --refutes--> v05\n- v01 --supports--> v08\n- v04 --refutes--> v00\n- v04 --supports--> v02\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"708c9c45ba0d7650691c4592ef98b02438ccb317e0d8cd1a3ae393e26dc20b37","schema":"livegraph.boolean-propagation.v1","spec_hash":"a43b75fd55d0b17f58edbef7ed2c28c2ddca6ac686efdc1f46836e06b47cb2ab","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":6,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-456977147b2873f58b0c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v08\n- v04 --refutes--> v05\n- v08 --supports--> v04\n- v02 --refutes--> v09\n- v03 --refutes--> v07\n- v07 --supports--> v01\n- v04 --supports--> v03\n- v04 --refutes--> v00\n- v09 --supports--> v06\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fcdd39b5c8a80b560ae3b1620f973b660168e9d0ce015c51ead9113287c58625","schema":"livegraph.boolean-propagation.v1","spec_hash":"456977147b2873f58b0c2cc26dc8becf23797b8742c34e3d7e219e0915f044d7","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-40f927905888ebfcca21","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v02\n- v06 --supports--> v08\n- v08 --refutes--> v00\n- v03 --refutes--> v01\n- v04 --supports--> v05\n- v06 --supports--> v09\n- v01 --supports--> v04\n- v08 --supports--> v07\n- v05 --supports--> v06\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bb83dfd04b69eeefdfb18b4ea095e995e9f2c35d364ee4d6d2d0723c542e11d2","schema":"livegraph.boolean-propagation.v1","spec_hash":"40f927905888ebfcca2142ccff4e5addbb6092d3bcf19f4c23bd92cb316a61bf","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-af9b53e8b962f49d106b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v00\n- v08 --supports--> v09\n- v01 --supports--> v04\n- v05 --refutes--> v06\n- v01 --refutes--> v07\n- v05 --supports--> v08\n- v09 --refutes--> v01\n- v00 --refutes--> v03\n- v02 --refutes--> v05\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d4f23491541f39143f8da0013e5d6e7c13d97c2f7fb986a3d58fef0a07cdf894","schema":"livegraph.boolean-propagation.v1","spec_hash":"af9b53e8b962f49d106bba7558740f6719a7c9b4043af9853d29d49cd7177f4d","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-7e91f012191b11b47fdd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v00\n- v09 --supports--> v06\n- v08 --supports--> v09\n- v06 --supports--> v01\n- v04 --refutes--> v05\n- v08 --supports--> v02\n- v03 --refutes--> v04\n- v08 --refutes--> v07\n- v05 --refutes--> v08\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"da24a46fa8f154ef075b73fc07f30b2dca292412aeaf166ff088ca2e43255760","schema":"livegraph.boolean-propagation.v1","spec_hash":"7e91f012191b11b47fdd0334f2582af834f4f123214d47ac1bb2b37d91ad5b8b","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-e5fbb12b3d547bc7445c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v05\n- v09 --supports--> v07\n- v04 --refutes--> v09\n- v02 --supports--> v00\n- v00 --supports--> v08\n- v05 --supports--> v02\n- v02 --supports--> v03\n- v09 --refutes--> v01\n- v09 --supports--> v06\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dab09695e0bb7540bdaff5b690bc8c87271ebee15cba837c0b53ed0eef194be7","schema":"livegraph.boolean-propagation.v1","spec_hash":"e5fbb12b3d547bc7445c899667541b9441987c63d49b477dfc555bc953fa0227","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-72c9339f971c3f1e3063","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v08\n- v00 --refutes--> v01\n- v09 --refutes--> v05\n- v04 --supports--> v06\n- v09 --supports--> v03\n- v07 --refutes--> v04\n- v00 --supports--> v07\n- v08 --refutes--> v00\n- v03 --supports--> v02\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bdb61333cf9b9b7b4ca2a9cff9e5434c6a8d6d187ace64cebbce143f0a3d0673","schema":"livegraph.boolean-propagation.v1","spec_hash":"72c9339f971c3f1e306316c330f032d336c5e25d515752ce52fe282d403ae11a","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-b23f4b77b34fe767e2c8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v08\n- v06 --refutes--> v07\n- v06 --refutes--> v09\n- v02 --refutes--> v05\n- v04 --supports--> v00\n- v03 --supports--> v04\n- v02 --refutes--> v06\n- v07 --supports--> v01\n- v01 --refutes--> v03\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"10d7df11718b522c2af67ccc0a3d88a658872d4326cba82966a88225e1ac0fd5","schema":"livegraph.boolean-propagation.v1","spec_hash":"b23f4b77b34fe767e2c87a3a0c7b53934c47bd09a2455e78c18537d31f0cda0f","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":6,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-038c476e98d97e749651","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v06\n- v09 --refutes--> v01\n- v04 --supports--> v02\n- v02 --refutes--> v00\n- v01 --refutes--> v03\n- v01 --supports--> v05\n- v07 --refutes--> v09\n- v08 --supports--> v04\n- v02 --supports--> v07\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"17f0d73aaa7f4922e309aab9897e7053640697c01c776812a650b9b3bde4f1e4","schema":"livegraph.boolean-propagation.v1","spec_hash":"038c476e98d97e7496513c0b61094ac764e58370291520f344475f53ac9a26f0","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-8b928120bb50d543abea","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v00\n- v02 --supports--> v01\n- v02 --supports--> v08\n- v00 --supports--> v04\n- v00 --supports--> v03\n- v01 --refutes--> v05\n- v09 --refutes--> v06\n- v05 --supports--> v09\n- v06 --refutes--> v07\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2f982cfadd2075efe9de5953c25919e30e8896233f56b3184015db0cf24b5c9a","schema":"livegraph.boolean-propagation.v1","spec_hash":"8b928120bb50d543abea1592ae7bcdf9fc90b261ecc989871767b5027e9755e5","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-21bbbf95a4a54bccceba","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v03\n- v01 --refutes--> v04\n- v07 --supports--> v00\n- v09 --supports--> v02\n- v06 --supports--> v01\n- v09 --refutes--> v05\n- v07 --refutes--> v09\n- v08 --refutes--> v07\n- v04 --supports--> v08\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"10d33d528a81285d05d0f797fb02bf1cba9a162002a2f4fe03c348390364d294","schema":"livegraph.boolean-propagation.v1","spec_hash":"21bbbf95a4a54bcccebad8c7590514efda20f765b6d89a91fc90e7663ab9dd9b","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-37848dbe14b4ad34b229","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v06\n- v00 --supports--> v04\n- v07 --supports--> v09\n- v06 --supports--> v03\n- v09 --refutes--> v02\n- v08 --supports--> v07\n- v01 --supports--> v00\n- v06 --supports--> v05\n- v09 --refutes--> v01\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"04bd50dbf9b4a446060ac9ef1e15722cab20968928485d13b80d88c64e2bd703","schema":"livegraph.boolean-propagation.v1","spec_hash":"37848dbe14b4ad34b229d0985ca060ae61ec0b82f8a013593c657a7f3732c604","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-de1138aa43830fc1024d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v08\n- v01 --supports--> v04\n- v00 --refutes--> v07\n- v02 --refutes--> v03\n- v08 --refutes--> v02\n- v03 --refutes--> v05\n- v00 --refutes--> v06\n- v03 --refutes--> v00\n- v03 --supports--> v09\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a2c1ff01ca9d0759a4febd976dba232c96596b2235634acf4f494e9aa7baf216","schema":"livegraph.boolean-propagation.v1","spec_hash":"de1138aa43830fc1024d72d28d7f25dc2221e819b61aa16feffec47e494739b7","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-92af92b40dfcb65a95be","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v01\n- v07 --supports--> v02\n- v00 --refutes--> v09\n- v09 --supports--> v06\n- v02 --supports--> v00\n- v03 --supports--> v08\n- v09 --supports--> v05\n- v09 --refutes--> v04\n- v08 --refutes--> v07\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"55a8467f73752a353e9280e510deed8addbf325fb0fa0d686b10cfeac5864d06","schema":"livegraph.boolean-propagation.v1","spec_hash":"92af92b40dfcb65a95be4b3d17d2e85bd5756b0db6d7c1bf9a3e12e2b9bd7ea1","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-aa2a937a17a1b9695f43","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v09\n- v05 --supports--> v06\n- v08 --supports--> v02\n- v03 --refutes--> v08\n- v05 --supports--> v07\n- v03 --refutes--> v01\n- v08 --supports--> v00\n- v00 --refutes--> v04\n- v09 --supports--> v03\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"466fb06711215c14e4f3eacc3c952f2cda49ecba9cea65e2ed9021a48e83e914","schema":"livegraph.boolean-propagation.v1","spec_hash":"aa2a937a17a1b9695f43b5f3f2fb5517c6c9106fccb92a1623ef58aafde39f46","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-34536f2ad87160e748da","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v06\n- v09 --supports--> v05\n- v03 --supports--> v00\n- v03 --supports--> v09\n- v04 --supports--> v03\n- v01 --supports--> v04\n- v05 --supports--> v02\n- v05 --refutes--> v07\n- v02 --refutes--> v08\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0d8cbaff94945e1aece30ba6e8c57e6ef50c9132df99c0acdf734ad095593f03","schema":"livegraph.boolean-propagation.v1","spec_hash":"34536f2ad87160e748da991bc384b2e002a3920bcff5ca0ec7f82a479e7d00df","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-580a3773cf96d11398a2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v02\n- v02 --supports--> v00\n- v06 --supports--> v01\n- v00 --supports--> v04\n- v05 --supports--> v07\n- v06 --supports--> v08\n- v00 --supports--> v09\n- v08 --supports--> v03\n- v04 --supports--> v05\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0d1a3713820c5a6585b4aec0438f1e1221d4cd10d76f5e3151614f04fe9db757","schema":"livegraph.boolean-propagation.v1","spec_hash":"580a3773cf96d11398a2fae968589004fd4e89265a562d8c65a4bcb9e6ec450c","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-5b6ade5a4d50e7e55fa8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v05\n- v09 --supports--> v07\n- v05 --supports--> v01\n- v08 --refutes--> v00\n- v08 --supports--> v04\n- v09 --refutes--> v03\n- v01 --refutes--> v06\n- v06 --supports--> v08\n- v00 --refutes--> v02\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"73843cb258239903fd6c98bd5fbc902b6327b6bbef3828155bbdbde3e759efbe","schema":"livegraph.boolean-propagation.v1","spec_hash":"5b6ade5a4d50e7e55fa8af8f508b59df6bfb432d82996e290bfce1fd6af5a359","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-0b203ffbff34a4c88cd9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v08\n- v01 --supports--> v00\n- v08 --supports--> v09\n- v05 --refutes--> v03\n- v09 --refutes--> v02\n- v08 --supports--> v04\n- v06 --supports--> v05\n- v07 --supports--> v06\n- v05 --refutes--> v01\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6df289872a0a9a844cd5a05f857b56a16e1868493f113bf27c136db22797288e","schema":"livegraph.boolean-propagation.v1","spec_hash":"0b203ffbff34a4c88cd968ae5c46b1a0d01337a78182f7266a74cff5f77a4015","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-550c8c28813fea76f8ad","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v03\n- v05 --refutes--> v08\n- v02 --supports--> v01\n- v06 --supports--> v07\n- v01 --refutes--> v05\n- v08 --refutes--> v09\n- v07 --supports--> v00\n- v05 --refutes--> v04\n- v07 --supports--> v02\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e881f904756427fdf0ab467d549d066e972d7fd5b9c87e63dfc5245613f88ac3","schema":"livegraph.boolean-propagation.v1","spec_hash":"550c8c28813fea76f8ad5227e525f89fc2c16c622eec92f1348a7ac322aca84c","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-beded241b82e081c11a6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v01\n- v09 --refutes--> v07\n- v06 --refutes--> v00\n- v01 --supports--> v05\n- v06 --refutes--> v08\n- v04 --supports--> v06\n- v01 --supports--> v02\n- v03 --refutes--> v04\n- v01 --refutes--> v03\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f1edcbb38b3d2b58fcaf0f0f8370818954222e0bbefa2ffed1d6b9dc9dfcb8b3","schema":"livegraph.boolean-propagation.v1","spec_hash":"beded241b82e081c11a650e3e0b048d66cbed3b2a304820f968ed9ed10ac5f31","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-1c1ce697dcfaeb4d09d8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v03\n- v04 --refutes--> v07\n- v04 --supports--> v06\n- v05 --refutes--> v09\n- v01 --supports--> v08\n- v09 --supports--> v01\n- v08 --refutes--> v02\n- v07 --refutes--> v05\n- v08 --refutes--> v00\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6213b37f48725bed00d5f4d471a4b2d8163aca2cfa1768390664cb143b09e6af","schema":"livegraph.boolean-propagation.v1","spec_hash":"1c1ce697dcfaeb4d09d8bf1856f3a9519f4a2e412f99fa8abce509186aaca647","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-984756c4b0499253811f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v09\n- v09 --supports--> v07\n- v03 --supports--> v04\n- v05 --refutes--> v02\n- v03 --refutes--> v08\n- v02 --refutes--> v01\n- v07 --supports--> v03\n- v05 --supports--> v06\n- v00 --refutes--> v05\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"285cbe1c293a8a231035665480d88487d31ced890d4242358490e2d52b045d90","schema":"livegraph.boolean-propagation.v1","spec_hash":"984756c4b0499253811f954c4bfb354cdb96a557619e5420cb33a3032a92aa14","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-49568aa79620f9fc7f00","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v07\n- v02 --refutes--> v08\n- v09 --supports--> v00\n- v08 --refutes--> v01\n- v00 --refutes--> v02\n- v08 --refutes--> v05\n- v01 --refutes--> v04\n- v07 --supports--> v06\n- v08 --supports--> v03\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4bcf7d0bf637cd1d78c03951df5b5e3d240c2d8153ed2db037b79ec6ce41fc5f","schema":"livegraph.boolean-propagation.v1","spec_hash":"49568aa79620f9fc7f00c34ffd00d61b5cb19d7a78443976a3d5bd143f30390b","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-6dfe67a9ae1d99dea24c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v00\n- v04 --supports--> v07\n- v09 --supports--> v05\n- v03 --supports--> v09\n- v03 --refutes--> v02\n- v07 --supports--> v01\n- v05 --supports--> v08\n- v05 --refutes--> v04\n- v06 --refutes--> v03\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2b5db0ac8558867ae11afc16242f64e908eb576e5bf4ef47d5878b81f1cc7b46","schema":"livegraph.boolean-propagation.v1","spec_hash":"6dfe67a9ae1d99dea24c8569d8a4c06932760dba28dffb6c1068c063351379bd","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-999700ce4d49fe3478a6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v08\n- v08 --refutes--> v02\n- v07 --refutes--> v06\n- v01 --supports--> v07\n- v00 --refutes--> v05\n- v09 --supports--> v01\n- v04 --refutes--> v03\n- v04 --refutes--> v09\n- v00 --refutes--> v04\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"309d3155418520446b8a031b278b6293f3a0e0816db1ca691064bc3c44fbc2b2","schema":"livegraph.boolean-propagation.v1","spec_hash":"999700ce4d49fe3478a6f38c9eb79fad4ae393d1ed991eeeeb2b2bc6890cd5d7","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-a71ad071f668a2f44ebc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v08\n- v03 --supports--> v01\n- v09 --supports--> v02\n- v05 --supports--> v09\n- v00 --supports--> v03\n- v00 --refutes--> v06\n- v00 --supports--> v07\n- v04 --supports--> v00\n- v08 --refutes--> v04\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f00d95826abb65683d642ddeadb0caa0c9af502cc804cf5c0455b55ffbbe25d6","schema":"livegraph.boolean-propagation.v1","spec_hash":"a71ad071f668a2f44ebc0ff559c1601bf7a95d5811792632a10c1b263c2eea4c","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-9a5ba4cc4a22d9f9037c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v00\n- v04 --supports--> v01\n- v00 --supports--> v09\n- v09 --supports--> v07\n- v03 --refutes--> v08\n- v03 --supports--> v04\n- v04 --supports--> v02\n- v09 --refutes--> v06\n- v07 --refutes--> v05\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2063791813238bd5fb31311d0ff622c7a635546bc3e5050493a343506637973a","schema":"livegraph.boolean-propagation.v1","spec_hash":"9a5ba4cc4a22d9f9037c95cfae8522e5dc93eb7c3a5337cf7f9ef63fb0125005","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-af609c99c018184836b0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v04\n- v06 --refutes--> v01\n- v00 --supports--> v03\n- v07 --supports--> v09\n- v03 --refutes--> v05\n- v08 --supports--> v07\n- v06 --refutes--> v08\n- v07 --supports--> v02\n- v09 --refutes--> v00\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c884cc4a5ee5dd2426756a0fde25129d3af16e43bb4f9185cc2ede7186decd1e","schema":"livegraph.boolean-propagation.v1","spec_hash":"af609c99c018184836b0dfcf1971ec799f5e98c3b44b7fef8396e6490025861d","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-407b90cc0fe242c666d5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v09\n- v09 --refutes--> v03\n- v06 --supports--> v00\n- v00 --refutes--> v08\n- v09 --refutes--> v02\n- v09 --supports--> v07\n- v04 --refutes--> v05\n- v07 --refutes--> v01\n- v05 --refutes--> v06\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"32aa9f3b7036dc5aac1bbf72eb851778687ca499718724ebad6cf67b243b9bec","schema":"livegraph.boolean-propagation.v1","spec_hash":"407b90cc0fe242c666d530d70eae50b4ef6d802dde2f8fce0e44e587f1460ead","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-7e1ee080046112707838","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v05\n- v03 --supports--> v04\n- v06 --supports--> v02\n- v06 --supports--> v09\n- v05 --refutes--> v07\n- v02 --supports--> v08\n- v05 --refutes--> v00\n- v08 --refutes--> v03\n- v01 --supports--> v06\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"895f37fd616bd700c5c90562b43529e4a2dd8d4615445f653dc2ddb1ae1c0baf","schema":"livegraph.boolean-propagation.v1","spec_hash":"7e1ee080046112707838b58a58ca5a2159b1bbe20bcb8f60a24858ed700beb26","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-b7d8d6fe0e4fafe2a83a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v02\n- v01 --supports--> v08\n- v05 --supports--> v06\n- v08 --supports--> v09\n- v09 --supports--> v00\n- v03 --refutes--> v01\n- v01 --supports--> v04\n- v01 --supports--> v07\n- v05 --supports--> v03\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7b88080ae8c3d8d0a505eb525cac6a7147568e7a42a9969919e9f2ee38c37ede","schema":"livegraph.boolean-propagation.v1","spec_hash":"b7d8d6fe0e4fafe2a83a8ac6a6d9e9555c34d90b0b361f57b9bebfd6dc329c8a","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-6c6259903799f30f73aa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v02\n- v09 --supports--> v00\n- v07 --refutes--> v01\n- v06 --supports--> v07\n- v01 --refutes--> v03\n- v00 --refutes--> v04\n- v01 --refutes--> v08\n- v04 --refutes--> v06\n- v07 --refutes--> v05\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4d197796bd4ffe0c9926d02aa8b429cb8c921d140425ee0442aa365d224ad6c4","schema":"livegraph.boolean-propagation.v1","spec_hash":"6c6259903799f30f73aa5534e49eece9983a6d7b666036bfc5d56c24d07b3781","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-0adf6336193782d28439","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v09\n- v02 --refutes--> v01\n- v06 --refutes--> v03\n- v01 --supports--> v00\n- v08 --supports--> v05\n- v01 --refutes--> v04\n- v03 --supports--> v02\n- v05 --refutes--> v06\n- v08 --supports--> v07\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e2f688a0f427c9ce6593a3415953240b7ee20a18ad6f8af7cb0e3d21e8962fa0","schema":"livegraph.boolean-propagation.v1","spec_hash":"0adf6336193782d2843930e409ebd9cab8297eeb9d6f645416dcee574c0a6cfa","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-9f12913b88ff9e00525c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v09\n- v07 --supports--> v03\n- v04 --refutes--> v01\n- v05 --supports--> v02\n- v07 --refutes--> v08\n- v09 --refutes--> v00\n- v00 --supports--> v07\n- v05 --refutes--> v06\n- v02 --refutes--> v04\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"edf96a8ec9bc35e64019e520fe12aa91f6547331dd760dfaa110784a437895a7","schema":"livegraph.boolean-propagation.v1","spec_hash":"9f12913b88ff9e00525c59b2b696fe17e24ba7d37b57e28e65a685083e730739","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-97fab80fbaa07ec58897","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v03\n- v07 --supports--> v01\n- v00 --supports--> v04\n- v01 --refutes--> v02\n- v06 --supports--> v00\n- v07 --supports--> v09\n- v01 --supports--> v08\n- v02 --refutes--> v06\n- v05 --supports--> v07\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4eb34996aa9756bf2f7a4b06d1e962de037adc3c62929630d0eddb11c281c8fc","schema":"livegraph.boolean-propagation.v1","spec_hash":"97fab80fbaa07ec588979f679f115cb282ee5e2bcc4d5c6f61080bee02210eb5","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":6,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-8cd19f2abc5689c5a42a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v04\n- v01 --supports--> v00\n- v00 --refutes--> v07\n- v09 --refutes--> v01\n- v09 --refutes--> v02\n- v07 --supports--> v03\n- v06 --refutes--> v09\n- v03 --supports--> v08\n- v07 --refutes--> v05\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"621697a82f2abf1de8695e8c0ebd26e16cd035d0a16d2320822f00709ed5e0d9","schema":"livegraph.boolean-propagation.v1","spec_hash":"8cd19f2abc5689c5a42afce8f58cecdca532b0705f703a8b418664a484ba217c","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-3e32274b92f046cc6f49","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v08\n- v02 --supports--> v09\n- v06 --supports--> v00\n- v09 --supports--> v07\n- v01 --supports--> v04\n- v05 --supports--> v01\n- v04 --supports--> v02\n- v09 --supports--> v06\n- v02 --refutes--> v03\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"51f4d40dbe23e60b22e43202a4ab3dad20b6214a5d2f4bedadce4166f49ed5df","schema":"livegraph.boolean-propagation.v1","spec_hash":"3e32274b92f046cc6f491f6e96e16441a648b3fd8b14467514ac8284cc03ab76","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-97207a8e00b6cb3e7e48","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v03\n- v04 --refutes--> v08\n- v03 --refutes--> v00\n- v00 --refutes--> v01\n- v04 --supports--> v02\n- v05 --refutes--> v07\n- v07 --supports--> v04\n- v04 --supports--> v09\n- v00 --supports--> v05\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"36bae3d34a2d6ee9c3478e71ef269d4765eb986e9d87e74aa7f49cffbd4da8e3","schema":"livegraph.boolean-propagation.v1","spec_hash":"97207a8e00b6cb3e7e48ba5f452fcc5cfe0a9c359439b82bbf490db1774fbd69","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":6,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-8b33bffeb74d5892b7ef","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v05\n- v09 --supports--> v06\n- v00 --refutes--> v07\n- v04 --supports--> v08\n- v07 --refutes--> v02\n- v09 --supports--> v03\n- v07 --supports--> v04\n- v05 --refutes--> v00\n- v03 --refutes--> v01\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5fe1c33989bedc2771641bdf0c1c45c10867bfb657f28d027d6a09fe6c75c0c7","schema":"livegraph.boolean-propagation.v1","spec_hash":"8b33bffeb74d5892b7ef713374fa556b4e006e38a96d4fc67f0057f758aca7f4","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-1d9979a8a494a1811125","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v07\n- v09 --refutes--> v03\n- v01 --supports--> v00\n- v07 --supports--> v06\n- v09 --supports--> v04\n- v02 --supports--> v09\n- v02 --refutes--> v05\n- v06 --refutes--> v02\n- v03 --refutes--> v08\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dd1e568e8ecccf0b47d56ed1677ba511721958bb254f057e7b17ca7949e7c2dd","schema":"livegraph.boolean-propagation.v1","spec_hash":"1d9979a8a494a1811125929e64401fb9f5d11b9d50bcb73e9e8f4c81b813e8a6","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-50bf5b46441ff7534e44","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v08\n- v06 --refutes--> v09\n- v06 --refutes--> v03\n- v07 --supports--> v06\n- v09 --refutes--> v05\n- v01 --supports--> v04\n- v08 --supports--> v00\n- v08 --refutes--> v02\n- v09 --supports--> v01\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"55758e687613a7690d2146452072eb2f1508a8d234e7264e8dfdd3f7374643d7","schema":"livegraph.boolean-propagation.v1","spec_hash":"50bf5b46441ff7534e44108a91c10055af1ab78215bf700594dc5031c806eff0","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-0f9de1abcf48b27db6c5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v01\n- v01 --supports--> v03\n- v02 --refutes--> v07\n- v01 --supports--> v00\n- v08 --supports--> v02\n- v00 --refutes--> v09\n- v09 --refutes--> v05\n- v09 --refutes--> v08\n- v06 --refutes--> v04\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"39b425f4103582cb14ef21e9f3705818a484c07bc341421c6037828664b051c5","schema":"livegraph.boolean-propagation.v1","spec_hash":"0f9de1abcf48b27db6c5c50f2226eb438750634a892ffa1b6a2d0a6da3b27f24","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-55dcd3183c0749fa3cc8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v01\n- v01 --refutes--> v03\n- v06 --refutes--> v05\n- v07 --supports--> v08\n- v06 --refutes--> v00\n- v05 --refutes--> v09\n- v06 --refutes--> v04\n- v07 --refutes--> v02\n- v02 --refutes--> v06\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"19910f76457d03c58d6de5849f99cd6aaecd5b9b9e2107cf625524c7755fd2c0","schema":"livegraph.boolean-propagation.v1","spec_hash":"55dcd3183c0749fa3cc80466094ad0235c1b3b0b8ba6ca2dec52b1ad9148e0cb","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-e47d334cfd8363dc088d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v03\n- v03 --refutes--> v02\n- v04 --refutes--> v07\n- v04 --supports--> v06\n- v09 --supports--> v00\n- v05 --refutes--> v09\n- v08 --refutes--> v05\n- v06 --supports--> v01\n- v03 --supports--> v04\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6210b969be3f35090f468889d828e64d9cb128769701f9dc5b6a2f8cffcf541c","schema":"livegraph.boolean-propagation.v1","spec_hash":"e47d334cfd8363dc088dc2a0cd88f3cfe0a77884608d0e8c58a47723ef212f40","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-f6d1145dc399ade477cb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v00\n- v01 --supports--> v08\n- v04 --supports--> v06\n- v04 --refutes--> v02\n- v09 --supports--> v01\n- v07 --refutes--> v03\n- v03 --refutes--> v04\n- v04 --refutes--> v09\n- v08 --refutes--> v05\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"095729aaceda4848f3df4ec939f7fee7a9f93f0c6d85fbe612044d912455428d","schema":"livegraph.boolean-propagation.v1","spec_hash":"f6d1145dc399ade477cba72a89fb8eb909bd235ae0dd0fca14125d20686c6b58","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-6dc4c584706eb7e30716","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v02\n- v02 --supports--> v08\n- v01 --refutes--> v07\n- v06 --refutes--> v05\n- v09 --supports--> v00\n- v08 --supports--> v04\n- v06 --supports--> v03\n- v01 --supports--> v09\n- v04 --refutes--> v06\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"752605cbeee57d46c1a4d31aa1a8a197b8d7ca2144354c82a9220648d6a6f5d9","schema":"livegraph.boolean-propagation.v1","spec_hash":"6dc4c584706eb7e30716a109b683be92d13888e66955418be8c1780e62e86a5e","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-ecb5a104f692716167d7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v03\n- v08 --refutes--> v09\n- v06 --refutes--> v00\n- v02 --supports--> v05\n- v01 --supports--> v02\n- v02 --supports--> v04\n- v09 --refutes--> v01\n- v04 --refutes--> v06\n- v09 --refutes--> v07\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ab905b26aa6c41e2f9048fef4e5382e6e9f75ab7e9a8e2926ad8d33bcbd3160e","schema":"livegraph.boolean-propagation.v1","spec_hash":"ecb5a104f692716167d71b305fbee178f1a279a548be318d9adc3efcbf9548b1","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-677de7601c1d5009f673","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v08\n- v06 --refutes--> v00\n- v05 --refutes--> v02\n- v04 --supports--> v07\n- v09 --refutes--> v03\n- v06 --refutes--> v01\n- v05 --refutes--> v06\n- v07 --refutes--> v05\n- v00 --supports--> v09\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e94b5ef4f70fb62eb853c8f6981f7daac3ca48d7e9543d1c69e485644674e59b","schema":"livegraph.boolean-propagation.v1","spec_hash":"677de7601c1d5009f6738142d97404644139b527ebe0a5603fd841f482321b23","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-300db762e22ff019f687","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v02\n- v07 --supports--> v05\n- v03 --supports--> v08\n- v00 --supports--> v03\n- v08 --refutes--> v01\n- v04 --refutes--> v09\n- v09 --refutes--> v07\n- v04 --refutes--> v06\n- v08 --refutes--> v04\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"73008ca2c668a5aad01cb1e1d63654871a5ece7172e45c3a3f0b0d66d9fb939c","schema":"livegraph.boolean-propagation.v1","spec_hash":"300db762e22ff019f687ec7190855bee320e85589ea63a7b18433bbc547e154b","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-a00c8a7e3b85a8e5b12c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v06\n- v07 --supports--> v04\n- v06 --supports--> v02\n- v06 --refutes--> v01\n- v01 --refutes--> v03\n- v03 --refutes--> v08\n- v04 --refutes--> v00\n- v07 --refutes--> v09\n- v04 --supports--> v05\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4bef85ab953d13b643dccdd20a56d50f07fd27ee8970e31527f27f0c7734f1b9","schema":"livegraph.boolean-propagation.v1","spec_hash":"a00c8a7e3b85a8e5b12c4c439140475155b4b450fc2b4b2b56ca8524f6d1f5f1","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":6,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-1a1fd7b763c732d545f3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v05\n- v05 --supports--> v00\n- v08 --supports--> v02\n- v07 --supports--> v04\n- v01 --refutes--> v09\n- v09 --supports--> v08\n- v05 --supports--> v06\n- v01 --supports--> v03\n- v00 --refutes--> v01\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"aaeedd299c5621884ae91c498c61ac6dc4d9a289aeeeb628b0812dddc6f10250","schema":"livegraph.boolean-propagation.v1","spec_hash":"1a1fd7b763c732d545f3886ab61d820793df03b5841e8309257f8d003f6edfc6","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-e84b7f64e38e05706387","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v01\n- v00 --supports--> v04\n- v07 --supports--> v08\n- v05 --refutes--> v06\n- v02 --supports--> v09\n- v08 --refutes--> v02\n- v06 --supports--> v00\n- v07 --supports--> v03\n- v02 --refutes--> v05\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8cc2cfa106760c991ba7a682b882bde0f91518bb68a233774caa13734efb1bb3","schema":"livegraph.boolean-propagation.v1","spec_hash":"e84b7f64e38e057063871a8a110d22d9f5058924571223e3809b94366556f483","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-bc1bba6a2de5fa065df0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v08\n- v07 --refutes--> v04\n- v05 --refutes--> v09\n- v04 --supports--> v01\n- v04 --supports--> v00\n- v06 --refutes--> v05\n- v08 --refutes--> v02\n- v04 --refutes--> v03\n- v01 --supports--> v06\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a807cd111dc11971745414c155cf4653a1a169695c7815cc6c1e25e8593e8360","schema":"livegraph.boolean-propagation.v1","spec_hash":"bc1bba6a2de5fa065df0e0bfc18afd98a0f1d5b95322d8320620c46b74123f49","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-a021f7321ffe90c2f4b1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v06\n- v03 --supports--> v02\n- v06 --refutes--> v07\n- v06 --supports--> v05\n- v00 --refutes--> v08\n- v02 --supports--> v00\n- v06 --refutes--> v09\n- v04 --refutes--> v03\n- v06 --refutes--> v01\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"753d91a886777a3699ded012fd67a25e963edb01c18ea64968756b752e167edf","schema":"livegraph.boolean-propagation.v1","spec_hash":"a021f7321ffe90c2f4b1929a10121c7cd09e05abdd414c9381d5bc55b24e0f54","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-d7a515e6bf60b90cb65b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v08\n- v05 --supports--> v07\n- v01 --supports--> v05\n- v07 --supports--> v02\n- v08 --refutes--> v01\n- v05 --refutes--> v03\n- v08 --supports--> v09\n- v00 --refutes--> v06\n- v02 --supports--> v04\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"32dd93f496deda970d7b9ab107a45934c1880363f822e83b2ce844b30e03030c","schema":"livegraph.boolean-propagation.v1","spec_hash":"d7a515e6bf60b90cb65bb9939b20a002c3bb921a694e606ab09eb99a6db3da0d","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-22b3c31bb9c90698e9cb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v01\n- v08 --supports--> v04\n- v01 --supports--> v07\n- v04 --supports--> v06\n- v02 --supports--> v03\n- v03 --refutes--> v00\n- v03 --supports--> v09\n- v07 --supports--> v02\n- v08 --supports--> v05\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bcc30b64bc496b7f4a08346734bb329c278c62e49403772cf460f2b202bdba28","schema":"livegraph.boolean-propagation.v1","spec_hash":"22b3c31bb9c90698e9cb859b99fe6f375dfa58a7004f99f8c264a011b95a1f6d","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-f1f5e3ba7f09fa74f16a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v03\n- v03 --refutes--> v00\n- v03 --refutes--> v04\n- v06 --refutes--> v01\n- v07 --supports--> v06\n- v00 --refutes--> v07\n- v07 --refutes--> v09\n- v05 --refutes--> v08\n- v05 --refutes--> v02\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f41c39fc6ca8a31041c4733dedf4ac71e2d78ba5f86a2473ba41bfb0ed9ec9e9","schema":"livegraph.boolean-propagation.v1","spec_hash":"f1f5e3ba7f09fa74f16a3d3a745c499dc23856ccec741f2a57d3cdf0ab04ef7b","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-64a7d99f1f153854ed44","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v04\n- v04 --refutes--> v00\n- v08 --supports--> v06\n- v01 --refutes--> v08\n- v02 --refutes--> v07\n- v08 --supports--> v03\n- v08 --supports--> v05\n- v08 --supports--> v09\n- v00 --supports--> v02\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f76e2bfc71aeb0638d08bcd49ef83fc9b38d6e92fbc129a561f2ceb525fb23a3","schema":"livegraph.boolean-propagation.v1","spec_hash":"64a7d99f1f153854ed443c7695af236daedd3dcc646ed74d37a530c70decb07e","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-18420a74f8d09d3e36f5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v08\n- v02 --supports--> v06\n- v06 --supports--> v05\n- v05 --refutes--> v04\n- v07 --supports--> v09\n- v09 --supports--> v02\n- v01 --refutes--> v00\n- v01 --supports--> v07\n- v01 --supports--> v03\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dbf76765d26b5ae147635cc6ebb4a8e57c71490ddabc43c6a6b69bc4d926b4ee","schema":"livegraph.boolean-propagation.v1","spec_hash":"18420a74f8d09d3e36f5dad6a6ea60181f32d12403d64fce9eb4c450788daed5","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-4e1949d78c881f39ecb2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v02\n- v00 --supports--> v07\n- v09 --supports--> v03\n- v08 --refutes--> v04\n- v05 --supports--> v06\n- v06 --supports--> v00\n- v00 --supports--> v09\n- v00 --supports--> v01\n- v02 --refutes--> v05\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2199dde64a7086703477aca95071c16150cfcf129627d2ac02b79b29070b4c01","schema":"livegraph.boolean-propagation.v1","spec_hash":"4e1949d78c881f39ecb20cdd84c28a8c608a65e09f6499f4aac8ed9ea3be2789","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-b8745a8b108c33fe09ed","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v05\n- v06 --supports--> v02\n- v07 --supports--> v09\n- v05 --supports--> v04\n- v06 --supports--> v00\n- v01 --supports--> v07\n- v08 --supports--> v03\n- v05 --refutes--> v01\n- v07 --supports--> v06\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"19aa12877844bd1023ee9a310fdeb4157d96682ff8d6840dbb088dd046503ac3","schema":"livegraph.boolean-propagation.v1","spec_hash":"b8745a8b108c33fe09ede92f1e3f9de1a3f39f454e2af70292f8b6c532021d13","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-40fa516ce14a1add4321","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v09\n- v07 --refutes--> v08\n- v02 --refutes--> v07\n- v03 --refutes--> v05\n- v03 --refutes--> v04\n- v07 --supports--> v00\n- v09 --refutes--> v03\n- v05 --supports--> v06\n- v08 --supports--> v01\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0ed07b49d6b12062c58b2ba33d4109e4304066762ce1274eff794c86d7d0bee9","schema":"livegraph.boolean-propagation.v1","spec_hash":"40fa516ce14a1add4321bc8b4e1c5a848d92820c2f5d4b875d18050006eed768","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-c61b9a0debc5cb3a2b87","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v03\n- v03 --refutes--> v00\n- v05 --supports--> v09\n- v03 --supports--> v01\n- v04 --refutes--> v02\n- v06 --supports--> v05\n- v06 --supports--> v07\n- v08 --supports--> v04\n- v01 --refutes--> v06\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3dc49ebdc6b03a5aee856590c3d41738ea605e56bdc53c0b8720b8d3284a5298","schema":"livegraph.boolean-propagation.v1","spec_hash":"c61b9a0debc5cb3a2b875f8559d0aaa3439d801687f1ed66c54bffee047f508e","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-4a15296303c28f0f0af3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v00\n- v00 --refutes--> v05\n- v08 --supports--> v02\n- v08 --supports--> v04\n- v07 --supports--> v09\n- v07 --refutes--> v03\n- v03 --supports--> v08\n- v05 --supports--> v07\n- v05 --supports--> v06\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fa12d636826760844d18e5e38a0986a18e1e9fe9dd35632545b454b87111c369","schema":"livegraph.boolean-propagation.v1","spec_hash":"4a15296303c28f0f0af388497fa4b0b9e69dd164a8997c7739b6acf72b078c79","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-72b28ddd63c8d3aa1bf0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v04\n- v01 --supports--> v03\n- v07 --supports--> v08\n- v05 --refutes--> v06\n- v01 --supports--> v02\n- v06 --supports--> v07\n- v00 --refutes--> v01\n- v08 --supports--> v09\n- v01 --refutes--> v05\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"006fb3a0a653cc4aa9f5213f44dc1086c5ac32bf85dc44fb2efbc29b14a33575","schema":"livegraph.boolean-propagation.v1","spec_hash":"72b28ddd63c8d3aa1bf0ee74717936402ce3d4b129bcdbf235673148da466da6","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-fae8fce6a80d115c7384","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v08\n- v04 --refutes--> v07\n- v05 --refutes--> v06\n- v01 --refutes--> v03\n- v06 --refutes--> v01\n- v06 --supports--> v02\n- v00 --refutes--> v04\n- v00 --supports--> v09\n- v07 --refutes--> v05\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"821edd0a87c9bbdec7bd545cb096bdf0f38f98dab77d9b2892b0572fb51eda54","schema":"livegraph.boolean-propagation.v1","spec_hash":"fae8fce6a80d115c7384a1650f1b2d5af45d13465799b03b567d1139eabf8a8c","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":6,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-c0227cdff23dd302341e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v02\n- v02 --refutes--> v03\n- v08 --supports--> v09\n- v05 --supports--> v06\n- v03 --refutes--> v05\n- v02 --supports--> v07\n- v04 --supports--> v01\n- v09 --supports--> v04\n- v05 --supports--> v00\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"46e8da2d28089d16103e7f369bd86e9b7d5ac9f6570a53dd4ad5ad2d2fd323a1","schema":"livegraph.boolean-propagation.v1","spec_hash":"c0227cdff23dd302341eceabc9cf6646e55cb84890b12c7adb108ed0b9746ffe","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-fa9d888e383ed44a6059","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v04\n- v07 --supports--> v00\n- v07 --refutes--> v09\n- v08 --refutes--> v03\n- v04 --supports--> v01\n- v01 --supports--> v07\n- v07 --refutes--> v06\n- v00 --supports--> v05\n- v08 --refutes--> v02\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5d343018b182d4c3d17e8ea157652fead1af50364cf24168b0a6404fe963af44","schema":"livegraph.boolean-propagation.v1","spec_hash":"fa9d888e383ed44a6059e86fef664a4c68b29fad73c63a7498c6d19a11320c9b","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-c78fbfcfeef3c57dd2d8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v09\n- v00 --supports--> v06\n- v09 --refutes--> v05\n- v06 --refutes--> v03\n- v06 --supports--> v02\n- v09 --refutes--> v04\n- v02 --refutes--> v08\n- v05 --refutes--> v00\n- v06 --supports--> v01\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"acb26b67dbbac8757e8b9ac2c6ce689149710dd8aedf23735ffe9754c6c777a7","schema":"livegraph.boolean-propagation.v1","spec_hash":"c78fbfcfeef3c57dd2d80b2a917e04b27a382e73da1900b12c674fbbc3861bc2","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-92b2fb19e1d7f141c070","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v03\n- v09 --supports--> v08\n- v00 --supports--> v09\n- v04 --refutes--> v07\n- v08 --refutes--> v06\n- v00 --supports--> v05\n- v07 --supports--> v00\n- v06 --refutes--> v01\n- v04 --supports--> v02\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b1d4003e2bd7bf4b0cffdfb4d3a06d6a1c046b8872ca15010869a28851b32e03","schema":"livegraph.boolean-propagation.v1","spec_hash":"92b2fb19e1d7f141c070469222679b68e19351afb1c29ed450f764bfbdbec7b0","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-3e4b2dea12674431501a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v06\n- v06 --supports--> v04\n- v05 --refutes--> v01\n- v05 --refutes--> v02\n- v08 --refutes--> v07\n- v01 --refutes--> v08\n- v00 --refutes--> v09\n- v04 --supports--> v03\n- v04 --supports--> v00\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"41c6a708dc59a15a302f271e2117f45a384629edcf614e9e9de888a86a13a7d3","schema":"livegraph.boolean-propagation.v1","spec_hash":"3e4b2dea12674431501add723a3f92b08110c2bf57918d0d029d021281796a36","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-62a1917e19f81ba35c04","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v02\n- v06 --refutes--> v00\n- v02 --refutes--> v08\n- v03 --refutes--> v01\n- v06 --refutes--> v05\n- v00 --supports--> v03\n- v07 --supports--> v06\n- v03 --refutes--> v09\n- v02 --refutes--> v07\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c6262b8f49f3ba7baa5bec6276264dd4a8bb547be34ff12119121ce93fb21837","schema":"livegraph.boolean-propagation.v1","spec_hash":"62a1917e19f81ba35c0481041eb6685049b9f518244802bbaea1d5450cbcc340","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":6,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-178a51311c2a5c5b60c5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v02\n- v08 --refutes--> v07\n- v02 --supports--> v00\n- v02 --refutes--> v04\n- v04 --supports--> v09\n- v08 --refutes--> v03\n- v09 --refutes--> v08\n- v05 --refutes--> v06\n- v02 --refutes--> v01\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8891135863bdec63e156b0777a6e6b557c283541f3981eb40ec68894cbdae3e7","schema":"livegraph.boolean-propagation.v1","spec_hash":"178a51311c2a5c5b60c5300e7598134c26e8799867d8ac0e28da19f238b8c24d","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-3da084c25e5a0bba5318","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v08\n- v02 --supports--> v07\n- v08 --refutes--> v00\n- v09 --refutes--> v05\n- v05 --supports--> v01\n- v08 --refutes--> v06\n- v05 --refutes--> v03\n- v06 --refutes--> v09\n- v05 --supports--> v04\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ad8b26d4a8f55a317f6e23bbdbfa1893242c63450f72e717cfef5195fa130a82","schema":"livegraph.boolean-propagation.v1","spec_hash":"3da084c25e5a0bba53187917a0e9d59420d48d8bc85f8fa5ff4a14c887e259f7","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-2b960f5bbaa601cbd855","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v05\n- v07 --refutes--> v09\n- v04 --supports--> v08\n- v08 --supports--> v07\n- v05 --supports--> v02\n- v05 --refutes--> v01\n- v07 --supports--> v00\n- v09 --supports--> v06\n- v09 --refutes--> v03\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b7bb34d8cea290ec76253f99068c523d712d3f45e7557716488839a12a048891","schema":"livegraph.boolean-propagation.v1","spec_hash":"2b960f5bbaa601cbd855e391e45948527151584c758fb4aca56ee020703e301d","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-affb779d463c3fed5924","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v09\n- v09 --refutes--> v01\n- v05 --refutes--> v08\n- v09 --supports--> v04\n- v04 --supports--> v07\n- v06 --refutes--> v05\n- v07 --refutes--> v03\n- v00 --supports--> v06\n- v07 --refutes--> v02\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1fbc773109a36ddda75f8d77f738750f0e8273d2cef346b0f2275abf7c14f6ed","schema":"livegraph.boolean-propagation.v1","spec_hash":"affb779d463c3fed5924e9ecb4bb2b78266e56b3dc4ef906fb5d55d0426b75ee","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-70ccf0092fe66efb6e41","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v09\n- v08 --refutes--> v02\n- v00 --supports--> v01\n- v00 --supports--> v05\n- v05 --supports--> v07\n- v04 --supports--> v00\n- v06 --refutes--> v03\n- v03 --refutes--> v08\n- v08 --supports--> v04\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e7e268007572b19fc98f6e44908d66e3c1b6e3083be5f193d32adba182da5f9a","schema":"livegraph.boolean-propagation.v1","spec_hash":"70ccf0092fe66efb6e41fe60ffd958cb4ac268fee6b4c24789706f452bcea45f","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-c08c6144855b2001d2f1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v08\n- v00 --refutes--> v07\n- v01 --supports--> v06\n- v03 --refutes--> v02\n- v04 --refutes--> v03\n- v05 --refutes--> v04\n- v07 --refutes--> v01\n- v04 --supports--> v09\n- v01 --supports--> v05\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ad71d108f643bdf5b428383d4edec74a1d161d75abaf4bb124239151045545b5","schema":"livegraph.boolean-propagation.v1","spec_hash":"c08c6144855b2001d2f1ce1ec8621d78759798afb6b7667c5b28c6bcb7497ee7","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":6,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-9525aa5f5d4dd0f67910","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v04\n- v00 --supports--> v06\n- v06 --supports--> v03\n- v01 --supports--> v02\n- v06 --refutes--> v09\n- v07 --supports--> v08\n- v05 --refutes--> v00\n- v03 --supports--> v07\n- v07 --supports--> v01\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5a58aa44aa37db2dc05ba425f05144f51d4a49d0233b1540a9ecf6ab452b9cee","schema":"livegraph.boolean-propagation.v1","spec_hash":"9525aa5f5d4dd0f679102b8e677c2331a00e61a32ec69febac729b25d46760ed","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":6,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-d6d904251199b5aff353","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v04\n- v07 --refutes--> v00\n- v07 --supports--> v02\n- v08 --supports--> v03\n- v05 --supports--> v01\n- v03 --refutes--> v06\n- v00 --supports--> v08\n- v05 --refutes--> v09\n- v06 --refutes--> v05\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4696a1da2c8889dfe7cca2273482aabf89fe9e0d580182c12fa83f2e4bc2820b","schema":"livegraph.boolean-propagation.v1","spec_hash":"d6d904251199b5aff353184c9720a055ba926b7d3311f6f6838c95d47181e79b","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-0eed32b63fbca01bea6b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v05\n- v03 --refutes--> v06\n- v08 --refutes--> v01\n- v06 --supports--> v00\n- v00 --supports--> v07\n- v02 --supports--> v04\n- v00 --refutes--> v09\n- v08 --supports--> v02\n- v07 --supports--> v08\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f47fd1e98cb5e1b2ecf33b202614ededc06d49e087527823d526f368c24ead31","schema":"livegraph.boolean-propagation.v1","spec_hash":"0eed32b63fbca01bea6bf4b6aadd5e4c7bed90142574e1297a898fed819d9b95","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-5788fa705c3bb83731af","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v02\n- v00 --supports--> v01\n- v00 --refutes--> v04\n- v02 --refutes--> v08\n- v06 --refutes--> v07\n- v08 --refutes--> v00\n- v04 --supports--> v03\n- v04 --supports--> v06\n- v06 --refutes--> v09\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1103ad4460c87e91faf0bf1f0ded6caa629d4cf3a86f18e9da3062fa12ef3feb","schema":"livegraph.boolean-propagation.v1","spec_hash":"5788fa705c3bb83731af7c604f5bb4edbef35514a2728ff4af377291f9b5bdbc","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-0062f3f07ca19f35d373","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v00\n- v00 --supports--> v05\n- v05 --refutes--> v02\n- v00 --refutes--> v09\n- v03 --supports--> v04\n- v06 --supports--> v08\n- v05 --refutes--> v07\n- v08 --supports--> v03\n- v03 --refutes--> v01\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dac404f2b192a6061f904a08f0fbeb2ff8b404bb3b1f6891fda8f3359d175a13","schema":"livegraph.boolean-propagation.v1","spec_hash":"0062f3f07ca19f35d373ad4ec52ce269bd78b05772d32d6eb54659930f07a883","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-826ea673620a4c25e4c0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v03\n- v06 --refutes--> v05\n- v08 --supports--> v09\n- v07 --refutes--> v06\n- v09 --supports--> v00\n- v06 --refutes--> v04\n- v00 --supports--> v07\n- v02 --supports--> v01\n- v06 --supports--> v02\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7c4ffddaddcf8d67094bf51c82ab53b03344fe4fd21be5815d226c9eb395d837","schema":"livegraph.boolean-propagation.v1","spec_hash":"826ea673620a4c25e4c04d99425724b6d3dbe5b7f190e3c1f13db65e44f237b5","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-31029d0587d7f317c9e8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v09\n- v01 --refutes--> v05\n- v04 --refutes--> v07\n- v03 --supports--> v02\n- v04 --refutes--> v06\n- v09 --supports--> v04\n- v06 --supports--> v00\n- v03 --refutes--> v08\n- v05 --supports--> v03\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2ffcf06efe9f5dbcb3cf2df5d7d8fa204e844eb89eb267c458c99d52f2b93338","schema":"livegraph.boolean-propagation.v1","spec_hash":"31029d0587d7f317c9e8a14cd6ab10ed5c37cffaca64f57f06a8823b1a753a21","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-7b9e2b6dfcdb30848431","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v07\n- v00 --supports--> v02\n- v09 --supports--> v00\n- v02 --supports--> v03\n- v07 --refutes--> v01\n- v07 --supports--> v04\n- v06 --supports--> v08\n- v02 --refutes--> v06\n- v06 --supports--> v05\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e366bbf243d44b48694ce5d8d4fcaf40d88b04803a216359a15271f0cc907561","schema":"livegraph.boolean-propagation.v1","spec_hash":"7b9e2b6dfcdb308484319be85a55a161e527840be189211093abb28bf426ec77","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-42c76a0cba0cf0222749","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v09\n- v04 --supports--> v02\n- v09 --supports--> v08\n- v04 --supports--> v01\n- v09 --refutes--> v07\n- v06 --refutes--> v04\n- v00 --supports--> v06\n- v07 --refutes--> v00\n- v09 --supports--> v03\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"52579542f7073306cd4f50297076918fbc729711bdfc49472b72fad91031b03d","schema":"livegraph.boolean-propagation.v1","spec_hash":"42c76a0cba0cf022274937b8e98836bba2e9492b5da3d9025aec22a04c13a3f6","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-67523629e0cb4fb99fd3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v06\n- v02 --supports--> v03\n- v09 --refutes--> v02\n- v08 --refutes--> v00\n- v03 --refutes--> v01\n- v02 --refutes--> v04\n- v02 --supports--> v07\n- v01 --refutes--> v05\n- v05 --refutes--> v08\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7d5707a36638ae9741449611e8d6c4273933ca3075535d099fb68a861733fe87","schema":"livegraph.boolean-propagation.v1","spec_hash":"67523629e0cb4fb99fd33f4dfd9668f5869ad1e1e248599ed2bacf360c665e8a","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-f92bc980201a4fad13c4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v02\n- v08 --refutes--> v01\n- v08 --supports--> v05\n- v02 --refutes--> v03\n- v09 --supports--> v08\n- v05 --refutes--> v07\n- v02 --refutes--> v00\n- v05 --refutes--> v04\n- v00 --supports--> v06\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"714dd549cc69b34a640ca7e315177da5fc401694838af4746a8b6e1b0be62ab0","schema":"livegraph.boolean-propagation.v1","spec_hash":"f92bc980201a4fad13c46832f5055d435ff761a88b05ac9bd6a7dba1d6e02d98","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-4f97f1553e77f65e460c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v06\n- v06 --supports--> v02\n- v03 --supports--> v08\n- v00 --supports--> v07\n- v05 --refutes--> v03\n- v06 --refutes--> v05\n- v05 --supports--> v09\n- v08 --refutes--> v01\n- v07 --refutes--> v04\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fb73bdffa6cbf1578ef0e47a371ab3f2202d1531ff9128cdc35b301731ab5878","schema":"livegraph.boolean-propagation.v1","spec_hash":"4f97f1553e77f65e460cfa199d966c701a5ff75d5f3871efff480b9fc52d25a8","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-e6aa7eaa10595af01e94","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v05\n- v00 --supports--> v09\n- v02 --refutes--> v06\n- v01 --supports--> v07\n- v03 --refutes--> v02\n- v07 --supports--> v04\n- v04 --refutes--> v08\n- v02 --supports--> v01\n- v04 --supports--> v00\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ed67972fbc2a8a18b454d440a076266cd1abbe3ad64b28f3fff889467b2bf057","schema":"livegraph.boolean-propagation.v1","spec_hash":"e6aa7eaa10595af01e949c9a08530bc9e40fe6d9100d7726b9d2ce918ac14dd8","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":6,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-fbc7967be951ff47775a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v02\n- v05 --refutes--> v01\n- v08 --refutes--> v09\n- v04 --supports--> v05\n- v05 --supports--> v00\n- v05 --supports--> v07\n- v03 --refutes--> v06\n- v06 --refutes--> v08\n- v01 --refutes--> v03\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b3cb7417566824ba650cc90d9058ed9bafa2209a8da5127c88085e44eb42a8d1","schema":"livegraph.boolean-propagation.v1","spec_hash":"fbc7967be951ff47775a247474688552a8c1424deb30ab422cf273e877fa6212","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-86d442ea9a1a3fa0f14e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v06\n- v05 --supports--> v03\n- v07 --refutes--> v01\n- v01 --refutes--> v05\n- v05 --supports--> v09\n- v08 --refutes--> v07\n- v08 --supports--> v00\n- v03 --refutes--> v04\n- v04 --supports--> v02\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"435c94426341749058343f641954bb92d799849946db3abc55152825f0321335","schema":"livegraph.boolean-propagation.v1","spec_hash":"86d442ea9a1a3fa0f14e3611ec90edcd3d094090ff226c552e1cb35f49aa9c35","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-3c18fed7dabde53ca8b0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v00\n- v04 --supports--> v05\n- v08 --refutes--> v03\n- v07 --supports--> v06\n- v04 --refutes--> v01\n- v09 --refutes--> v04\n- v03 --refutes--> v07\n- v03 --supports--> v02\n- v06 --supports--> v09\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bb6feb632adba9b1cb21d1652076aaa25f80291f9d398aabc0b4851d5b1dad97","schema":"livegraph.boolean-propagation.v1","spec_hash":"3c18fed7dabde53ca8b0ddf75af4da43b195f00e99a97b001589c0a7d5e31e3b","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-25d189418678a1bca0d5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v07\n- v08 --refutes--> v04\n- v09 --refutes--> v05\n- v05 --refutes--> v02\n- v04 --supports--> v01\n- v06 --supports--> v09\n- v09 --supports--> v00\n- v04 --supports--> v03\n- v02 --refutes--> v08\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a23b92ed90fc2e5ade1af84ee990c2b9ac9494a87b0075686466d187a32a7b01","schema":"livegraph.boolean-propagation.v1","spec_hash":"25d189418678a1bca0d50e20b7ccf55f740fbf336ae3585fa6f01d127d99d8e8","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-951055ee230aafafd6f8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v08\n- v06 --refutes--> v00\n- v03 --refutes--> v06\n- v09 --refutes--> v02\n- v00 --refutes--> v09\n- v09 --refutes--> v05\n- v01 --supports--> v03\n- v09 --refutes--> v04\n- v07 --refutes--> v01\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0908d144c17c4cd9b34a19fd9b9430d58ab70d747a320eece9a131816c435b60","schema":"livegraph.boolean-propagation.v1","spec_hash":"951055ee230aafafd6f893c2bf0cb2767abc6fa2131cda19b7677d3304dbf237","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-c660aabb892ba65ab3e0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v08\n- v06 --refutes--> v04\n- v04 --refutes--> v05\n- v08 --supports--> v09\n- v09 --refutes--> v06\n- v04 --refutes--> v01\n- v06 --supports--> v02\n- v03 --refutes--> v07\n- v03 --supports--> v00\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b61a6bd1fdcb963f342a26734d739f70655027d8fdd92a898a7ce7573a963583","schema":"livegraph.boolean-propagation.v1","spec_hash":"c660aabb892ba65ab3e008601d634ac8caa07185b8a176173b14d09bfc1cb3ff","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-dab1b8c9ecace35e80ae","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v09\n- v03 --supports--> v07\n- v09 --supports--> v08\n- v02 --supports--> v01\n- v07 --refutes--> v05\n- v09 --refutes--> v04\n- v08 --refutes--> v02\n- v01 --refutes--> v00\n- v01 --refutes--> v06\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d584fe3853dbfddd9030ed64d8b5698dfe6c3566c2d5ad76088131864a64f4b9","schema":"livegraph.boolean-propagation.v1","spec_hash":"dab1b8c9ecace35e80ae0041a61296457e7bf8e0954548eeb909841df7f60aa5","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-497cafca7aa98d40f013","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v07\n- v08 --refutes--> v03\n- v08 --refutes--> v01\n- v03 --supports--> v04\n- v04 --refutes--> v02\n- v04 --supports--> v00\n- v08 --refutes--> v09\n- v00 --supports--> v05\n- v07 --supports--> v08\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0cc13a158211ddc9a8ab141c45903cb57207140fba920adf81e25ac0607e260c","schema":"livegraph.boolean-propagation.v1","spec_hash":"497cafca7aa98d40f013ba541081157df00560faa12c24ce61b68e47469d0745","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-5193c611bed184a7d13a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v03\n- v01 --supports--> v09\n- v07 --supports--> v06\n- v03 --refutes--> v00\n- v07 --refutes--> v04\n- v00 --refutes--> v07\n- v02 --refutes--> v01\n- v00 --supports--> v08\n- v01 --refutes--> v05\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"718c56e17fc13ee1a5a552811d988ca09885f9e72347b2d06565be5cbe900a74","schema":"livegraph.boolean-propagation.v1","spec_hash":"5193c611bed184a7d13a2bac5877e73e5d0af36799dc404f7dcfd0959dddf23a","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":6,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-19785cbd758dd3eb04e3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v04\n- v02 --refutes--> v00\n- v05 --supports--> v08\n- v03 --refutes--> v09\n- v02 --supports--> v07\n- v05 --refutes--> v01\n- v07 --refutes--> v06\n- v09 --refutes--> v02\n- v08 --supports--> v03\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c411bb1233685bffab81318b5d0b0216c87adcde28582ded6bac8de53fac04a0","schema":"livegraph.boolean-propagation.v1","spec_hash":"19785cbd758dd3eb04e3824312479eb18362f3325bea05ef8b4b9721337f4781","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-b83c0c13ffd4857bce17","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v05\n- v09 --refutes--> v02\n- v00 --supports--> v09\n- v01 --refutes--> v03\n- v07 --refutes--> v00\n- v00 --supports--> v06\n- v05 --refutes--> v01\n- v07 --supports--> v04\n- v02 --refutes--> v08\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6a145a2dbab21617fc1af09578a330d2f3d39b30ce9179c934684a366217b299","schema":"livegraph.boolean-propagation.v1","spec_hash":"b83c0c13ffd4857bce177d79ee1686172e37777b6433a8a88e858702d4f4ee26","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-6ff430522035c6239bc1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v09\n- v05 --supports--> v02\n- v04 --refutes--> v01\n- v04 --refutes--> v00\n- v07 --refutes--> v04\n- v00 --supports--> v03\n- v07 --refutes--> v08\n- v06 --supports--> v05\n- v03 --supports--> v06\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6e13849f3692dad9c08e7b68ece6f7707d19943a33f214c8a4945b2e1ad42c9c","schema":"livegraph.boolean-propagation.v1","spec_hash":"6ff430522035c6239bc1e7ccc4fec92e05266b55ee8ea2d111ba80667463a200","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":6,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-1cd9dd7fa80cd64d1d2f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v03\n- v07 --supports--> v09\n- v02 --refutes--> v05\n- v02 --refutes--> v04\n- v01 --refutes--> v07\n- v0a --supports--> v06\n- v09 --refutes--> v0a\n- v0a --supports--> v00\n- v00 --refutes--> v02\n- v04 --supports--> v08\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5bf21fadaec49cbbff3c96b3e7302f78f01401ebec9623ac19962baefc9e8489","schema":"livegraph.boolean-propagation.v1","spec_hash":"1cd9dd7fa80cd64d1d2ffeb68aedeb3fc13c0f9bf1361b5ea9015c78f3f9da10","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-9d088a7d39bebb832341","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v09\n- v00 --refutes--> v04\n- v03 --supports--> v00\n- v08 --refutes--> v0a\n- v02 --supports--> v03\n- v02 --refutes--> v07\n- v01 --supports--> v05\n- v09 --refutes--> v08\n- v08 --supports--> v01\n- v00 --supports--> v06\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"483ede5511098ad0bfea14e46fe8914092fbf6d61dd705bffbe576314b41bd94","schema":"livegraph.boolean-propagation.v1","spec_hash":"9d088a7d39bebb83234162ffd9ac555be12f675926bd032282dbaeafe2dbb550","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-73c2ac8cb2b844a49fae","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v09\n- v0a --refutes--> v02\n- v09 --supports--> v01\n- v01 --refutes--> v05\n- v00 --refutes--> v08\n- v01 --supports--> v03\n- v06 --refutes--> v0a\n- v05 --refutes--> v06\n- v0a --refutes--> v07\n- v09 --refutes--> v04\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"11af873d5405071e08e6ecfacee2b6b3981f3b3aae1cf99399817e8270f9e5b6","schema":"livegraph.boolean-propagation.v1","spec_hash":"73c2ac8cb2b844a49fae3a02782f715a284a908f4f252526d385a431f73739ff","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-085ba45272cbde7027ff","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v04\n- v01 --refutes--> v02\n- v03 --refutes--> v01\n- v07 --refutes--> v06\n- v07 --supports--> v09\n- v01 --refutes--> v05\n- v0a --refutes--> v00\n- v08 --refutes--> v0a\n- v04 --supports--> v07\n- v06 --refutes--> v08\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b342a53490678f2fef45d4d31faa4290faf48a82e69b9ec7cdf813b1f2bd3790","schema":"livegraph.boolean-propagation.v1","spec_hash":"085ba45272cbde7027ff31a381fd025f6c69567e27096935d36f1ded735a2d84","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-6ed84456bb9f9286c7a4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v07\n- v00 --supports--> v0a\n- v0a --supports--> v06\n- v07 --refutes--> v08\n- v06 --refutes--> v05\n- v09 --refutes--> v00\n- v06 --supports--> v01\n- v04 --refutes--> v02\n- v0a --refutes--> v03\n- v04 --supports--> v09\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e6341c13e5fc86d1e2e67c209d987170454fc64a76518a15a042a715dcbeb8c6","schema":"livegraph.boolean-propagation.v1","spec_hash":"6ed84456bb9f9286c7a4051722321a947170701de9863d636c0af12f2c93fa45","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-5205a7d98bd743a6a6c3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v08\n- v06 --supports--> v02\n- v00 --refutes--> v06\n- v03 --refutes--> v00\n- v08 --refutes--> v05\n- v00 --refutes--> v09\n- v00 --refutes--> v07\n- v04 --supports--> v03\n- v04 --supports--> v0a\n- v02 --supports--> v01\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8934082637cabdeac145bd34fe01c7b8185840c656102227385f3c65cbccff62","schema":"livegraph.boolean-propagation.v1","spec_hash":"5205a7d98bd743a6a6c32f2c204c956d9747a6a5b5ea09972659a32b155a431e","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-917ae01b3f60796c6087","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v08\n- v01 --supports--> v0a\n- v06 --supports--> v09\n- v04 --refutes--> v01\n- v06 --supports--> v00\n- v00 --refutes--> v04\n- v0a --refutes--> v05\n- v05 --refutes--> v07\n- v03 --refutes--> v06\n- v05 --refutes--> v02\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f97781ab1f00efa58f4fc538603835f5e6daacb0ca2efac7eebff04912e4a993","schema":"livegraph.boolean-propagation.v1","spec_hash":"917ae01b3f60796c608782201a7a61711de9adc90e5c9522610f5d9569d729fa","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-8670c5bbb5d64d8a093f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v06\n- v0a --refutes--> v03\n- v03 --refutes--> v09\n- v05 --refutes--> v02\n- v01 --supports--> v05\n- v00 --supports--> v08\n- v02 --supports--> v0a\n- v03 --refutes--> v04\n- v00 --refutes--> v07\n- v07 --supports--> v01\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5615197a5602df0af724d0de3bdc8c68d02622c9185f3c362839894ef2cdda6a","schema":"livegraph.boolean-propagation.v1","spec_hash":"8670c5bbb5d64d8a093f19ca66920d7d6280822b6098ca79da072eeebef341cf","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-480ef8ee4752841f7c8e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v01\n- v04 --refutes--> v05\n- v01 --refutes--> v06\n- v04 --supports--> v00\n- v01 --refutes--> v07\n- v03 --refutes--> v0a\n- v06 --refutes--> v08\n- v01 --refutes--> v02\n- v05 --refutes--> v03\n- v08 --refutes--> v09\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"54f018242d3046534bc0c17f952c610bdf6ec7e77203bc32417e07d01ffdfd79","schema":"livegraph.boolean-propagation.v1","spec_hash":"480ef8ee4752841f7c8e5417b24d9c8e84cfca2c433a8b2ae4f5badd141f5ddb","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-4a2b0494d8fdfb9834e6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v0a\n- v04 --supports--> v01\n- v01 --supports--> v02\n- v09 --supports--> v00\n- v07 --refutes--> v06\n- v0a --refutes--> v03\n- v07 --refutes--> v08\n- v0a --refutes--> v05\n- v07 --supports--> v04\n- v03 --refutes--> v09\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b5b9b7833c42b865880e6b58fb35af19bfb6bd50c4ee6e7b3ead3171ca8476b2","schema":"livegraph.boolean-propagation.v1","spec_hash":"4a2b0494d8fdfb9834e68b57010fc67a4626906507668fc2fefd2efcf5c7461e","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-abe94d761a96c8c5d30c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v0a\n- v00 --refutes--> v05\n- v02 --refutes--> v00\n- v05 --refutes--> v01\n- v07 --refutes--> v09\n- v09 --supports--> v06\n- v05 --supports--> v03\n- v04 --supports--> v07\n- v03 --supports--> v04\n- v00 --supports--> v08\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e18b15bad92acc2bb964d49e6ca1bfffd89a4ff9ed7cae98a7d0ad9165d66cc4","schema":"livegraph.boolean-propagation.v1","spec_hash":"abe94d761a96c8c5d30cfbe470c44c769ed2ee497621153572092b06ebb612f5","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-ecdb4ebfc11fba1d2154","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v07\n- v06 --supports--> v04\n- v01 --supports--> v09\n- v0a --refutes--> v02\n- v06 --supports--> v01\n- v02 --supports--> v03\n- v09 --refutes--> v05\n- v03 --refutes--> v08\n- v05 --refutes--> v00\n- v05 --supports--> v0a\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d86abdfc9b114259cd02562a178f10dc6a0b0df79b405a171e831266c435dc67","schema":"livegraph.boolean-propagation.v1","spec_hash":"ecdb4ebfc11fba1d2154b37366095cbea6e87c63ed60fe50a1f0f3f1a26edc07","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-14a0feb35dd5f4b4189d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v02\n- v01 --refutes--> v00\n- v0a --supports--> v09\n- v07 --refutes--> v06\n- v06 --refutes--> v05\n- v07 --refutes--> v03\n- v08 --supports--> v01\n- v02 --supports--> v08\n- v00 --refutes--> v0a\n- v08 --refutes--> v04\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2424d8d3720c3e05427433fafc250c284140ebf522285c91288b8aa568529c50","schema":"livegraph.boolean-propagation.v1","spec_hash":"14a0feb35dd5f4b4189d584e37335868c420199f5a966be77972fda91f49ac3d","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-707580574e91e8881fe7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v0a\n- v03 --refutes--> v08\n- v0a --refutes--> v07\n- v09 --supports--> v02\n- v07 --refutes--> v01\n- v03 --refutes--> v00\n- v07 --refutes--> v06\n- v06 --supports--> v03\n- v08 --supports--> v05\n- v05 --refutes--> v04\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ece5d2ed37999324846e47481f3fe5421cf259bed966daff6667060ab8a2d82f","schema":"livegraph.boolean-propagation.v1","spec_hash":"707580574e91e8881fe7470332e94514ce210696e96996e38e927158de36035f","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-13ccdd9f9c37454118fd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v07\n- v0a --refutes--> v05\n- v0a --supports--> v04\n- v02 --supports--> v06\n- v09 --supports--> v03\n- v00 --supports--> v02\n- v07 --supports--> v01\n- v01 --supports--> v0a\n- v0a --refutes--> v08\n- v09 --supports--> v00\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"049fb7588ec6a0a6544750ac736be186a8329c90b2a7da85b2c988596c9320c2","schema":"livegraph.boolean-propagation.v1","spec_hash":"13ccdd9f9c37454118fd3397cb4ea2f1cea4a79fcd58b6ff2477c8da9580a2b2","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-84e4e3cba5d4bfcf203d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v09\n- v0a --supports--> v07\n- v01 --refutes--> v05\n- v08 --supports--> v06\n- v09 --refutes--> v02\n- v0a --refutes--> v08\n- v09 --refutes--> v0a\n- v05 --refutes--> v03\n- v05 --supports--> v00\n- v06 --supports--> v01\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3edc5b40595d984a15f72c748b1069bac3e5c0b5d0109be611e9ec24cb345c43","schema":"livegraph.boolean-propagation.v1","spec_hash":"84e4e3cba5d4bfcf203df4d8131bc703d314f42f26d5e3f364cf64e03b15e179","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-16c63e24d48d16b358da","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v05\n- v02 --supports--> v03\n- v02 --refutes--> v07\n- v04 --refutes--> v08\n- v03 --refutes--> v00\n- v01 --refutes--> v06\n- v08 --refutes--> v02\n- v00 --refutes--> v0a\n- v04 --refutes--> v09\n- v06 --refutes--> v04\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5dd987aec7674639b6d166fef8c53834c389349f2359ce8798f0625612b70507","schema":"livegraph.boolean-propagation.v1","spec_hash":"16c63e24d48d16b358da960ce91686c6cb67a39e0d8e0e66ddbefec735108bdb","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":7,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-ec415823628b1792f3e6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v08\n- v06 --refutes--> v09\n- v00 --refutes--> v0a\n- v05 --refutes--> v07\n- v07 --supports--> v01\n- v06 --refutes--> v04\n- v07 --supports--> v03\n- v08 --refutes--> v02\n- v01 --supports--> v06\n- v02 --supports--> v00\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f065456c43f95eaf8214c79dc70223782c3078ba35c410e84fce37bced9ebeb4","schema":"livegraph.boolean-propagation.v1","spec_hash":"ec415823628b1792f3e6ed7078c51f21195a144ba69d543fda2942bc32d2a24e","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":7,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-c64301bf9a5d7117d425","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v02\n- v07 --supports--> v08\n- v08 --refutes--> v00\n- v09 --refutes--> v07\n- v00 --refutes--> v03\n- v08 --refutes--> v01\n- v03 --refutes--> v05\n- v02 --refutes--> v06\n- v02 --refutes--> v09\n- v03 --refutes--> v04\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a40933194c7c49240fd0144e00694209f5be1d7d7a64fc2b705923b82be5f914","schema":"livegraph.boolean-propagation.v1","spec_hash":"c64301bf9a5d7117d4252d92aaf00afa1f8aefa6fc306ebc140b889686fe89ab","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-dacfd5b16376aca782ab","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v08\n- v09 --supports--> v06\n- v01 --supports--> v07\n- v06 --refutes--> v0a\n- v00 --refutes--> v03\n- v08 --refutes--> v00\n- v03 --refutes--> v01\n- v09 --supports--> v04\n- v01 --supports--> v02\n- v02 --supports--> v05\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"597bd062416e1a7918ed14d4cd674d742d5b10161616368000ba384d474c1134","schema":"livegraph.boolean-propagation.v1","spec_hash":"dacfd5b16376aca782abb0e0a3378b9e43b1ca1244ce981f0abfbcf06fd12104","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-42b93055ce612bbc933b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v05\n- v04 --refutes--> v00\n- v05 --supports--> v03\n- v06 --refutes--> v08\n- v0a --refutes--> v04\n- v07 --refutes--> v09\n- v04 --supports--> v01\n- v03 --supports--> v0a\n- v05 --supports--> v02\n- v00 --supports--> v06\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3bfe6df1f516f2d3dd955f0dde8fe3bade5876fb2efa2bcc9b99b8f885fec7eb","schema":"livegraph.boolean-propagation.v1","spec_hash":"42b93055ce612bbc933ba53efe104e3c4a511130d4d7e1f44479c7c34529e2cf","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-a7e86abc284908e913d7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v09\n- v04 --supports--> v0a\n- v0a --refutes--> v01\n- v02 --refutes--> v05\n- v05 --refutes--> v07\n- v00 --refutes--> v03\n- v05 --supports--> v04\n- v00 --supports--> v06\n- v06 --refutes--> v02\n- v01 --supports--> v08\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ae96f7224782533ab931df7dcc0190d30bfb08806885a8a22a7e610e3aca2ad6","schema":"livegraph.boolean-propagation.v1","spec_hash":"a7e86abc284908e913d7f1b43be5884372ddc716445b4c110af54abdbd1f4ac6","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-3a13f40a34b24593c4c1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v07\n- v04 --supports--> v00\n- v07 --supports--> v06\n- v01 --supports--> v0a\n- v04 --supports--> v05\n- v03 --refutes--> v02\n- v03 --supports--> v08\n- v06 --supports--> v09\n- v09 --refutes--> v01\n- v01 --refutes--> v04\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"420670d8fc11d83964c32ae55bc6350792ad70e88b468ff74330eca9298b59e0","schema":"livegraph.boolean-propagation.v1","spec_hash":"3a13f40a34b24593c4c11cdc36c3b464693e5f00d20ebd9b39f08498a0a08765","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-f2b897126cfe3a4e4305","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v09\n- v09 --supports--> v05\n- v07 --refutes--> v04\n- v00 --supports--> v02\n- v07 --refutes--> v00\n- v06 --supports--> v0a\n- v01 --supports--> v03\n- v01 --refutes--> v06\n- v00 --refutes--> v08\n- v05 --supports--> v07\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3948aa92ad970902337620f0ded2ba78c1789609011af228b5a74f4de493f78c","schema":"livegraph.boolean-propagation.v1","spec_hash":"f2b897126cfe3a4e4305475e6a1c8a919612b45654dd257d5c18b4bdf290fa5c","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-b1d01069b2d130998039","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v04\n- v06 --supports--> v03\n- v04 --refutes--> v06\n- v09 --supports--> v02\n- v02 --refutes--> v0a\n- v04 --refutes--> v07\n- v01 --supports--> v08\n- v08 --refutes--> v00\n- v02 --refutes--> v01\n- v02 --supports--> v05\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"04db9eca5059d4646418b6ccccc3c4e568c31f37f61aeb60697309ab67bbf0a1","schema":"livegraph.boolean-propagation.v1","spec_hash":"b1d01069b2d130998039496bc2f474c55ec87de83d993a89ec7b7b18a1c87df1","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-fe751ee9225dff60bd4f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v05\n- v04 --supports--> v01\n- v01 --refutes--> v08\n- v05 --refutes--> v06\n- v07 --supports--> v03\n- v09 --refutes--> v07\n- v01 --supports--> v0a\n- v03 --supports--> v00\n- v04 --refutes--> v02\n- v08 --refutes--> v09\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"df0be2df15f647e143822cc6d04881ad826fa73ad75ff7be525afbe1f4fd8b64","schema":"livegraph.boolean-propagation.v1","spec_hash":"fe751ee9225dff60bd4f86ae3750cea0f653fb712591508ed76c7561bbdfddf7","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-1ee7113641d22ee80e09","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v08\n- v02 --refutes--> v04\n- v00 --supports--> v02\n- v06 --supports--> v00\n- v08 --refutes--> v0a\n- v08 --refutes--> v06\n- v09 --refutes--> v07\n- v01 --supports--> v09\n- v08 --supports--> v05\n- v04 --supports--> v03\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"66867cf3e731759d56755d2744ae62d0e3c821369042008793c5a3a0a6fcccc4","schema":"livegraph.boolean-propagation.v1","spec_hash":"1ee7113641d22ee80e09e3b25c42af81d4b51929caee722f7bbcfca4053abab5","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-1859d5b9df050b0af153","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v05\n- v02 --refutes--> v03\n- v02 --supports--> v06\n- v00 --supports--> v0a\n- v04 --supports--> v08\n- v04 --supports--> v01\n- v0a --refutes--> v04\n- v07 --supports--> v09\n- v06 --refutes--> v07\n- v01 --refutes--> v02\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f2aa0dabae513d80abfab9ecb213439ed513cf0371e8134d745f982f2d1e1651","schema":"livegraph.boolean-propagation.v1","spec_hash":"1859d5b9df050b0af1539e7cc1d48f34d3e35b41a5158f1e8bac569e6e333ad8","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-65922216a55223460ab0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v0a\n- v06 --refutes--> v00\n- v05 --supports--> v01\n- v03 --supports--> v09\n- v03 --refutes--> v07\n- v09 --supports--> v06\n- v05 --supports--> v02\n- v00 --supports--> v04\n- v08 --supports--> v05\n- v04 --refutes--> v08\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a6b16e790147f1b4f072684e6eed0e6a37256dbee608f9618d8091e20e063979","schema":"livegraph.boolean-propagation.v1","spec_hash":"65922216a55223460ab0acd0636da33d6d234e7ebe88ad7e47346e9a2aff6aeb","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-d10b74969c12fde62680","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v02\n- v00 --supports--> v07\n- v07 --supports--> v01\n- v03 --supports--> v06\n- v05 --supports--> v09\n- v08 --refutes--> v0a\n- v01 --supports--> v08\n- v05 --supports--> v00\n- v05 --refutes--> v04\n- v02 --refutes--> v03\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bced7e76274a3a4ba9ad671c3f1643d376fb3e2ea66bca4734ff0aa75662f5d6","schema":"livegraph.boolean-propagation.v1","spec_hash":"d10b74969c12fde62680af433adb9b2d3ebefb52db4a6253b9f4e3f274e85e35","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":7,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-9648baf63c9ba8ab69c4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v05\n- v03 --supports--> v0a\n- v03 --refutes--> v04\n- v00 --refutes--> v02\n- v0a --refutes--> v01\n- v06 --refutes--> v08\n- v05 --refutes--> v07\n- v08 --supports--> v00\n- v05 --supports--> v09\n- v07 --refutes--> v03\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0ea61f9eb8d0c7e46a3ca8599be56e4463efda25781b2e098f20deaf77fbbb07","schema":"livegraph.boolean-propagation.v1","spec_hash":"9648baf63c9ba8ab69c428557efaec0ba75291a0a7e0dac5b522227220470eac","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-7b88eeb44ae6a8ba13fa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v05\n- v09 --refutes--> v06\n- v07 --supports--> v09\n- v05 --refutes--> v03\n- v03 --supports--> v0a\n- v00 --supports--> v07\n- v09 --supports--> v04\n- v07 --supports--> v08\n- v03 --refutes--> v00\n- v04 --refutes--> v02\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"87a0b7f36761a92c85411fedf22336f5afe06d32af7ca25c2c76598cf8fe7d87","schema":"livegraph.boolean-propagation.v1","spec_hash":"7b88eeb44ae6a8ba13fabc62cc39f3ab99fb1ff3e5338559811ac23912f6f34e","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-592dc19c5d53a5deff1b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v03\n- v04 --refutes--> v0a\n- v08 --refutes--> v06\n- v05 --refutes--> v00\n- v03 --supports--> v08\n- v05 --supports--> v01\n- v00 --supports--> v04\n- v0a --supports--> v07\n- v08 --supports--> v02\n- v08 --refutes--> v09\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7a9794dee1cb8f68242fb0ce45e57d172f495eb559718f5f61b290518f454d3e","schema":"livegraph.boolean-propagation.v1","spec_hash":"592dc19c5d53a5deff1b94daf05cec3c69e1a24377842dbca5deca34978c9820","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-fa22aaea520a684f3e35","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v0a\n- v02 --refutes--> v08\n- v00 --refutes--> v03\n- v09 --supports--> v00\n- v07 --refutes--> v04\n- v01 --refutes--> v09\n- v06 --supports--> v01\n- v07 --refutes--> v02\n- v08 --supports--> v06\n- v08 --refutes--> v05\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4508091f58b1f82a3ac9b3161e7c4ab68cfcf7808085618db0eaa7861d6fe5f8","schema":"livegraph.boolean-propagation.v1","spec_hash":"fa22aaea520a684f3e353027ac54d6884aa9cfd4160d4e5a66050f3819e6dd97","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-d79b97b35af66d12e945","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v07\n- v01 --refutes--> v0a\n- v07 --refutes--> v09\n- v09 --refutes--> v01\n- v05 --supports--> v02\n- v02 --supports--> v04\n- v02 --refutes--> v06\n- v0a --refutes--> v00\n- v09 --refutes--> v03\n- v0a --supports--> v08\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1ad9d56e90248ea7d631f321168c00fc9329cd28dd9d56abeec6d54ae9cac317","schema":"livegraph.boolean-propagation.v1","spec_hash":"d79b97b35af66d12e9454915506e44f219188bceb0c47abf765f9ec17c4cdf69","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-e200ea7c875356d1bdb6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v01\n- v07 --refutes--> v03\n- v06 --supports--> v08\n- v0a --refutes--> v07\n- v09 --supports--> v0a\n- v03 --refutes--> v05\n- v05 --refutes--> v02\n- v02 --supports--> v06\n- v06 --supports--> v00\n- v06 --supports--> v04\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"de32db4eb03332d4644e93dcfa09ae91e32661f5b1d5e13d95512a01dccc229b","schema":"livegraph.boolean-propagation.v1","spec_hash":"e200ea7c875356d1bdb6c5a1fd01c1b37106f09d467bd9bf9679e1ddd48ac04b","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-6ba33bb46f7197227520","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v00\n- v04 --supports--> v08\n- v0a --refutes--> v06\n- v0a --refutes--> v09\n- v04 --supports--> v01\n- v05 --refutes--> v0a\n- v02 --refutes--> v03\n- v00 --refutes--> v05\n- v03 --supports--> v04\n- v02 --refutes--> v07\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ea9a258e6c289d35b2e3e3e166cdcaac56acd2d2cd6b416e66cbb069e9c992a9","schema":"livegraph.boolean-propagation.v1","spec_hash":"6ba33bb46f71972275201c72414b17a34f823c2ec2dc2b4f0c078a76863cce86","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-c5680da5751fd419ae3d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v02\n- v00 --refutes--> v01\n- v07 --refutes--> v06\n- v05 --refutes--> v09\n- v00 --supports--> v03\n- v08 --refutes--> v00\n- v05 --refutes--> v07\n- v03 --refutes--> v05\n- v03 --supports--> v0a\n- v04 --supports--> v08\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"19a28aa8c6105fb8310a080ce5e2221ac7a19ba3443f3881e8f72833638c92aa","schema":"livegraph.boolean-propagation.v1","spec_hash":"c5680da5751fd419ae3d8004e082902fec82b2535d36baed3e9779554cc92a45","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-42f5ae308210fc6c29c5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v02\n- v06 --refutes--> v05\n- v04 --refutes--> v0a\n- v09 --refutes--> v01\n- v04 --refutes--> v08\n- v02 --refutes--> v09\n- v02 --supports--> v03\n- v05 --supports--> v04\n- v09 --supports--> v07\n- v00 --supports--> v06\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"75e4512fc9b7367d82831693eb78c77c6b33a892a9b9eda5207835c185cc10d8","schema":"livegraph.boolean-propagation.v1","spec_hash":"42f5ae308210fc6c29c5bea3e9e86405591d74db3b47a0161230bd8ab1dbbb2d","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-4c1f2b92dc1b8b9e5cbc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v07\n- v08 --supports--> v04\n- v02 --refutes--> v00\n- v09 --supports--> v08\n- v02 --refutes--> v0a\n- v05 --refutes--> v09\n- v02 --refutes--> v05\n- v06 --refutes--> v01\n- v04 --refutes--> v03\n- v01 --refutes--> v02\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"274affe7f2f79b5042e88057e50dc36b403a18c04d3ad32f75ca372d97431242","schema":"livegraph.boolean-propagation.v1","spec_hash":"4c1f2b92dc1b8b9e5cbc547bd403a44351ee641e7973587fcb911cc6db57f427","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-ecfb7d5da1a4ccebcb98","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v05\n- v00 --supports--> v04\n- v03 --supports--> v00\n- v05 --supports--> v01\n- v06 --supports--> v02\n- v0a --refutes--> v06\n- v06 --supports--> v08\n- v04 --supports--> v0a\n- v06 --refutes--> v09\n- v05 --refutes--> v07\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c9b387c533e296aa18a7492d78d9a7cf691518d1661a7f0ccffd0341057b67be","schema":"livegraph.boolean-propagation.v1","spec_hash":"ecfb7d5da1a4ccebcb9872cb83390dd546bb7d87c55bf16a7f85c9b5480af695","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-26799a89d65b69c31993","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v00\n- v09 --refutes--> v01\n- v02 --supports--> v03\n- v09 --supports--> v06\n- v05 --refutes--> v02\n- v0a --refutes--> v07\n- v08 --refutes--> v05\n- v03 --supports--> v04\n- v03 --refutes--> v09\n- v09 --supports--> v0a\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"544a6b07ec80e3a67a51a8034378f95ecf4dd6653219bb8007338a1d65014e27","schema":"livegraph.boolean-propagation.v1","spec_hash":"26799a89d65b69c31993b292b7a590bfd2ba9a37544ba600a3eec138b0a96c14","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-720e0b3dc1d54ab97d4c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v03\n- v07 --supports--> v04\n- v09 --supports--> v0a\n- v01 --supports--> v05\n- v0a --supports--> v02\n- v03 --refutes--> v07\n- v01 --refutes--> v00\n- v07 --supports--> v08\n- v0a --supports--> v06\n- v02 --refutes--> v01\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a2a13adf792821408ce28c63ef2738512c86211f92df6b8b33134de148c68d2d","schema":"livegraph.boolean-propagation.v1","spec_hash":"720e0b3dc1d54ab97d4cd09c4ff91cc53aed914a45a71ae4732000c37c0eec5d","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-f63c9fe12522652e5a4d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v04\n- v06 --supports--> v08\n- v0a --refutes--> v03\n- v01 --supports--> v0a\n- v02 --refutes--> v00\n- v07 --supports--> v05\n- v03 --supports--> v09\n- v03 --refutes--> v02\n- v05 --supports--> v01\n- v02 --supports--> v06\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5caf46e4c5a32d0207c9992bc4eaa3facb1f9bfad4199604fcffcb38446a2bd3","schema":"livegraph.boolean-propagation.v1","spec_hash":"f63c9fe12522652e5a4de2e0a38d5ceaad64ab0bbf86bfb1849cea45e766f198","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-16694c0a224dfc70fae7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v07\n- v03 --supports--> v04\n- v09 --refutes--> v03\n- v04 --refutes--> v0a\n- v05 --supports--> v06\n- v06 --supports--> v08\n- v06 --refutes--> v01\n- v0a --refutes--> v00\n- v0a --supports--> v02\n- v00 --refutes--> v05\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e8f2a9e06c79d468c7b6adce9cc3c5dbc2df56b70802dcdb2d2b211713940d02","schema":"livegraph.boolean-propagation.v1","spec_hash":"16694c0a224dfc70fae7bca1573cf7be1b7dbf9cd58a0b881301814bb805dd10","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-aaf39dd6d5a92d3f714c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v00\n- v07 --supports--> v09\n- v06 --refutes--> v04\n- v0a --supports--> v08\n- v0a --supports--> v06\n- v09 --refutes--> v03\n- v09 --refutes--> v01\n- v05 --supports--> v07\n- v00 --supports--> v05\n- v00 --supports--> v02\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"42ed2bd610ad57af5dcbafbf82525df6da934776fc2c6c7da2842a47ef033c0e","schema":"livegraph.boolean-propagation.v1","spec_hash":"aaf39dd6d5a92d3f714c3a72a7a7f8b73b7d3dcc1866eb50df7677f792e93a76","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-a1dedc565a71a18eafb4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v01\n- v06 --refutes--> v07\n- v01 --supports--> v06\n- v07 --refutes--> v04\n- v04 --supports--> v00\n- v00 --refutes--> v02\n- v05 --supports--> v0a\n- v04 --refutes--> v03\n- v06 --refutes--> v08\n- v06 --refutes--> v09\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"03b1cb5d4e81141d4f732bcc8020bf564eae3b06a55bf086fba44cc52f87f8ff","schema":"livegraph.boolean-propagation.v1","spec_hash":"a1dedc565a71a18eafb423b64a3e3a7d0e4c5ffa4a4b060e75b2c694ab9514da","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-086fc96a79febe491378","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v09\n- v09 --refutes--> v07\n- v03 --supports--> v0a\n- v07 --supports--> v03\n- v07 --supports--> v08\n- v03 --supports--> v00\n- v06 --refutes--> v02\n- v01 --refutes--> v05\n- v06 --supports--> v01\n- v00 --supports--> v06\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c79c1bbac5f1e024cf1e501fdf2f7f8b3faf34d16f461ae2f82d07b5f326ca76","schema":"livegraph.boolean-propagation.v1","spec_hash":"086fc96a79febe4913780b7f67fad66f78725cce33dfa42a0029352f624357ed","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-39a86d3efbea8c52af07","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v04\n- v01 --supports--> v0a\n- v0a --supports--> v07\n- v08 --supports--> v01\n- v06 --supports--> v00\n- v05 --supports--> v08\n- v00 --supports--> v05\n- v01 --supports--> v09\n- v03 --supports--> v06\n- v01 --refutes--> v02\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0c49d7129b34bb5853b5de3fa3d7cf6f52695566e5c6d85c65989235d644529c","schema":"livegraph.boolean-propagation.v1","spec_hash":"39a86d3efbea8c52af075b4041a6c4d7fd4f3996744f6b4b4f8fdb727697c0f3","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-d6ab2be44cf18ca25729","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v09\n- v09 --refutes--> v07\n- v08 --refutes--> v06\n- v02 --supports--> v0a\n- v03 --supports--> v01\n- v06 --supports--> v00\n- v01 --supports--> v02\n- v02 --supports--> v04\n- v02 --refutes--> v05\n- v06 --supports--> v03\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a1c0c91f7ccc6a7c96e6a894e0c945dc5fcbbb1b3b2fbbf1c4e6d1f5971aaa41","schema":"livegraph.boolean-propagation.v1","spec_hash":"d6ab2be44cf18ca257294297c6366f46a4b8fb6a5b9b9878ecc908e1095d2439","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-ffa0ad40fe8ca82538a1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v02\n- v00 --refutes--> v04\n- v02 --supports--> v03\n- v01 --refutes--> v0a\n- v00 --supports--> v08\n- v06 --refutes--> v07\n- v0a --refutes--> v00\n- v01 --supports--> v09\n- v02 --refutes--> v01\n- v05 --refutes--> v06\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9d60018bd45c9ef897a59c2c3f5196eb4959043d16a5ae54ccbd9a429b944b27","schema":"livegraph.boolean-propagation.v1","spec_hash":"ffa0ad40fe8ca82538a19c9683e4bd2e6a276dceba5668f622a40ad30187ce25","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-02b668ac6d62f320752c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v02\n- v01 --supports--> v04\n- v09 --refutes--> v05\n- v00 --supports--> v07\n- v09 --supports--> v06\n- v02 --supports--> v09\n- v06 --refutes--> v00\n- v01 --supports--> v08\n- v04 --refutes--> v0a\n- v07 --supports--> v03\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6ff7018b5edf74310bf38ebe765ab147801a20c33afce1ece7180de80448dae5","schema":"livegraph.boolean-propagation.v1","spec_hash":"02b668ac6d62f320752caf223b97218de0738c3ab352314bdda34166aa9f2246","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-1ef76a8615073206d984","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v01\n- v00 --refutes--> v02\n- v03 --refutes--> v09\n- v04 --supports--> v00\n- v05 --refutes--> v08\n- v09 --refutes--> v06\n- v07 --supports--> v03\n- v00 --supports--> v05\n- v08 --supports--> v07\n- v03 --refutes--> v0a\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"693fad12d62228d880a40fbdbde8fc48f7b9538a4f97d58e4371a7c4f2fecef4","schema":"livegraph.boolean-propagation.v1","spec_hash":"1ef76a8615073206d9843598aa1eaf309db5ce5102890a76a2e49150a59c5b30","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-c7c1e027c128ae36775a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v07\n- v02 --refutes--> v00\n- v01 --supports--> v04\n- v01 --supports--> v08\n- v00 --refutes--> v0a\n- v0a --supports--> v09\n- v03 --refutes--> v06\n- v05 --supports--> v02\n- v0a --supports--> v01\n- v07 --supports--> v03\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bcb413aa0896f03203109a69839980c5693e75d77f7c7acbe45bae4ca487ee44","schema":"livegraph.boolean-propagation.v1","spec_hash":"c7c1e027c128ae36775aa9f912835271a561003ed312643fd442c32b9485c8d2","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-8a4644be8b4af6e43d58","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v07\n- v01 --supports--> v03\n- v06 --refutes--> v04\n- v05 --refutes--> v06\n- v02 --refutes--> v0a\n- v02 --supports--> v08\n- v04 --refutes--> v02\n- v00 --supports--> v01\n- v03 --refutes--> v05\n- v03 --refutes--> v09\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ce635a19db63940339c47bcc5236df3744913a97f3f29199c478cb0a39276106","schema":"livegraph.boolean-propagation.v1","spec_hash":"8a4644be8b4af6e43d5896457a8ab8b02dd6ae1c7ee16652326dc64fc4726c96","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":7,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-90a13d80d22f0620ba41","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v04\n- v02 --supports--> v07\n- v00 --supports--> v06\n- v01 --supports--> v0a\n- v00 --refutes--> v09\n- v09 --refutes--> v03\n- v08 --refutes--> v05\n- v04 --refutes--> v02\n- v09 --supports--> v01\n- v07 --refutes--> v08\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"940bcc0742d86a6f29332d3e7b2d76d618a2a1ff36f1730f980b49b11189a4b2","schema":"livegraph.boolean-propagation.v1","spec_hash":"90a13d80d22f0620ba41edd56f6e4f14e6969ac27a1e838eb625b79c11c402ff","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-35d3f36f179d17d14ced","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v08\n- v00 --refutes--> v01\n- v00 --refutes--> v05\n- v06 --refutes--> v09\n- v09 --supports--> v04\n- v03 --supports--> v0a\n- v05 --supports--> v07\n- v04 --supports--> v02\n- v0a --refutes--> v00\n- v04 --refutes--> v03\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"86a112c19d5b32c0db09f88fed646cbe6d6c927c422799dace009d86160eed54","schema":"livegraph.boolean-propagation.v1","spec_hash":"35d3f36f179d17d14cedfb4426c4cbdb62f2df834475c05df041eb592da89e39","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-087fd6fe65f1880abb48","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v02\n- v04 --refutes--> v07\n- v05 --supports--> v09\n- v07 --refutes--> v05\n- v09 --refutes--> v06\n- v03 --supports--> v04\n- v02 --refutes--> v01\n- v03 --supports--> v00\n- v03 --refutes--> v0a\n- v02 --refutes--> v08\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4f6b5e2f65a5679dbd07984ce4ddbf26d26194d42eef652beb8cb81657d203da","schema":"livegraph.boolean-propagation.v1","spec_hash":"087fd6fe65f1880abb4830edbc6089bdb9f39320833c4bf12d0da3a4d0e92d51","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-25f87cad7702edc88e10","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v08\n- v03 --supports--> v00\n- v09 --supports--> v03\n- v05 --supports--> v02\n- v09 --refutes--> v07\n- v01 --supports--> v04\n- v08 --supports--> v0a\n- v03 --refutes--> v06\n- v04 --refutes--> v05\n- v06 --refutes--> v01\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3261cc5fdd9823ac4257495dafa41848551d653faf6197b53cbdaf99618578d3","schema":"livegraph.boolean-propagation.v1","spec_hash":"25f87cad7702edc88e104fffdf1d8ab09fbcfd98f713ad20e7d888619ec7a033","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-744e959520f2765d267a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v01\n- v05 --refutes--> v09\n- v01 --supports--> v02\n- v03 --refutes--> v04\n- v06 --supports--> v05\n- v01 --refutes--> v08\n- v08 --refutes--> v03\n- v03 --supports--> v07\n- v00 --supports--> v06\n- v07 --refutes--> v0a\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3bab6308a20548806401aa866c15d1970890bb2868f00072a1df65fa167f6b02","schema":"livegraph.boolean-propagation.v1","spec_hash":"744e959520f2765d267aaa916fc414b8a03457762f4b71d26cfcf6741c8d5c99","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":7,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-48cc2fc4f2848140d2a1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v00\n- v00 --refutes--> v02\n- v00 --supports--> v09\n- v00 --supports--> v03\n- v03 --supports--> v07\n- v04 --refutes--> v08\n- v01 --supports--> v04\n- v05 --supports--> v06\n- v06 --supports--> v01\n- v06 --supports--> v0a\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4e3715ed2170085c1219cf7934c8b1d46e261e2796795fef545067bde3c80eb0","schema":"livegraph.boolean-propagation.v1","spec_hash":"48cc2fc4f2848140d2a167c7e2d5cef33cd41422a49537909862f82a56cbdb37","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":7,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-fe1608e590b3ef8b5495","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v00\n- v09 --supports--> v02\n- v0a --supports--> v01\n- v08 --supports--> v0a\n- v06 --supports--> v04\n- v02 --supports--> v08\n- v09 --supports--> v03\n- v04 --refutes--> v05\n- v04 --supports--> v07\n- v01 --refutes--> v06\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"937665bbd72dd1711a1d53a98bfbf65464203383473c61f44c9314b98b801770","schema":"livegraph.boolean-propagation.v1","spec_hash":"fe1608e590b3ef8b5495007c3f5ad58a0c363bd67d7e65d38674b4c7d1ccb89e","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-66d8f5c40b37807a5375","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v04\n- v05 --supports--> v09\n- v04 --supports--> v06\n- v05 --refutes--> v03\n- v09 --supports--> v07\n- v07 --refutes--> v01\n- v02 --supports--> v0a\n- v0a --refutes--> v05\n- v00 --refutes--> v08\n- v04 --supports--> v02\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ce403ba27b8a2f27d82fcb001794a15c0c8a47e33f083f380e3600abf3a12c19","schema":"livegraph.boolean-propagation.v1","spec_hash":"66d8f5c40b37807a5375409f07f0baa6377e13a1c68e70a457bd759ac8538159","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-bc126d88daa81fb185f1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v00\n- v07 --refutes--> v01\n- v05 --supports--> v07\n- v0a --supports--> v05\n- v06 --supports--> v08\n- v07 --refutes--> v06\n- v09 --refutes--> v02\n- v03 --supports--> v0a\n- v09 --refutes--> v03\n- v09 --supports--> v04\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4a7568b0bdacccab72e0f85602bacb9cadb499848256fa7253816c321b67a904","schema":"livegraph.boolean-propagation.v1","spec_hash":"bc126d88daa81fb185f1012988059d7fd4a4e16cf0cb5705019c8762a89cf538","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-a185bbef36c6308d5da8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v09\n- v08 --refutes--> v07\n- v08 --refutes--> v04\n- v05 --refutes--> v01\n- v00 --refutes--> v05\n- v09 --refutes--> v03\n- v03 --supports--> v08\n- v02 --supports--> v06\n- v00 --supports--> v0a\n- v01 --refutes--> v02\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e58ffe8a3f96fd967ea3e7690695934ffa10129b0e2bc5466376566e2cd92034","schema":"livegraph.boolean-propagation.v1","spec_hash":"a185bbef36c6308d5da8e1c39bf2e20b1610724de9f9679701bcecf63b21e0ff","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-bfdc3e39930f7aaa9e25","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v0a\n- v01 --supports--> v05\n- v01 --supports--> v07\n- v00 --supports--> v04\n- v02 --supports--> v01\n- v02 --refutes--> v08\n- v0a --supports--> v02\n- v09 --refutes--> v00\n- v06 --refutes--> v09\n- v04 --supports--> v03\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"adfa5460a50951abfaeb809030bdec61f293851c192a0f3230999c17b0ca6726","schema":"livegraph.boolean-propagation.v1","spec_hash":"bfdc3e39930f7aaa9e252cabef905ca40d5da34fc29354611a8d75fb3446d8c5","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-d3af805c85aeb0302b12","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v09\n- v0a --refutes--> v05\n- v04 --refutes--> v08\n- v0a --refutes--> v02\n- v08 --supports--> v03\n- v06 --refutes--> v0a\n- v04 --refutes--> v07\n- v09 --refutes--> v01\n- v01 --refutes--> v06\n- v08 --refutes--> v00\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d644401010b11ca62c8a211f00846f5375071b5c8b9819d7bf028c50143696e5","schema":"livegraph.boolean-propagation.v1","spec_hash":"d3af805c85aeb0302b120f7da405187d431595ade465fdb75f24be2183c51234","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-87ccd8a31504b1e1fd09","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v03\n- v08 --refutes--> v06\n- v06 --supports--> v05\n- v00 --refutes--> v01\n- v0a --refutes--> v00\n- v05 --refutes--> v09\n- v00 --refutes--> v04\n- v01 --refutes--> v08\n- v02 --refutes--> v0a\n- v05 --refutes--> v07\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"00f8d344c1c3f4b95fa41768b83fed53ed3c42ea81b35c9435ae36701f6d2d37","schema":"livegraph.boolean-propagation.v1","spec_hash":"87ccd8a31504b1e1fd094d78f92214a31f93ce402cfa84d3fcf606180e2e2403","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-8b7793667da6a6e574ad","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v02\n- v02 --refutes--> v0a\n- v07 --refutes--> v01\n- v02 --refutes--> v00\n- v03 --supports--> v08\n- v09 --supports--> v06\n- v03 --refutes--> v09\n- v0a --refutes--> v03\n- v07 --supports--> v04\n- v05 --supports--> v07\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e00e2f2ef806d8b6d28c77fe50cec3ddfe7b19f563f169af7c23d3f37d6e97c4","schema":"livegraph.boolean-propagation.v1","spec_hash":"8b7793667da6a6e574ad4eb7e2954213df7f9e3638c88040354c928905a983e2","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-5a5adc2dd4e92e93a636","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v01\n- v02 --supports--> v08\n- v03 --supports--> v00\n- v06 --refutes--> v0a\n- v08 --refutes--> v07\n- v06 --supports--> v09\n- v06 --supports--> v05\n- v03 --supports--> v02\n- v01 --refutes--> v06\n- v0a --refutes--> v04\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f12d503bcdadfc43b0c11b1d89881d2e089e15c39bff7780ed2bd71e47b63c14","schema":"livegraph.boolean-propagation.v1","spec_hash":"5a5adc2dd4e92e93a636498b8f8e21e02fbe3130c3fa3e0910f018f692a6b50a","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-b130ff36752e574ef24e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v08\n- v01 --refutes--> v03\n- v03 --refutes--> v04\n- v06 --supports--> v01\n- v06 --supports--> v05\n- v04 --refutes--> v0a\n- v04 --refutes--> v07\n- v07 --refutes--> v00\n- v02 --refutes--> v09\n- v00 --refutes--> v02\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"713459b879906f9f5d1b7fdb703caee9e382146e7ec19f4178c7c4f46e051ed8","schema":"livegraph.boolean-propagation.v1","spec_hash":"b130ff36752e574ef24e1c30aa61c94f942d250edb1c04f595bd10e5388dc2b4","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":7,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-0412587fbd059ec8ee55","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v03\n- v05 --refutes--> v04\n- v03 --supports--> v07\n- v02 --supports--> v01\n- v00 --supports--> v09\n- v00 --supports--> v08\n- v07 --refutes--> v02\n- v05 --refutes--> v06\n- v08 --supports--> v05\n- v03 --supports--> v0a\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5fbca644bf46ec555ed8481fc5bd99daca67127a9c7e8e1598b55a48568115ba","schema":"livegraph.boolean-propagation.v1","spec_hash":"0412587fbd059ec8ee55d3ea2284a8343c859d7d7d683d393aa8e5aa6182e6f6","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":7,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-555ccd2e7c9f0965e0b3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v09\n- v03 --refutes--> v0a\n- v05 --supports--> v02\n- v01 --refutes--> v04\n- v0a --supports--> v05\n- v07 --refutes--> v08\n- v05 --refutes--> v00\n- v07 --supports--> v03\n- v04 --refutes--> v06\n- v02 --refutes--> v01\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fabd34121ccf4093e295e042e06a4caf908972a23be9a70cb618a00635ca3a59","schema":"livegraph.boolean-propagation.v1","spec_hash":"555ccd2e7c9f0965e0b355021bccb5b648f65de667f16b84d9b012b52a67e9f5","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-1be0c83565e238329d09","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v09\n- v07 --supports--> v03\n- v02 --refutes--> v05\n- v07 --refutes--> v04\n- v06 --refutes--> v07\n- v09 --refutes--> v02\n- v05 --refutes--> v01\n- v05 --supports--> v00\n- v07 --refutes--> v0a\n- v00 --supports--> v08\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e9854aa31f618fe06a3d94dcd95f44707022c7c547e0fcb0225a81a84df99566","schema":"livegraph.boolean-propagation.v1","spec_hash":"1be0c83565e238329d095d10d522a1c603b4cdb393e0e5b455b5e15ad0942961","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-7259c8ff950413ceb018","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v0a\n- v04 --supports--> v06\n- v02 --refutes--> v05\n- v00 --supports--> v04\n- v02 --refutes--> v09\n- v05 --supports--> v00\n- v0a --refutes--> v03\n- v04 --refutes--> v01\n- v08 --refutes--> v07\n- v07 --refutes--> v02\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0c8c67f8978e615ac01a5e673e3738186e327142d090c4b48f630d8822529edc","schema":"livegraph.boolean-propagation.v1","spec_hash":"7259c8ff950413ceb018ff5e0e2544a7a84ca9a11c9b66808ae1846807ea099c","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-4a1612dcef337d57b482","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v05\n- v01 --supports--> v06\n- v03 --refutes--> v00\n- v0a --supports--> v01\n- v05 --refutes--> v03\n- v0a --supports--> v08\n- v06 --supports--> v02\n- v03 --refutes--> v07\n- v04 --refutes--> v09\n- v04 --refutes--> v0a\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d9498bd012008f43d9f96054d5bed7414032c5020f73f9faaae936779deefedf","schema":"livegraph.boolean-propagation.v1","spec_hash":"4a1612dcef337d57b48274009a3f878cef713053495d6b1fff37ab11ac609bec","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-9d08e31a8003cd3cb537","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v04\n- v07 --refutes--> v0a\n- v00 --supports--> v01\n- v01 --refutes--> v02\n- v08 --supports--> v07\n- v06 --supports--> v00\n- v0a --supports--> v06\n- v09 --supports--> v05\n- v01 --refutes--> v09\n- v01 --supports--> v03\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9799d262dede3163021267d77540e03ff10199d7c0d674bfffffa6b1cd06f918","schema":"livegraph.boolean-propagation.v1","spec_hash":"9d08e31a8003cd3cb5370bcc0a7886c2b94a4913fc575e84feb06d5947e06db0","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-16e5bee63d7c2bb00e39","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v00\n- v07 --refutes--> v09\n- v05 --refutes--> v01\n- v02 --refutes--> v07\n- v05 --refutes--> v04\n- v09 --supports--> v0a\n- v08 --supports--> v02\n- v06 --refutes--> v08\n- v0a --supports--> v03\n- v0a --refutes--> v05\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2bcda5bf826a6b9f7fba1e3d45e8432c44c5e1a98e9efd73ff27a7ea56636487","schema":"livegraph.boolean-propagation.v1","spec_hash":"16e5bee63d7c2bb00e3912bce2ddcd9ddc0397574f4ca85170c3a425c973eba2","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-882b80421c996f00ff94","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v00\n- v02 --supports--> v04\n- v08 --supports--> v03\n- v03 --supports--> v06\n- v0a --refutes--> v02\n- v09 --refutes--> v05\n- v08 --refutes--> v07\n- v09 --refutes--> v01\n- v01 --supports--> v08\n- v06 --supports--> v0a\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c0a7edbe31fcd65758f2614539cdf900796a22878d7c0fe6dd7f5ddd0f81bb1a","schema":"livegraph.boolean-propagation.v1","spec_hash":"882b80421c996f00ff943ec7780e6fb8f0a4462263f4dc78ba5aa071d98daacf","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-22cf77f36588d04183dc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v06\n- v05 --refutes--> v0a\n- v0a --refutes--> v03\n- v04 --supports--> v07\n- v08 --supports--> v00\n- v09 --refutes--> v05\n- v02 --supports--> v09\n- v08 --supports--> v01\n- v04 --refutes--> v02\n- v03 --refutes--> v08\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2fa44a6afc330abba7f1cc80a1adaab485a92b98d791ad5a06faf189b1ec2c70","schema":"livegraph.boolean-propagation.v1","spec_hash":"22cf77f36588d04183dcfdeb2b86dd64f7b55aacc6442dd50b5cd1bb591668b8","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-85bff6cbaff51eacefcd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v0a\n- v08 --supports--> v09\n- v06 --supports--> v02\n- v06 --supports--> v03\n- v09 --refutes--> v04\n- v09 --refutes--> v01\n- v02 --refutes--> v05\n- v09 --refutes--> v00\n- v0a --refutes--> v06\n- v05 --supports--> v07\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c79a8b575d7b034b2b3649b468b1bf8967428bffa6e51609601e6366f1e8f364","schema":"livegraph.boolean-propagation.v1","spec_hash":"85bff6cbaff51eacefcdb9c1fe50cd8343f33984ba5fc73f4e240fd89aa706c7","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-793b1d0fda8970375e0f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v08\n- v0a --supports--> v07\n- v00 --supports--> v04\n- v06 --supports--> v00\n- v03 --supports--> v06\n- v0a --refutes--> v05\n- v02 --supports--> v0a\n- v01 --supports--> v02\n- v0a --refutes--> v03\n- v09 --refutes--> v01\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"428f1b77bd0baf55614220f1fb63188e0fafe26ab50eae4afdf1cdf520c216c7","schema":"livegraph.boolean-propagation.v1","spec_hash":"793b1d0fda8970375e0f8b366933f628106b75a8dadf5f1da177c8b3e2955cc8","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-f6c406f897a7edb92401","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v06\n- v04 --refutes--> v07\n- v08 --refutes--> v05\n- v07 --supports--> v09\n- v0a --refutes--> v02\n- v05 --refutes--> v01\n- v03 --refutes--> v04\n- v0a --supports--> v03\n- v07 --refutes--> v08\n- v00 --refutes--> v0a\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"65670210c93b6f0f4fbe53c148b5379c35996d06c42f9d48623634ee1aac8f50","schema":"livegraph.boolean-propagation.v1","spec_hash":"f6c406f897a7edb924015e687e203b913d201c1aa2cba1877d91b16406578ad4","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-1532ae283c25f1f4b417","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v01\n- v03 --supports--> v08\n- v03 --refutes--> v06\n- v00 --supports--> v04\n- v02 --supports--> v0a\n- v06 --supports--> v09\n- v09 --supports--> v02\n- v02 --refutes--> v00\n- v03 --refutes--> v07\n- v01 --supports--> v05\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"689d0a7e0583050a22b1c93106a5814aee20b0053a1e28406161c6fb3cf6cb8b","schema":"livegraph.boolean-propagation.v1","spec_hash":"1532ae283c25f1f4b417edfffaceddc0835fb2e4212fdd619c841e9014408775","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-302cee78b0c6e249cbb3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v02\n- v09 --supports--> v05\n- v08 --supports--> v01\n- v06 --supports--> v00\n- v02 --refutes--> v0a\n- v05 --refutes--> v03\n- v05 --supports--> v07\n- v07 --supports--> v04\n- v02 --supports--> v06\n- v06 --refutes--> v09\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7c994279e041773a219e7bae2c8a6e3b43ca6cc06c74f3f7a0887925175f1a59","schema":"livegraph.boolean-propagation.v1","spec_hash":"302cee78b0c6e249cbb3aa1a304c80e65bf9b1582c9414d458a27020d3af9ae8","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-3b3c1497dfdcb7a3c911","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v0a\n- v06 --supports--> v01\n- v01 --supports--> v07\n- v08 --refutes--> v00\n- v04 --refutes--> v08\n- v05 --refutes--> v03\n- v07 --refutes--> v09\n- v05 --refutes--> v02\n- v03 --supports--> v06\n- v07 --refutes--> v04\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a5b62025f64d901c37abe8d8486f2b9d14ead657c9c251387801fa7f37945dc2","schema":"livegraph.boolean-propagation.v1","spec_hash":"3b3c1497dfdcb7a3c91111f8e2deb957cac492c025d5957229bbe686b2b120cc","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":7,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-bf4a6512bd5e9e7e87a9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v04\n- v04 --refutes--> v08\n- v03 --refutes--> v01\n- v00 --supports--> v0a\n- v04 --supports--> v09\n- v08 --supports--> v02\n- v00 --refutes--> v06\n- v07 --refutes--> v00\n- v08 --supports--> v05\n- v05 --refutes--> v07\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f66270b079da8e98ec08c2cade7a0792d00bf0bee119524fddf67d23611668d8","schema":"livegraph.boolean-propagation.v1","spec_hash":"bf4a6512bd5e9e7e87a926db8e404bed68f3106e5c562c5c55d295075cc97a1a","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":7,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-66fd3a8e427633a84f55","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v08\n- v03 --refutes--> v05\n- v02 --supports--> v07\n- v05 --refutes--> v09\n- v09 --refutes--> v06\n- v0a --refutes--> v00\n- v00 --supports--> v04\n- v0a --supports--> v01\n- v06 --refutes--> v0a\n- v02 --supports--> v03\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e27e159a8b1226e2d99d20709d364b72d24eabf6cfe6d15d2e783a95207ebe64","schema":"livegraph.boolean-propagation.v1","spec_hash":"66fd3a8e427633a84f55b51ec633b898b62bc1ca1829c50d2efb414c6ebd8e61","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-ce48ed02f7a779546fb3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v04\n- v03 --refutes--> v01\n- v09 --refutes--> v0a\n- v08 --supports--> v00\n- v03 --refutes--> v05\n- v00 --supports--> v07\n- v02 --refutes--> v03\n- v0a --supports--> v06\n- v00 --refutes--> v09\n- v01 --refutes--> v08\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"273e7c5254afda5386a49fcad65e9af23ab5dcabc7e6a850a4a0546089639060","schema":"livegraph.boolean-propagation.v1","spec_hash":"ce48ed02f7a779546fb3120042fd1af81b156f0e2398b2eff5a084ae2d10179c","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-48cd73943b6b5436ea30","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v00\n- v09 --supports--> v02\n- v0a --refutes--> v04\n- v04 --supports--> v06\n- v01 --supports--> v07\n- v04 --supports--> v08\n- v07 --refutes--> v05\n- v05 --refutes--> v0a\n- v06 --refutes--> v09\n- v09 --refutes--> v03\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5aa905e815bbb42116ea48b67c95fd4dc75ca8a647fc1198d5437fd9f9be309d","schema":"livegraph.boolean-propagation.v1","spec_hash":"48cd73943b6b5436ea3071e8433e8bda6b454a399971f7e5fce75d72f2130ba9","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-c31d18b10817cb3a705d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v06\n- v04 --refutes--> v09\n- v03 --supports--> v08\n- v09 --supports--> v01\n- v05 --supports--> v0a\n- v0a --supports--> v03\n- v00 --supports--> v02\n- v09 --refutes--> v05\n- v00 --refutes--> v07\n- v08 --refutes--> v00\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6e407c4844273dd273cc0f31b7f03894111e0dcb83e1d66a9ac7d5265f336762","schema":"livegraph.boolean-propagation.v1","spec_hash":"c31d18b10817cb3a705dba23f575a41b6a8c5fc8a7f353fbdad0419f32af66f6","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-5d40b321555e98ce9f80","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v03\n- v09 --supports--> v0a\n- v04 --supports--> v05\n- v08 --refutes--> v06\n- v04 --refutes--> v07\n- v01 --supports--> v04\n- v03 --refutes--> v02\n- v08 --refutes--> v00\n- v07 --supports--> v08\n- v02 --refutes--> v09\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bc8a6ffd82276e5e55d96294560c6d748a7e270b2771cd3921130a52ad4c6bcb","schema":"livegraph.boolean-propagation.v1","spec_hash":"5d40b321555e98ce9f80bb90de6926e42f4a82d1ba5dfd8bc896df71de7d095a","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-43d54dbb32ebfb287770","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v03\n- v07 --refutes--> v02\n- v0a --supports--> v07\n- v06 --refutes--> v08\n- v07 --refutes--> v00\n- v01 --supports--> v0a\n- v07 --supports--> v05\n- v03 --supports--> v06\n- v03 --refutes--> v04\n- v09 --supports--> v01\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e7016fd60def8abcfb9ecf9f59bb8e86696517a30cc66f374bb04ad94f1fe741","schema":"livegraph.boolean-propagation.v1","spec_hash":"43d54dbb32ebfb2877701ac88ee3eda8d43af007fb219c15204cf79644b424a3","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-18bf75dcbabe97b24c9d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v04\n- v07 --refutes--> v05\n- v06 --supports--> v09\n- v02 --supports--> v06\n- v07 --refutes--> v01\n- v07 --supports--> v00\n- v04 --refutes--> v08\n- v03 --refutes--> v0a\n- v0a --supports--> v07\n- v06 --supports--> v03\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2091b0bfde1fa2c06036d8e755b57450e4b2347c4245b9b7c232eb2b8c69f69a","schema":"livegraph.boolean-propagation.v1","spec_hash":"18bf75dcbabe97b24c9d5e92d4fdaac4ad576e3b97e65b3524ab68209826fb9e","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-a4411783797fe57b254a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v03\n- v01 --supports--> v07\n- v0a --refutes--> v09\n- v09 --refutes--> v01\n- v08 --supports--> v00\n- v04 --supports--> v02\n- v01 --refutes--> v05\n- v02 --supports--> v06\n- v02 --refutes--> v0a\n- v07 --supports--> v08\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5f46c758c12d5ea9cb6657c19b99095bea2ddd7c8de47faf3d05db8f51288e17","schema":"livegraph.boolean-propagation.v1","spec_hash":"a4411783797fe57b254a1514f1d70b8a2a7c7e8c00ed80c6a8e38d5fca845ea4","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-fd0434de5341793f7518","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v00\n- v06 --refutes--> v09\n- v04 --supports--> v02\n- v05 --supports--> v0a\n- v07 --refutes--> v05\n- v0a --supports--> v03\n- v06 --supports--> v01\n- v05 --refutes--> v08\n- v00 --supports--> v04\n- v01 --supports--> v07\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"be34042075e774dbb1d5936b0b9d32e53abcd4f3a0eb63edeb44d06adbdb6e6f","schema":"livegraph.boolean-propagation.v1","spec_hash":"fd0434de5341793f75183ff01999937987f2bdf200bd1bc144b6260449b4103e","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-e68e7f1c038ecaaf5431","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v00\n- v00 --refutes--> v0a\n- v09 --refutes--> v04\n- v01 --refutes--> v09\n- v0a --refutes--> v01\n- v04 --refutes--> v07\n- v01 --supports--> v05\n- v03 --refutes--> v06\n- v0a --supports--> v08\n- v06 --supports--> v02\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"761eaef34a38eb3ca415f99063c88c9cca10b971248fe0cb57e93fc654a21d90","schema":"livegraph.boolean-propagation.v1","spec_hash":"e68e7f1c038ecaaf543157b88421a775292a1a6c8ea42b7cf45c31a571609f1c","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-f68917fc4a03cb11af17","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v09\n- v03 --refutes--> v0a\n- v01 --refutes--> v07\n- v0a --supports--> v04\n- v04 --supports--> v01\n- v08 --supports--> v05\n- v01 --supports--> v06\n- v01 --refutes--> v02\n- v04 --supports--> v00\n- v05 --refutes--> v03\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"388cf9676f779dd98b69629ff70219b00913d0585e6179d4e0a9edd0a7ada9ff","schema":"livegraph.boolean-propagation.v1","spec_hash":"f68917fc4a03cb11af1716987e678cf073ba4c8cf88caf4bff7a5050a582dd4b","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-cb40654b52868ec434c1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v06\n- v00 --supports--> v08\n- v04 --supports--> v03\n- v0a --refutes--> v02\n- v05 --supports--> v01\n- v04 --refutes--> v07\n- v02 --supports--> v09\n- v01 --supports--> v04\n- v09 --supports--> v05\n- v04 --supports--> v00\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"61c65d569ac42caedb9baba64cb6ee14a7e453eae27ce7c99470b8ed66d18a2f","schema":"livegraph.boolean-propagation.v1","spec_hash":"cb40654b52868ec434c1626ba6b8cc900ee192aaae76c56f75dbe7b96cd34a6d","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-b2fc456a1d7042bdebac","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v08\n- v01 --refutes--> v07\n- v03 --supports--> v01\n- v00 --supports--> v0a\n- v02 --refutes--> v03\n- v09 --supports--> v02\n- v06 --refutes--> v00\n- v00 --refutes--> v09\n- v07 --refutes--> v05\n- v07 --supports--> v04\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a031fcbacb73e4336cd81befab88c958bc66897b21c2831fe73a8981eeb1a12d","schema":"livegraph.boolean-propagation.v1","spec_hash":"b2fc456a1d7042bdebac57d3787951dde015351eafe8888f7c82fef3a5d5cda8","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-8aeb04d859b168aec3d7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v02\n- v04 --refutes--> v0a\n- v02 --supports--> v00\n- v08 --refutes--> v07\n- v0a --refutes--> v01\n- v09 --supports--> v03\n- v01 --supports--> v09\n- v0a --refutes--> v06\n- v03 --refutes--> v08\n- v02 --refutes--> v05\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"221f505f1b6af221f5d6ffe627e51579ae651b68aa8fe3797dd397676a97a3db","schema":"livegraph.boolean-propagation.v1","spec_hash":"8aeb04d859b168aec3d7c0086a2c0fe9982ab01c5e4fc9f51923aa4917b5de37","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-dfb4645148e211e6233d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v05\n- v03 --supports--> v06\n- v03 --refutes--> v01\n- v05 --refutes--> v0a\n- v09 --supports--> v07\n- v0a --supports--> v03\n- v02 --supports--> v08\n- v01 --supports--> v02\n- v04 --supports--> v09\n- v0a --supports--> v00\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e95be04437e66c1988a61da0b94322840938ca61e09b9b6efe058c90632bad77","schema":"livegraph.boolean-propagation.v1","spec_hash":"dfb4645148e211e6233d4475817d05521686dd466ad1f640a1503dc9400bb3dd","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-2eb664e26290b1888197","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v07\n- v06 --supports--> v09\n- v09 --supports--> v00\n- v09 --refutes--> v03\n- v00 --refutes--> v0a\n- v06 --refutes--> v04\n- v06 --refutes--> v02\n- v01 --supports--> v05\n- v0a --refutes--> v08\n- v08 --refutes--> v01\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2e7a69ccbc4e49593dbdb2f29465c043a64889b408b0afa224887fe488afc6cb","schema":"livegraph.boolean-propagation.v1","spec_hash":"2eb664e26290b188819700024231177f6b5244f3c864e8707a9adbdac6ae85f4","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-cb3be188d88f06454e0d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v00\n- v04 --supports--> v06\n- v00 --supports--> v05\n- v07 --supports--> v04\n- v07 --refutes--> v09\n- v06 --refutes--> v03\n- v02 --refutes--> v01\n- v0a --supports--> v08\n- v06 --supports--> v0a\n- v02 --refutes--> v07\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"da24816bdd157c8939069234608c00d5d5ffd70da0070396cb5cb9d2aff2a09e","schema":"livegraph.boolean-propagation.v1","spec_hash":"cb3be188d88f06454e0dbda4a17f5841cf94e407b92619d3991afba4d8a6828d","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-dd0fab52c8e6c73ca273","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v03\n- v0a --supports--> v07\n- v04 --refutes--> v01\n- v01 --refutes--> v00\n- v06 --refutes--> v04\n- v00 --refutes--> v09\n- v08 --refutes--> v06\n- v07 --refutes--> v05\n- v01 --refutes--> v02\n- v00 --refutes--> v0a\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bf4923bb246a5700db3ccd4cccb11f2c479f785029e717e7745da478870bba46","schema":"livegraph.boolean-propagation.v1","spec_hash":"dd0fab52c8e6c73ca273ff1f20e863b5a5f5484c7c7f39c38b92c38e60e548f2","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-1a100df7d926ecede45d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v00\n- v03 --refutes--> v05\n- v05 --supports--> v08\n- v03 --refutes--> v09\n- v02 --supports--> v07\n- v06 --refutes--> v0a\n- v02 --refutes--> v01\n- v08 --refutes--> v06\n- v04 --supports--> v02\n- v00 --refutes--> v04\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6079541bf584bd8e8b0b1ed2cfe14c80866cf6c1307a326176610533875e53c4","schema":"livegraph.boolean-propagation.v1","spec_hash":"1a100df7d926ecede45df99a78ede2813f26af503edefd64880bc7ff5b47b930","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":7,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-940f7d80e31181aa9737","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v07\n- v0a --refutes--> v04\n- v08 --supports--> v02\n- v06 --refutes--> v09\n- v06 --refutes--> v00\n- v01 --refutes--> v0a\n- v05 --supports--> v06\n- v04 --supports--> v05\n- v06 --refutes--> v08\n- v03 --supports--> v01\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2757fc8887ab54ec7189133dfe30c3efe116c4e8ad69de256dae9380b1112317","schema":"livegraph.boolean-propagation.v1","spec_hash":"940f7d80e31181aa9737bf2d17099121a3cf833f3b8eb917d8df9b0d32f02543","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-8cd27d2cdbbba7091821","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v01\n- v06 --supports--> v00\n- v04 --refutes--> v05\n- v02 --supports--> v08\n- v08 --refutes--> v06\n- v0a --supports--> v09\n- v04 --supports--> v07\n- v05 --supports--> v0a\n- v03 --refutes--> v02\n- v0a --supports--> v03\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dba6a5f9f194538885b724ab7fa14c23be8136ec5d88257ac9450336cd009255","schema":"livegraph.boolean-propagation.v1","spec_hash":"8cd27d2cdbbba7091821d48fa74a68feca9ee44124b51589f26c4533c027a02f","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-e731db0081be4a3c8d36","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v07\n- v01 --refutes--> v02\n- v02 --supports--> v05\n- v05 --refutes--> v08\n- v08 --supports--> v09\n- v0a --supports--> v01\n- v04 --refutes--> v0a\n- v02 --refutes--> v06\n- v0a --supports--> v00\n- v09 --supports--> v03\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f9b41f702d521e9874e4e47fd52983396528356f7623f24355aa886d8a054cc2","schema":"livegraph.boolean-propagation.v1","spec_hash":"e731db0081be4a3c8d362e8425735de9933b65a66536b16e65d347bb75062318","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-08f3e36ab8eb655e2a9e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v04\n- v05 --refutes--> v09\n- v04 --supports--> v08\n- v0a --supports--> v01\n- v00 --refutes--> v02\n- v08 --supports--> v07\n- v07 --refutes--> v0a\n- v04 --supports--> v06\n- v02 --supports--> v05\n- v00 --refutes--> v03\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"59cff44872fa2e10459f184fc4129f9f65ac7ff7ca553e7fb8539fc0575d8bba","schema":"livegraph.boolean-propagation.v1","spec_hash":"08f3e36ab8eb655e2a9ec6805064c2687a2c24104807e484a5a3fd7c5142c524","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-5a76941de1b89d91316b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v02\n- v04 --supports--> v00\n- v07 --supports--> v06\n- v03 --refutes--> v05\n- v04 --refutes--> v09\n- v02 --refutes--> v07\n- v08 --refutes--> v03\n- v0a --supports--> v04\n- v07 --supports--> v01\n- v06 --supports--> v0a\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"838667254d9140c7186b5c9420a7eeb35b4f56f0ea6fa332482f929977abbbb7","schema":"livegraph.boolean-propagation.v1","spec_hash":"5a76941de1b89d91316bf0fdf16249d470810dc634b07c91549cf8f5aa85db13","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-51a6077a64ed1d745eaf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v02\n- v09 --refutes--> v08\n- v09 --supports--> v0a\n- v05 --supports--> v09\n- v05 --supports--> v03\n- v07 --refutes--> v04\n- v07 --supports--> v01\n- v01 --refutes--> v06\n- v08 --refutes--> v00\n- v00 --refutes--> v07\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"aa61d0a5b8ae6b9ad13855b11cbccd66e3db06d05fd0903813778bcc5cf7b8d6","schema":"livegraph.boolean-propagation.v1","spec_hash":"51a6077a64ed1d745eaf181260497fe936a4b7e6e330a16cc74fea8d1a72162d","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-ad7f2c0b7327cb1b3a7d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v04\n- v08 --refutes--> v0a\n- v0a --refutes--> v01\n- v0a --supports--> v07\n- v00 --refutes--> v09\n- v04 --supports--> v08\n- v00 --supports--> v05\n- v09 --refutes--> v06\n- v01 --supports--> v02\n- v00 --supports--> v03\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f8738043d6d02ffca7e72c3e0a218ac77bbd28a9a31ce36cdee6415f9beb1aba","schema":"livegraph.boolean-propagation.v1","spec_hash":"ad7f2c0b7327cb1b3a7d478f0cedd45de2872bcb26e97cafe138aa36a8785496","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-c4e54be6da7b6554b566","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v09\n- v02 --supports--> v06\n- v05 --refutes--> v03\n- v07 --supports--> v02\n- v09 --refutes--> v08\n- v03 --refutes--> v07\n- v05 --refutes--> v01\n- v05 --supports--> v0a\n- v05 --refutes--> v04\n- v06 --supports--> v00\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1acc5c597cd4520b966c215a9a39ef7f0329208a079fd40e5f3d18d69aeb394f","schema":"livegraph.boolean-propagation.v1","spec_hash":"c4e54be6da7b6554b5666daec262c2ecff2910968d08cc4ac268d147e90f2706","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-6912748dd910cca9a40e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v02\n- v00 --supports--> v08\n- v08 --refutes--> v03\n- v04 --supports--> v07\n- v06 --refutes--> v09\n- v07 --supports--> v00\n- v05 --refutes--> v04\n- v07 --supports--> v0a\n- v09 --refutes--> v05\n- v09 --supports--> v01\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1e2b095e4145a340d3e463b3b5993fa139fa1cefbb0d5b6cc351d1e8ed4ce463","schema":"livegraph.boolean-propagation.v1","spec_hash":"6912748dd910cca9a40e86fa5f29097bfbc9feabe8f46264acb88f3ab56b1f50","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-4b092d69cd932e393622","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v06\n- v06 --refutes--> v05\n- v04 --refutes--> v07\n- v03 --supports--> v09\n- v06 --refutes--> v01\n- v06 --refutes--> v0a\n- v09 --supports--> v00\n- v00 --supports--> v04\n- v03 --refutes--> v08\n- v01 --refutes--> v02\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"72874b611389dcd4bb6d37c1705ddfa3835997603dbd0f71a7e62bba4d6e8115","schema":"livegraph.boolean-propagation.v1","spec_hash":"4b092d69cd932e3936227af8ee512d5b0145aa3eeecffbf3277f5f13b6911987","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-8a352056f79e7e918443","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v00\n- v08 --supports--> v01\n- v01 --supports--> v0a\n- v06 --supports--> v08\n- v00 --supports--> v03\n- v01 --supports--> v02\n- v05 --refutes--> v06\n- v07 --supports--> v05\n- v08 --supports--> v04\n- v05 --refutes--> v09\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"63d647fcbb587783ec12012bf5159857f01734746f1a887b578c33e85801b001","schema":"livegraph.boolean-propagation.v1","spec_hash":"8a352056f79e7e918443334dfb73bac5fc9b7ed78acb56f51867aff50431ea1f","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-6a4257f14c45d360e6e4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v00\n- v01 --supports--> v02\n- v09 --supports--> v06\n- v00 --refutes--> v08\n- v00 --refutes--> v0a\n- v04 --supports--> v01\n- v07 --supports--> v05\n- v05 --refutes--> v09\n- v00 --refutes--> v03\n- v03 --supports--> v07\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"239a9575ca01a3e9f222944c1148ccdda339b8cfe8ef344fc3d40a9792002a57","schema":"livegraph.boolean-propagation.v1","spec_hash":"6a4257f14c45d360e6e40d19649a7879729d3820bd3cd852ce69cf515daa42d4","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-08e13c3571ca6354a749","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v06\n- v07 --refutes--> v09\n- v03 --supports--> v08\n- v03 --supports--> v00\n- v03 --supports--> v01\n- v02 --supports--> v07\n- v05 --supports--> v03\n- v04 --refutes--> v02\n- v08 --supports--> v0a\n- v0a --refutes--> v04\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7bb4a3b46f3259105432865399b395732adf88e34dd6ba76585355c9f04fd93c","schema":"livegraph.boolean-propagation.v1","spec_hash":"08e13c3571ca6354a7497de82a4a44bae016b40c334915ab1133788b427f60dc","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-2cf80413a702ba1bf11b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v01\n- v09 --supports--> v03\n- v06 --refutes--> v0a\n- v03 --supports--> v00\n- v02 --supports--> v05\n- v06 --supports--> v07\n- v04 --supports--> v08\n- v00 --refutes--> v02\n- v0a --supports--> v04\n- v04 --supports--> v09\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4d478e3ad1a2ef1af64b4d36b4e772f3f20ccd39cc091aa025ecbe4dbbdacccb","schema":"livegraph.boolean-propagation.v1","spec_hash":"2cf80413a702ba1bf11b688f7ce47fba232f3821f42418ddb37795bf0b67f701","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-29bba276b9f374c27cf1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v06\n- v05 --refutes--> v02\n- v00 --supports--> v07\n- v07 --supports--> v05\n- v04 --refutes--> v00\n- v03 --supports--> v04\n- v02 --refutes--> v08\n- v04 --refutes--> v09\n- v01 --refutes--> v0a\n- v01 --supports--> v03\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fa38b038c3a163220d58f2a3b0f89d1db5aeb663f2fec950045a228c1835ecb7","schema":"livegraph.boolean-propagation.v1","spec_hash":"29bba276b9f374c27cf140daf5d627ced14fce9f9e07ecb5086e8486f8b5df31","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":7,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-328acd12741b2b1917c8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v08\n- v05 --refutes--> v00\n- v05 --refutes--> v03\n- v01 --refutes--> v04\n- v06 --supports--> v02\n- v05 --refutes--> v09\n- v0a --supports--> v01\n- v04 --supports--> v07\n- v09 --supports--> v0a\n- v02 --supports--> v05\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"80fb0ce8aac1afa709e8d71168ae3d63e9c591a23c48df600fb5d6f019f287b2","schema":"livegraph.boolean-propagation.v1","spec_hash":"328acd12741b2b1917c8437808993d86548321e41a4bb204e40003a6dc0bd5d3","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-4e2998c515544fddd49f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v08\n- v04 --refutes--> v07\n- v02 --supports--> v09\n- v09 --refutes--> v00\n- v03 --refutes--> v01\n- v05 --supports--> v02\n- v03 --refutes--> v06\n- v09 --refutes--> v0a\n- v0a --supports--> v03\n- v06 --supports--> v04\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f65433abeac18efed66995fa8e4c4f5636fa414a81632954969a26c19c4e1805","schema":"livegraph.boolean-propagation.v1","spec_hash":"4e2998c515544fddd49ff87057fc696b0d091f4d5d47d072835ad5d38411efa7","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-c9815f7d75d1ffe8316c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v01\n- v05 --supports--> v07\n- v07 --supports--> v09\n- v02 --refutes--> v04\n- v07 --supports--> v03\n- v00 --refutes--> v05\n- v09 --refutes--> v02\n- v0a --supports--> v00\n- v02 --refutes--> v06\n- v08 --supports--> v0a\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0fd7278fa7d2c252331e93a770e2d1101672e1c18091b353bb2c5b7ab2f6da59","schema":"livegraph.boolean-propagation.v1","spec_hash":"c9815f7d75d1ffe8316c8789ddee763bb3946bf4300b2716ac290c9f51fc8a0c","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-001d33092f051f0e0210","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v0a\n- v05 --refutes--> v02\n- v09 --supports--> v08\n- v09 --supports--> v06\n- v03 --refutes--> v00\n- v00 --supports--> v01\n- v02 --refutes--> v03\n- v09 --refutes--> v07\n- v06 --refutes--> v05\n- v02 --supports--> v04\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f955af8bdc674111e0f0f1697666a8b4f1468ddfb6378e50ea729dab67491425","schema":"livegraph.boolean-propagation.v1","spec_hash":"001d33092f051f0e02100f697404121e55850b31232cd94a2a41d0004d07602d","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-90e040e54a188d164a0c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v04\n- v07 --supports--> v00\n- v02 --supports--> v0a\n- v01 --refutes--> v02\n- v00 --supports--> v06\n- v05 --refutes--> v09\n- v09 --refutes--> v08\n- v02 --refutes--> v03\n- v0a --refutes--> v05\n- v09 --refutes--> v07\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e2c59643c7bf8caf35cf94471f44730c0c743a98ca9c451cd10b7566fb0b78ab","schema":"livegraph.boolean-propagation.v1","spec_hash":"90e040e54a188d164a0c122b02e97a65106b62bbb0d85a0de4d3dcb700f9673c","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-6deeaf405a5658a8edda","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v04\n- v05 --supports--> v01\n- v04 --supports--> v09\n- v03 --supports--> v08\n- v09 --refutes--> v00\n- v05 --supports--> v0a\n- v01 --supports--> v06\n- v09 --supports--> v07\n- v01 --refutes--> v02\n- v00 --refutes--> v03\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dfdcff1846a7053b3a1f6f9f2855e1bc204e6fb97a73a1945a7c9c5e2e90e35f","schema":"livegraph.boolean-propagation.v1","spec_hash":"6deeaf405a5658a8eddae8e4dcee1bf8efe5a9e6dedeb06fa15a7dede0ed0587","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-5c166a08b816f4836e3d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v01\n- v05 --refutes--> v00\n- v00 --supports--> v03\n- v02 --refutes--> v04\n- v00 --supports--> v07\n- v08 --supports--> v02\n- v06 --refutes--> v0a\n- v00 --refutes--> v09\n- v03 --supports--> v08\n- v01 --supports--> v06\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a13e46e8cac73150ad4d81c71a85685846db11d06cd4a52993893f82448ef766","schema":"livegraph.boolean-propagation.v1","spec_hash":"5c166a08b816f4836e3dff21e22385a11b996bea5a841278a3ae32f3752e7843","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":7,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-10c66beeec7cc196b29e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v0a\n- v07 --refutes--> v06\n- v08 --supports--> v05\n- v0a --supports--> v04\n- v07 --supports--> v01\n- v09 --supports--> v08\n- v04 --refutes--> v02\n- v04 --supports--> v07\n- v05 --refutes--> v03\n- v09 --refutes--> v00\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"799f122f864017b70b2fba8156d57267d7fc8b2b9ce371698d9a011b8cdf85a7","schema":"livegraph.boolean-propagation.v1","spec_hash":"10c66beeec7cc196b29ec424a06b719680e77f99189879bd3fa88a391d0e9c43","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-67862de2f083c041b83b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v02\n- v06 --refutes--> v07\n- v09 --supports--> v08\n- v00 --refutes--> v01\n- v00 --supports--> v09\n- v06 --refutes--> v00\n- v08 --refutes--> v03\n- v08 --refutes--> v05\n- v02 --supports--> v0a\n- v0a --refutes--> v06\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3a8e9e8f1c55e1764a2ea32caf9759b2112e450697b55234c7510a1085680615","schema":"livegraph.boolean-propagation.v1","spec_hash":"67862de2f083c041b83beb057c00015878a0ad12ad5f49017669973e03103b9d","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-3b8d6a6b8d4c857b9321","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v04\n- v09 --supports--> v08\n- v04 --refutes--> v09\n- v06 --supports--> v07\n- v07 --refutes--> v03\n- v04 --supports--> v0a\n- v03 --supports--> v02\n- v06 --refutes--> v00\n- v08 --supports--> v06\n- v07 --refutes--> v05\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a58de4df10e2e3f97f8f0866e11ed28d5b67eb275487234b8913a7c90b42196d","schema":"livegraph.boolean-propagation.v1","spec_hash":"3b8d6a6b8d4c857b9321ffd8386d199324af5c3805699b4e90a37617593f69f0","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-322a30e9bc204c1fe327","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v01\n- v02 --refutes--> v04\n- v05 --supports--> v00\n- v04 --refutes--> v09\n- v00 --refutes--> v03\n- v03 --supports--> v02\n- v09 --refutes--> v08\n- v02 --supports--> v06\n- v07 --refutes--> v05\n- v09 --supports--> v0a\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3dafdceb921cd9f30eb37d4bf3b1282fa9faf31fd2d4119cb4465c672d73845e","schema":"livegraph.boolean-propagation.v1","spec_hash":"322a30e9bc204c1fe32727ab79d3f6315d499f4d3660bb1db3429a03f79f7100","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-2431577887f4da348b94","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v04\n- v01 --refutes--> v05\n- v06 --supports--> v07\n- v07 --supports--> v03\n- v05 --supports--> v00\n- v06 --refutes--> v08\n- v09 --refutes--> v06\n- v02 --supports--> v09\n- v0a --supports--> v02\n- v07 --refutes--> v01\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"37cce5c1a9a3af702d1591dcfa3dc8d7fbe01e90801d8607714225a860283135","schema":"livegraph.boolean-propagation.v1","spec_hash":"2431577887f4da348b9414fe6276de8337ec77cd89fff3f8f4d8f17c5fadf11e","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-57f2451a5adba3aa3596","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v02\n- v02 --supports--> v06\n- v03 --supports--> v05\n- v00 --refutes--> v04\n- v02 --refutes--> v09\n- v08 --supports--> v01\n- v09 --supports--> v08\n- v09 --supports--> v07\n- v0a --refutes--> v03\n- v05 --refutes--> v00\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"172b00d3ba0109321549954a83309c61933cc9073ab13b85b8c3c1d7cb7c4d24","schema":"livegraph.boolean-propagation.v1","spec_hash":"57f2451a5adba3aa3596fe976b3650bc8a3ba20734a9087582019859a9fe64ce","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-e043ed053bda326d78fd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v04\n- v09 --supports--> v0a\n- v02 --supports--> v01\n- v01 --supports--> v08\n- v07 --supports--> v03\n- v0a --refutes--> v00\n- v00 --refutes--> v06\n- v01 --supports--> v07\n- v06 --supports--> v02\n- v06 --refutes--> v05\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2b042b4d70bf2014b15c7197dc9f32bfb89f99329c38ae66dbe65110a2cf5075","schema":"livegraph.boolean-propagation.v1","spec_hash":"e043ed053bda326d78fda98bb0be24539e7fc433a3cc70060712bb35376f8335","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-bb8741cf98891700ad2a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v05\n- v09 --refutes--> v06\n- v04 --supports--> v07\n- v00 --refutes--> v0a\n- v04 --supports--> v08\n- v0a --supports--> v03\n- v0a --supports--> v09\n- v08 --supports--> v00\n- v05 --refutes--> v04\n- v02 --refutes--> v01\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0485303dedff897f02b01d2091b30cdee6a5612b375e5c6b536ea6e1482f897c","schema":"livegraph.boolean-propagation.v1","spec_hash":"bb8741cf98891700ad2ace20319cd46dc6085f5f9b9219768256a388694026a2","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-c037a1e032e2175c6449","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v05\n- v01 --supports--> v07\n- v07 --refutes--> v0a\n- v06 --supports--> v02\n- v05 --supports--> v06\n- v05 --supports--> v04\n- v09 --supports--> v00\n- v05 --refutes--> v08\n- v02 --supports--> v03\n- v09 --refutes--> v01\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"137c09865a57c269c687267dabef835225845ec78b8e2999c1db87c80cfe96e5","schema":"livegraph.boolean-propagation.v1","spec_hash":"c037a1e032e2175c64491d6d695318c40ef63dc3da9de5398b3ca0d99445bc78","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-03f32a5369051bde1426","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v02\n- v04 --refutes--> v05\n- v05 --refutes--> v00\n- v08 --supports--> v07\n- v02 --refutes--> v09\n- v09 --refutes--> v01\n- v09 --supports--> v06\n- v04 --supports--> v03\n- v01 --supports--> v08\n- v04 --refutes--> v0a\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"058667f8f719fa5c9a4926ac4c6a13843c657e6e8cd01c26ebe2d653a7efcc50","schema":"livegraph.boolean-propagation.v1","spec_hash":"03f32a5369051bde1426a130783e928c85f5058cb1df0c70be4bab036a3d4ea1","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-b0b6839877cdb86dca43","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v04\n- v00 --refutes--> v05\n- v05 --refutes--> v07\n- v06 --supports--> v0a\n- v0a --refutes--> v01\n- v0a --supports--> v03\n- v0a --refutes--> v08\n- v07 --refutes--> v06\n- v06 --refutes--> v02\n- v04 --refutes--> v00\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fbaf259eee77ef5c540f5dc05c463b05990fe2e71ff73a78d4c84721def1e80c","schema":"livegraph.boolean-propagation.v1","spec_hash":"b0b6839877cdb86dca43248c5c2b8aa75cf667156a5a3f456bc087527e9badad","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-8f55c29f1712f421fb4f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v08\n- v02 --refutes--> v00\n- v00 --supports--> v06\n- v09 --refutes--> v05\n- v04 --supports--> v01\n- v06 --supports--> v09\n- v07 --refutes--> v04\n- v0a --refutes--> v02\n- v09 --supports--> v07\n- v04 --refutes--> v03\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"27cdc0eee3dd6aca7c8a5c768434fcfb1ea5e2d7b0128a462b4eb25902323d7e","schema":"livegraph.boolean-propagation.v1","spec_hash":"8f55c29f1712f421fb4f99f7f40b767dd296fe9b4210cbeed81861ceb7430490","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-b7b810848cc5593043e4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v02\n- v09 --refutes--> v01\n- v07 --refutes--> v04\n- v0a --supports--> v07\n- v04 --refutes--> v06\n- v05 --supports--> v09\n- v07 --supports--> v00\n- v09 --refutes--> v03\n- v01 --refutes--> v08\n- v06 --refutes--> v05\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5cbccf4ae5369d0939a70daf48ed4eaffe06f21e2589b798da4cc4295cab47f4","schema":"livegraph.boolean-propagation.v1","spec_hash":"b7b810848cc5593043e4cee7a5de11e9ead57e0504bc1d8837e666ed184112ee","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-40776dc4203347fbd0cc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v09\n- v09 --refutes--> v01\n- v07 --refutes--> v02\n- v0a --supports--> v08\n- v07 --refutes--> v04\n- v00 --supports--> v0a\n- v07 --supports--> v05\n- v08 --supports--> v03\n- v02 --supports--> v00\n- v08 --refutes--> v06\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"277048e72465024499333e46827c121b01267ccb2ad111b3c1f6b24c1b2081c4","schema":"livegraph.boolean-propagation.v1","spec_hash":"40776dc4203347fbd0cc183a3770fd7f44098f1cca7794d1e6312ac682124dfc","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-02995f891e5a91084d10","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v06\n- v0a --supports--> v03\n- v07 --refutes--> v00\n- v08 --refutes--> v02\n- v07 --supports--> v0a\n- v01 --supports--> v09\n- v02 --supports--> v01\n- v03 --supports--> v08\n- v04 --refutes--> v07\n- v03 --refutes--> v05\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0e6523d673465159891aa8d46ac22dbc7305643ab688d25a6945ff57fe27b93c","schema":"livegraph.boolean-propagation.v1","spec_hash":"02995f891e5a91084d106b91631c35f898449e9cd486b10dc1723374aae5a375","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-f4ecbb8139f6921277c0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v08\n- v02 --refutes--> v04\n- v0a --supports--> v03\n- v02 --supports--> v05\n- v07 --supports--> v0a\n- v09 --supports--> v02\n- v07 --refutes--> v00\n- v06 --supports--> v09\n- v0a --supports--> v01\n- v08 --supports--> v07\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0fcf01aa8f2460e94fbc3449cccbab13495ea62fcb2c77843acdeb02548de1bd","schema":"livegraph.boolean-propagation.v1","spec_hash":"f4ecbb8139f6921277c0a0b3a615a04cb0ded5796ddc4ed1217f77946e671a64","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-4b7c68ad2077ecce8630","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v01\n- v03 --refutes--> v06\n- v04 --supports--> v09\n- v02 --refutes--> v08\n- v01 --supports--> v04\n- v0a --supports--> v05\n- v06 --supports--> v00\n- v04 --supports--> v07\n- v02 --supports--> v03\n- v04 --refutes--> v02\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c62f4c8715e3f6e024a71b048cbbce63d375ea1f50c78f2363c37f7aa8887e35","schema":"livegraph.boolean-propagation.v1","spec_hash":"4b7c68ad2077ecce86300c53b20cdda6689a9cb3afc1646aacfd37e009a92723","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-e2e899e9873ce4cff3a6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v01\n- v09 --refutes--> v0a\n- v04 --supports--> v07\n- v0a --supports--> v06\n- v01 --supports--> v08\n- v05 --supports--> v00\n- v0a --supports--> v05\n- v02 --refutes--> v04\n- v01 --supports--> v02\n- v05 --supports--> v03\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"37f0726f6341bb5791520e87b0eeda4b1fc1658fb383ea85d1c6527eb7a53b2c","schema":"livegraph.boolean-propagation.v1","spec_hash":"e2e899e9873ce4cff3a6342ef2a5eb619f4322c2168aea1b65cbf8b8bd417697","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-2b48ee79b7c5b8928e1c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v04\n- v05 --refutes--> v03\n- v06 --supports--> v05\n- v00 --supports--> v0a\n- v00 --refutes--> v06\n- v07 --refutes--> v02\n- v08 --refutes--> v00\n- v05 --supports--> v01\n- v04 --supports--> v07\n- v02 --refutes--> v09\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a885bdb2cef1a22e1ee80f16ceffeb615e2e64a112841fdcd481c154815fee0a","schema":"livegraph.boolean-propagation.v1","spec_hash":"2b48ee79b7c5b8928e1cbca81e931376c6d54be99fd4506dbcb90459eda20999","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":7,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-b2377c36c2ee485ce08f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v08\n- v00 --refutes--> v03\n- v07 --refutes--> v00\n- v01 --supports--> v06\n- v04 --refutes--> v07\n- v06 --supports--> v09\n- v0a --refutes--> v04\n- v00 --refutes--> v05\n- v06 --refutes--> v02\n- v05 --refutes--> v01\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"85cd024180979f3c845f99c0a90bccccd0e0eff43e59dbe80b432258f67f7479","schema":"livegraph.boolean-propagation.v1","spec_hash":"b2377c36c2ee485ce08feec97b89951677096238d1cd1733e5b6248b92f446bc","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-ebdd28155fd77ab4f90e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v01\n- v04 --supports--> v05\n- v09 --refutes--> v08\n- v01 --refutes--> v02\n- v07 --refutes--> v04\n- v06 --refutes--> v0a\n- v09 --refutes--> v00\n- v07 --refutes--> v03\n- v08 --supports--> v07\n- v02 --refutes--> v09\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f130e9e4dfe7af0d00827652dedef7516ec3bfa1bee2de6767af857369f476b4","schema":"livegraph.boolean-propagation.v1","spec_hash":"ebdd28155fd77ab4f90ea753a6af5db11e55901afcf1227c15693effabab046d","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-cd6e5a767da92522ce0d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v02\n- v03 --refutes--> v04\n- v07 --refutes--> v0a\n- v09 --refutes--> v07\n- v04 --supports--> v00\n- v01 --supports--> v08\n- v05 --supports--> v01\n- v00 --refutes--> v09\n- v07 --supports--> v05\n- v03 --supports--> v06\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"759a6d850396bf0c30d14cf04baec79774847087bc54fbb7f0853d0deaadad70","schema":"livegraph.boolean-propagation.v1","spec_hash":"cd6e5a767da92522ce0d946ca6884f0c64ea9bcbbf2b358cbf31d5f6ff2fc4c0","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-e2d134d9d982e37ebf43","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v01\n- v04 --refutes--> v03\n- v03 --refutes--> v02\n- v02 --supports--> v06\n- v06 --supports--> v0a\n- v09 --supports--> v00\n- v0a --refutes--> v09\n- v00 --supports--> v05\n- v06 --supports--> v08\n- v0a --supports--> v07\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d5711149d97a1377cff3f019c3ec26e1c1c272d6a77f7550f7e086931f360a55","schema":"livegraph.boolean-propagation.v1","spec_hash":"e2d134d9d982e37ebf434f6cfe3ae30e7e29e7ee804e677caf8d96e4a66b777b","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-9de1e307121895b0b276","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v08\n- v05 --supports--> v00\n- v06 --refutes--> v03\n- v04 --refutes--> v02\n- v07 --supports--> v05\n- v06 --refutes--> v04\n- v0a --supports--> v06\n- v01 --supports--> v0a\n- v04 --supports--> v09\n- v00 --supports--> v01\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2595bffda05859f6512d61f0963a6cc1c051467b5962951f2009c69aaef7d145","schema":"livegraph.boolean-propagation.v1","spec_hash":"9de1e307121895b0b276478bdd1157c7532e791a0699c39012304dbe6721bcee","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-0db6c5daa05a184075b5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v09\n- v06 --refutes--> v07\n- v04 --supports--> v00\n- v03 --refutes--> v04\n- v07 --supports--> v0a\n- v08 --supports--> v02\n- v09 --supports--> v03\n- v05 --supports--> v06\n- v04 --supports--> v05\n- v05 --refutes--> v01\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"11f82e58cd5897d4d87ecb040398438b669c8551050edf5402a20fc09fe913fc","schema":"livegraph.boolean-propagation.v1","spec_hash":"0db6c5daa05a184075b5b895c23da1b21fea227177b6b86f5171701df8394e82","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-cf8b76293246e0061544","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v06\n- v06 --refutes--> v04\n- v03 --refutes--> v05\n- v0a --supports--> v07\n- v03 --supports--> v0a\n- v07 --refutes--> v01\n- v06 --refutes--> v03\n- v08 --refutes--> v02\n- v00 --supports--> v09\n- v00 --supports--> v08\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"eb738830401672c846dc3c558c7f113b39f32d56b225a841fceabb2c6a8239b0","schema":"livegraph.boolean-propagation.v1","spec_hash":"cf8b76293246e006154460b5baf0770f83a745cea10610933cfc21c14102c156","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-af926b5834e841505041","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v08\n- v01 --refutes--> v07\n- v06 --supports--> v02\n- v09 --refutes--> v04\n- v07 --supports--> v05\n- v03 --refutes--> v0a\n- v09 --refutes--> v06\n- v05 --supports--> v09\n- v01 --refutes--> v00\n- v0a --supports--> v01\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"217a9b59e3459e50b67de1f206e541ddaf0850c1ba99024d500ffdc30708a1f2","schema":"livegraph.boolean-propagation.v1","spec_hash":"af926b5834e8415050412949f0a430941a19fb661973e79ca355bbceadf47a4d","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-b102cc85f98a3008081a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v03\n- v02 --supports--> v08\n- v08 --supports--> v01\n- v03 --refutes--> v00\n- v02 --supports--> v04\n- v02 --supports--> v06\n- v00 --refutes--> v02\n- v00 --supports--> v07\n- v01 --refutes--> v05\n- v09 --refutes--> v0a\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"78ff9ab34fbec037395b98c6d7b6922f09568cc3f2934baa8f5b1cb9df0ae0cf","schema":"livegraph.boolean-propagation.v1","spec_hash":"b102cc85f98a3008081ad1146af6391f79d2f1b0a87e44f42e31963bf04a98fd","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-ea0629968391d55883de","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v07\n- v00 --refutes--> v04\n- v09 --supports--> v06\n- v05 --refutes--> v0a\n- v00 --supports--> v09\n- v0a --supports--> v08\n- v06 --supports--> v03\n- v08 --refutes--> v02\n- v06 --supports--> v01\n- v03 --supports--> v05\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6b76684361edaa599f11f11e1e9b4e8229d11a5ca99abc5a5e7dbe9a3b32eb0e","schema":"livegraph.boolean-propagation.v1","spec_hash":"ea0629968391d55883de9d1311c64b74599aba552bdec695737f1609d644c6fd","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-1e4c68e5ecf652bf5e7c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v08\n- v0a --refutes--> v04\n- v08 --supports--> v00\n- v04 --refutes--> v05\n- v06 --supports--> v02\n- v04 --refutes--> v01\n- v08 --supports--> v09\n- v05 --refutes--> v07\n- v00 --supports--> v06\n- v08 --supports--> v03\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"159d4e3e7cc494b4ca3e7fa20ccb1bbb7e47cf5f45b6767430331d6c0151df87","schema":"livegraph.boolean-propagation.v1","spec_hash":"1e4c68e5ecf652bf5e7c88b33339fccc0c9f87bfc584199d42c24f60668df349","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-b84ab1bf50c91bea2701","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v00\n- v04 --supports--> v09\n- v00 --refutes--> v01\n- v06 --supports--> v02\n- v01 --refutes--> v06\n- v04 --refutes--> v05\n- v06 --supports--> v04\n- v08 --refutes--> v0a\n- v05 --refutes--> v03\n- v08 --refutes--> v07\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"28bfaf27758109ecdc063b49702d8f04dd4b324ee0c1c417a3bf1ccb2138c7d7","schema":"livegraph.boolean-propagation.v1","spec_hash":"b84ab1bf50c91bea270121831eee5e8c0ddaf7874ad7c777b2c86a3022c52b17","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-f728bf96bbdfcf05bf50","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v00\n- v07 --refutes--> v05\n- v00 --supports--> v04\n- v05 --supports--> v06\n- v03 --refutes--> v0a\n- v02 --supports--> v09\n- v03 --refutes--> v07\n- v09 --refutes--> v03\n- v03 --supports--> v01\n- v05 --refutes--> v08\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e5a687847778240e5ff502ec4d9b677b38885c18955c1d12464346dfacc7e5e0","schema":"livegraph.boolean-propagation.v1","spec_hash":"f728bf96bbdfcf05bf50fcd1f6196299a17eef91e966270c58fc67b84b4b5f2f","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-96e8aa5b934dd7712f4e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v02\n- v09 --supports--> v06\n- v0a --supports--> v01\n- v01 --refutes--> v08\n- v02 --refutes--> v00\n- v07 --supports--> v03\n- v01 --supports--> v04\n- v06 --refutes--> v07\n- v07 --supports--> v05\n- v00 --supports--> v09\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7cc86394c60b50d7037c6664564faa4925fbd48df2bfdfaedef27e1f0adfef5d","schema":"livegraph.boolean-propagation.v1","spec_hash":"96e8aa5b934dd7712f4e9340189c39a489263b4f233010adc866877f2b802782","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-7b350416d10b7240e9d5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v0a\n- v09 --supports--> v08\n- v00 --refutes--> v09\n- v01 --refutes--> v06\n- v00 --refutes--> v03\n- v0a --supports--> v07\n- v02 --refutes--> v04\n- v07 --supports--> v00\n- v08 --supports--> v02\n- v00 --supports--> v05\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"61595ba8b39ff32e676015ef998c77d206d22661bd4e7712c4fa4c680bfe8ba7","schema":"livegraph.boolean-propagation.v1","spec_hash":"7b350416d10b7240e9d5ba429dca69667789f47043b6383c52eb7c90cb44a678","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-f7333fec48e0d46f9458","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v08\n- v09 --supports--> v00\n- v09 --refutes--> v06\n- v02 --refutes--> v01\n- v07 --supports--> v04\n- v01 --refutes--> v09\n- v08 --supports--> v03\n- v05 --refutes--> v02\n- v03 --refutes--> v05\n- v05 --supports--> v0a\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ccdfb74e4038ec3a8b0c4ea4c81a94f1117318370cb4d82059c6ec7060c53d61","schema":"livegraph.boolean-propagation.v1","spec_hash":"f7333fec48e0d46f94581d0a3ed66d9b0dc06c827081d00302dfecd5d78a7f2c","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-006ebf6f1d6358ddef9b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v03\n- v08 --supports--> v09\n- v03 --supports--> v06\n- v07 --refutes--> v05\n- v04 --supports--> v08\n- v05 --supports--> v04\n- v01 --supports--> v0a\n- v04 --refutes--> v02\n- v05 --supports--> v00\n- v08 --supports--> v01\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"04084c0af3888e83a9bdb4594d58629891f283216794bd4cb3315f356b03224a","schema":"livegraph.boolean-propagation.v1","spec_hash":"006ebf6f1d6358ddef9b94fdd9606d6cb70c927369aad70c2acb4fcb1e5e7a7d","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-ae17275d1bb52a15e5cd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v03\n- v00 --refutes--> v09\n- v00 --refutes--> v07\n- v08 --supports--> v01\n- v0a --refutes--> v05\n- v07 --refutes--> v0a\n- v08 --refutes--> v02\n- v02 --refutes--> v00\n- v07 --supports--> v04\n- v03 --refutes--> v08\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6f89d83626ead1357c3a72b685f04cbeb7ce6aaf8f2a053c688ae580de632e4b","schema":"livegraph.boolean-propagation.v1","spec_hash":"ae17275d1bb52a15e5cd96bd0e9bb52a34096a76e5e109f165ab064d6958bc21","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-4b4838a3971855079145","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v09\n- v09 --supports--> v03\n- v00 --supports--> v06\n- v06 --supports--> v07\n- v01 --refutes--> v00\n- v04 --refutes--> v0a\n- v00 --supports--> v02\n- v06 --supports--> v04\n- v04 --refutes--> v08\n- v08 --supports--> v05\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"17a1fd5273d9657bf7e06901c7510428da0bb0069b63b7849e358a523c265ba7","schema":"livegraph.boolean-propagation.v1","spec_hash":"4b4838a3971855079145bf7809f1d82aea700bda0813c9cceb7eb53a3b9e6148","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-f09cefcb5b05198c135f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v06\n- v05 --supports--> v00\n- v00 --refutes--> v01\n- v07 --refutes--> v08\n- v01 --supports--> v03\n- v09 --supports--> v07\n- v01 --refutes--> v04\n- v01 --refutes--> v02\n- v0a --supports--> v05\n- v04 --refutes--> v09\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"90241061dd693b1a491b3911521ea4d83b25f1ac54b5d46955f619c5b9cc62ca","schema":"livegraph.boolean-propagation.v1","spec_hash":"f09cefcb5b05198c135ff30c982ffd140ca8ec4e1bfdd9d335c357de497b4816","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-3729c9f2026e3b55372a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v08\n- v01 --refutes--> v07\n- v04 --refutes--> v05\n- v01 --supports--> v03\n- v04 --supports--> v06\n- v02 --supports--> v0a\n- v09 --supports--> v00\n- v05 --refutes--> v02\n- v08 --refutes--> v01\n- v03 --refutes--> v04\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1fed4299c449850e201756ccf1131621f05830c61479fef5fc2767979c86677c","schema":"livegraph.boolean-propagation.v1","spec_hash":"3729c9f2026e3b55372a459533d88889e6c37ef960ed5b468040d330603b14d7","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-ca7cfe4411721a03e0ef","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v06\n- v01 --refutes--> v02\n- v03 --supports--> v08\n- v06 --refutes--> v03\n- v07 --supports--> v01\n- v03 --refutes--> v00\n- v01 --supports--> v05\n- v08 --supports--> v04\n- v01 --supports--> v0a\n- v09 --supports--> v07\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b1c5ed24684392911004e1b409ac7cb2a68fd02490829607c33b60c277dec9d1","schema":"livegraph.boolean-propagation.v1","spec_hash":"ca7cfe4411721a03e0ef7c9d2b8c3aac7f6143740f2cea840f6bce087e2f4ac4","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-b0555b64bf6fc0aafdec","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v01\n- v02 --supports--> v05\n- v07 --refutes--> v09\n- v01 --supports--> v06\n- v01 --supports--> v04\n- v06 --refutes--> v0a\n- v06 --supports--> v03\n- v03 --supports--> v02\n- v05 --refutes--> v07\n- v01 --refutes--> v00\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1c9f33d48e752f0b0ba8f090d6a4e2f58e1be2586478247d68711b9443f5e374","schema":"livegraph.boolean-propagation.v1","spec_hash":"b0555b64bf6fc0aafdec795083edd47f4660011b88546818e8c1bb13d23a2b11","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":7,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-7f487a277cca66dd2000","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v07\n- v06 --refutes--> v01\n- v09 --refutes--> v0a\n- v04 --refutes--> v06\n- v00 --refutes--> v08\n- v04 --refutes--> v03\n- v06 --refutes--> v02\n- v09 --supports--> v04\n- v07 --supports--> v00\n- v08 --supports--> v05\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1cbda1dc114feba6e9e397ce3fa6be3380bd5cd15f3434ae75f7c58d353bd212","schema":"livegraph.boolean-propagation.v1","spec_hash":"7f487a277cca66dd2000490bea969e738ff20b6d3b7e0882af91a273d71dbd3c","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-401345d80e3893f15090","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v02\n- v0a --supports--> v06\n- v03 --supports--> v0a\n- v06 --supports--> v09\n- v02 --refutes--> v05\n- v04 --supports--> v00\n- v00 --supports--> v03\n- v00 --supports--> v07\n- v03 --supports--> v01\n- v09 --supports--> v08\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1ed7830e143452e5a6de451a3d5193f3d526c60866bd0c504a2487bdefdd6504","schema":"livegraph.boolean-propagation.v1","spec_hash":"401345d80e3893f150900be1c11a80dbed27edb573df95cc7ec4105b06932ab4","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-655d8d72fc81dfc45d9a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v00\n- v08 --refutes--> v09\n- v06 --refutes--> v04\n- v0a --refutes--> v07\n- v06 --supports--> v03\n- v0a --supports--> v05\n- v01 --refutes--> v02\n- v09 --supports--> v06\n- v02 --refutes--> v0a\n- v05 --refutes--> v08\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"69bfb9fc4958a117e7313cc84cce7a6e005d316578229eb6dc8b9546f472fb8e","schema":"livegraph.boolean-propagation.v1","spec_hash":"655d8d72fc81dfc45d9a515d2e6839aafa5532c82b4b850bc435d9dfd32ad94e","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-a7339c0f710bf1f8ed53","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v08\n- v0a --supports--> v02\n- v04 --refutes--> v06\n- v09 --supports--> v0a\n- v0a --supports--> v01\n- v08 --supports--> v00\n- v02 --supports--> v07\n- v04 --refutes--> v09\n- v02 --refutes--> v03\n- v00 --supports--> v04\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"eeac86a139cef8746e9b51a939131553f41e634ffc5653922d6b2490a9a3f0de","schema":"livegraph.boolean-propagation.v1","spec_hash":"a7339c0f710bf1f8ed5321034f8ff33f2c6955d2a6ba5f79481ceecd219b67f9","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-ed3f45ac08be48a6ce6b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v01\n- v03 --refutes--> v05\n- v0a --supports--> v08\n- v04 --refutes--> v03\n- v00 --supports--> v09\n- v05 --supports--> v0a\n- v09 --refutes--> v06\n- v08 --refutes--> v00\n- v05 --refutes--> v02\n- v0a --refutes--> v07\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"29239cd4538477cd6a08ad5b19720f64997d1e845a1c233549723f88a0ea8cbd","schema":"livegraph.boolean-propagation.v1","spec_hash":"ed3f45ac08be48a6ce6b81a25feadeefe05b75cb4e1ff9dd835858413d0b10ff","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-3b9908b0cf9d0c914e9b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v07\n- v05 --supports--> v06\n- v01 --refutes--> v0a\n- v06 --refutes--> v08\n- v02 --supports--> v04\n- v02 --supports--> v01\n- v00 --supports--> v09\n- v07 --supports--> v03\n- v07 --supports--> v05\n- v06 --refutes--> v00\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2262f28061f0ca1efc095ed817879fa56b30db990e067d5205b9a6aa8d99666f","schema":"livegraph.boolean-propagation.v1","spec_hash":"3b9908b0cf9d0c914e9bf5f35a43fcb030d23933be8c3c57f5a667618ee1f18d","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-f34d14ebbda8337509bb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v06\n- v08 --refutes--> v02\n- v08 --refutes--> v07\n- v08 --supports--> v0a\n- v04 --supports--> v00\n- v0a --supports--> v01\n- v05 --refutes--> v03\n- v09 --refutes--> v05\n- v05 --refutes--> v04\n- v01 --refutes--> v09\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c1bb987326825d580b4735a88330bf2fe91f9992bbb56402ddcf4b57ccfc88fe","schema":"livegraph.boolean-propagation.v1","spec_hash":"f34d14ebbda8337509bb25f59dd827ed009b77267f9f54f59a7b93057082812c","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-584186b75bfa62cc5554","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v09\n- v00 --refutes--> v06\n- v00 --refutes--> v05\n- v04 --supports--> v0a\n- v08 --refutes--> v07\n- v09 --supports--> v08\n- v03 --supports--> v02\n- v03 --supports--> v04\n- v05 --refutes--> v01\n- v08 --supports--> v03\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"23e67a65755f0d6392b4ae92c66a540d8067413559d6e8844e4c090c27c0a09d","schema":"livegraph.boolean-propagation.v1","spec_hash":"584186b75bfa62cc55541a665ba2248f9dd30ac5a12992eb7c861642b3c62da7","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-9d7e4e6c0a978e01926a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v0a\n- v03 --refutes--> v04\n- v03 --refutes--> v06\n- v00 --supports--> v02\n- v05 --refutes--> v07\n- v05 --refutes--> v09\n- v07 --refutes--> v01\n- v08 --refutes--> v03\n- v02 --refutes--> v08\n- v0a --refutes--> v05\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"16deb325939a5feddbdef75d698ffb1fd6cba279f53e0f9a83e140232506af1f","schema":"livegraph.boolean-propagation.v1","spec_hash":"9d7e4e6c0a978e01926a9665528d17d6c971d2095ede29cbd58e81546744053e","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-74761ff1d61e3f4b5ee2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v06\n- v06 --refutes--> v04\n- v05 --supports--> v03\n- v01 --supports--> v05\n- v09 --refutes--> v02\n- v00 --supports--> v0a\n- v03 --supports--> v00\n- v09 --supports--> v07\n- v00 --supports--> v09\n- v00 --refutes--> v08\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6ab4e1c54ce3dde3181075de25b1c8d70e5dcdad78f71bb1e9c7f2a234add4e9","schema":"livegraph.boolean-propagation.v1","spec_hash":"74761ff1d61e3f4b5ee2e12cf82da491828e86f8d25d4303e5e321e6e7129cf5","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-2d862c8130a44a284f17","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v01\n- v04 --supports--> v0a\n- v02 --refutes--> v07\n- v02 --supports--> v04\n- v09 --supports--> v05\n- v01 --refutes--> v09\n- v08 --refutes--> v06\n- v01 --supports--> v03\n- v06 --supports--> v02\n- v00 --supports--> v08\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b13a6a9581042d6f56db0588eac51d612c8437abc32083297e4bfa3171798e5d","schema":"livegraph.boolean-propagation.v1","spec_hash":"2d862c8130a44a284f171be1b4d6baf13baaa349ca7fdae82cb3cbd7385a4d9b","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-e8fd139e39636926c6c0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v06\n- v0a --refutes--> v05\n- v00 --supports--> v08\n- v03 --refutes--> v09\n- v09 --supports--> v02\n- v01 --supports--> v03\n- v08 --supports--> v07\n- v07 --supports--> v0a\n- v09 --supports--> v04\n- v09 --refutes--> v00\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"73e4f1e79b9d927a663ab671d008aeb6979f30e28ccc330e15020925c0023619","schema":"livegraph.boolean-propagation.v1","spec_hash":"e8fd139e39636926c6c01d15667c20deb37b0245f1bb72c906d4b0a355263cc1","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-dd6d4d60528637d4df6b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v08\n- v0a --refutes--> v09\n- v05 --supports--> v0a\n- v04 --refutes--> v01\n- v07 --supports--> v02\n- v05 --supports--> v00\n- v06 --supports--> v05\n- v02 --supports--> v06\n- v0a --supports--> v03\n- v03 --refutes--> v04\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ec62072bde52811e480198e77494696f0d4200f05df0a5ff1619b84a4b7f3826","schema":"livegraph.boolean-propagation.v1","spec_hash":"dd6d4d60528637d4df6b144ae20942d8dd137d8c2cf744de16e5e0448ef74037","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-b56b08be8e73435b5332","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v05\n- v09 --refutes--> v06\n- v09 --supports--> v00\n- v08 --refutes--> v04\n- v03 --supports--> v09\n- v05 --supports--> v02\n- v04 --supports--> v01\n- v09 --supports--> v07\n- v05 --refutes--> v0a\n- v02 --refutes--> v08\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bd4c66d3e079922489ef70d5e187d0d0652424bda6fb0210478e1e220c832413","schema":"livegraph.boolean-propagation.v1","spec_hash":"b56b08be8e73435b533246ab7a53e57382784d97b12e9e60ec14d9de56773627","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-b178bf7d5bac58b3e068","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v0a\n- v08 --supports--> v05\n- v03 --supports--> v09\n- v0a --supports--> v04\n- v09 --refutes--> v08\n- v06 --supports--> v01\n- v09 --refutes--> v07\n- v09 --supports--> v02\n- v08 --supports--> v06\n- v04 --supports--> v03\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"407a8ba546eb09157321920c5f09d56aab2d570d848615e46a92880ea3445452","schema":"livegraph.boolean-propagation.v1","spec_hash":"b178bf7d5bac58b3e068fb346e2e692880e2562261931e343561c922ad03baf7","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-dc83a74b2caa6bcad08e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v06\n- v06 --supports--> v0a\n- v05 --supports--> v08\n- v00 --supports--> v03\n- v05 --refutes--> v04\n- v09 --supports--> v00\n- v06 --supports--> v02\n- v0a --refutes--> v05\n- v03 --refutes--> v07\n- v07 --refutes--> v01\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"139241189b0486169f865308d92f0a34ffe7db7d79d0928c2b8a37b7eddd8026","schema":"livegraph.boolean-propagation.v1","spec_hash":"dc83a74b2caa6bcad08e4c4e6399a2cd42b462d98b00e0f0af1dddcf1eda8610","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-e7543424f77059ec68a9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v06\n- v00 --supports--> v03\n- v08 --refutes--> v00\n- v06 --refutes--> v07\n- v03 --supports--> v05\n- v00 --refutes--> v09\n- v06 --refutes--> v02\n- v03 --supports--> v0a\n- v02 --refutes--> v08\n- v05 --refutes--> v01\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ee767e93aba95182304ab46fd9fb2a3bec2b46a333e7c36cd3f9c8b1076a52e0","schema":"livegraph.boolean-propagation.v1","spec_hash":"e7543424f77059ec68a9c115cd44450b7957ba0064850fe5fa087761fb41bdae","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-dab7e76b31304b449626","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v00\n- v07 --supports--> v0a\n- v01 --refutes--> v09\n- v09 --refutes--> v05\n- v04 --supports--> v01\n- v05 --supports--> v03\n- v03 --refutes--> v06\n- v08 --supports--> v04\n- v08 --refutes--> v02\n- v03 --supports--> v07\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"80ab1c56e2df4dfda1939402e0593c20b9bb18d4dd2b09ad0549e5e65987afb2","schema":"livegraph.boolean-propagation.v1","spec_hash":"dab7e76b31304b449626e63598266b808f5ae70ecce8eadd10ef9da9507b2b34","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-5e8cfe2ce327c57978aa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v05\n- v09 --refutes--> v02\n- v04 --supports--> v08\n- v06 --refutes--> v04\n- v00 --refutes--> v01\n- v05 --refutes--> v00\n- v01 --supports--> v03\n- v00 --refutes--> v0a\n- v03 --refutes--> v09\n- v00 --supports--> v07\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9555e960095f38c862f41b44e6c79a1ed9eac38de708974e64fdb34dec7c483b","schema":"livegraph.boolean-propagation.v1","spec_hash":"5e8cfe2ce327c57978aa8e68092814ebeee554990484ed294f7acc9d9a37657a","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-bee5a10c57b792ff2eb7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v09\n- v09 --refutes--> v01\n- v08 --refutes--> v06\n- v04 --supports--> v08\n- v07 --refutes--> v00\n- v01 --refutes--> v03\n- v01 --refutes--> v02\n- v08 --supports--> v07\n- v01 --refutes--> v0a\n- v02 --supports--> v05\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1497401a9a5c1daba90c6fb443888ad345081cfc4636c4ef57b7e55027ff3e8d","schema":"livegraph.boolean-propagation.v1","spec_hash":"bee5a10c57b792ff2eb7d1542971ca8d64e6c3ccb9aa57e4a3bec7d5276a8890","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-1eb1eca4c6a555fc759d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v08\n- v06 --refutes--> v0a\n- v09 --refutes--> v05\n- v09 --refutes--> v04\n- v01 --refutes--> v03\n- v05 --supports--> v01\n- v0a --refutes--> v02\n- v08 --supports--> v09\n- v08 --supports--> v07\n- v05 --refutes--> v00\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a1151ea5868e90c30d7bbfbdf40ffee6068f44151d157644c8d8b09319cdb66f","schema":"livegraph.boolean-propagation.v1","spec_hash":"1eb1eca4c6a555fc759d1c9f70fd4bb80f03d5c1dcc809f41ab291ee3559c587","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-7a5e923958e8e4d0cac9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v07\n- v04 --refutes--> v06\n- v09 --supports--> v05\n- v00 --refutes--> v0a\n- v04 --supports--> v03\n- v03 --refutes--> v00\n- v00 --supports--> v01\n- v02 --refutes--> v04\n- v08 --refutes--> v02\n- v01 --refutes--> v09\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c16f4929d639d817546348eb6b38d7c23151f35bec8172798244afd1009618d7","schema":"livegraph.boolean-propagation.v1","spec_hash":"7a5e923958e8e4d0cac9dc24496a8bbc2e7e632d6526feddb0592de49db76e18","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-d41ad3ff9d73186161cb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v03\n- v00 --supports--> v06\n- v02 --supports--> v0a\n- v06 --supports--> v01\n- v08 --supports--> v00\n- v01 --supports--> v02\n- v0a --refutes--> v09\n- v06 --supports--> v04\n- v0a --refutes--> v07\n- v00 --supports--> v05\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"40e27f48ddf503a2ba8a00369a452e932e194566bf4641f6217e4d28adde0a0d","schema":"livegraph.boolean-propagation.v1","spec_hash":"d41ad3ff9d73186161cb4f2674474b1530910870ab4bae9632ffe95ca2c7d4b4","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-dda007d92420d01824e6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v02\n- v06 --refutes--> v07\n- v07 --supports--> v00\n- v02 --supports--> v05\n- v0a --refutes--> v01\n- v04 --refutes--> v09\n- v02 --refutes--> v08\n- v00 --supports--> v04\n- v09 --refutes--> v0a\n- v00 --supports--> v03\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bad29668ce960776724233bc56abf98286728f88a25ac7b02139559f2e3b15f4","schema":"livegraph.boolean-propagation.v1","spec_hash":"dda007d92420d01824e6a2747c026a39702e846f8159bd90807a8259f61ead96","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-069bbc746c9891946803","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v05\n- v06 --refutes--> v0a\n- v04 --supports--> v01\n- v06 --supports--> v03\n- v08 --supports--> v07\n- v07 --refutes--> v06\n- v06 --refutes--> v02\n- v01 --supports--> v08\n- v09 --refutes--> v04\n- v04 --refutes--> v00\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e795be595f0cec87489e9b193ec691be9669d3bccff010b3ba12d8b920f291cb","schema":"livegraph.boolean-propagation.v1","spec_hash":"069bbc746c9891946803e547ba644098dbaa5de59d250b20166389adbe0241a6","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-de0295b49f73b474f4bf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v04\n- v04 --refutes--> v09\n- v03 --supports--> v00\n- v01 --supports--> v07\n- v07 --refutes--> v08\n- v04 --refutes--> v05\n- v04 --supports--> v06\n- v08 --supports--> v02\n- v09 --supports--> v01\n- v08 --refutes--> v03\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c1ed0cbf7012f7d69dfaae5af06edba318fccb0e055f27d97d9a49687075990f","schema":"livegraph.boolean-propagation.v1","spec_hash":"de0295b49f73b474f4bfe610f781a4142b393a6d0a27df87e581b997399300a7","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-a8115129de350d96aa01","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v09\n- v05 --refutes--> v06\n- v06 --refutes--> v03\n- v02 --refutes--> v04\n- v06 --supports--> v0a\n- v09 --supports--> v08\n- v07 --supports--> v02\n- v0a --supports--> v00\n- v06 --supports--> v01\n- v08 --supports--> v05\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"01cf21a541bb1a6300c4b995f58f38b4f78fecc43b3cd74f12e42700c1afd2c3","schema":"livegraph.boolean-propagation.v1","spec_hash":"a8115129de350d96aa014fd23d3510efbc4cf2d693a95e41c64945ce713368c0","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-fc77924addf9047d00a2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v04\n- v00 --supports--> v05\n- v04 --refutes--> v00\n- v06 --refutes--> v03\n- v02 --refutes--> v07\n- v09 --refutes--> v0a\n- v09 --supports--> v02\n- v07 --refutes--> v08\n- v05 --supports--> v09\n- v05 --refutes--> v01\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"770af95600bdef4139ebdd2f2b91dfe8a1fe2fe43729122f22a5ca30c83c3548","schema":"livegraph.boolean-propagation.v1","spec_hash":"fc77924addf9047d00a2d2c1aa0a61944ac37affb98042fc64432333b419a4da","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-5898cc42c305092ea3f2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v08\n- v07 --supports--> v06\n- v03 --supports--> v0a\n- v09 --supports--> v03\n- v03 --refutes--> v05\n- v04 --refutes--> v02\n- v04 --supports--> v09\n- v0a --supports--> v00\n- v03 --refutes--> v01\n- v08 --supports--> v04\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b0ca9011142650f480e35f16827a82459cfaa30eac720acff1566d0147259767","schema":"livegraph.boolean-propagation.v1","spec_hash":"5898cc42c305092ea3f2bc7228375e865229f3176cf7eff094d84ec1f4cffcc8","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-bcc0009dbc4c329571f6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v03\n- v07 --refutes--> v01\n- v09 --refutes--> v04\n- v08 --supports--> v00\n- v07 --refutes--> v05\n- v06 --supports--> v08\n- v05 --refutes--> v0a\n- v0a --refutes--> v02\n- v08 --refutes--> v09\n- v02 --supports--> v06\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"24706ddcf22c3d6791134e2cdfdd024cd06bd33bbb44637d2201ddae49e41ba4","schema":"livegraph.boolean-propagation.v1","spec_hash":"bcc0009dbc4c329571f6e8829bd8e2494ef8beb403b5b5ef50cfe7f207045eb1","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-3851d8f89b3ad5ab2803","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v0a\n- v06 --refutes--> v05\n- v00 --refutes--> v03\n- v05 --refutes--> v07\n- v04 --supports--> v01\n- v01 --refutes--> v08\n- v05 --supports--> v00\n- v06 --refutes--> v02\n- v09 --refutes--> v04\n- v08 --supports--> v06\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"042f9ee03adbedb98f35ba31ce106bbca46c02481d67547df66e3004bc494aac","schema":"livegraph.boolean-propagation.v1","spec_hash":"3851d8f89b3ad5ab2803ac38bb1cba99d1fa25c921d858bf6f0bbd3f875d9e99","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-ace5017ef8413fa0fa8e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v08\n- v09 --supports--> v04\n- v02 --refutes--> v09\n- v06 --refutes--> v03\n- v03 --refutes--> v07\n- v0a --refutes--> v02\n- v04 --refutes--> v00\n- v00 --supports--> v06\n- v03 --refutes--> v01\n- v03 --supports--> v05\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"767422d2e3faba4e44150c1f8bd5e1db078b72b00431a669d2db86765c377e27","schema":"livegraph.boolean-propagation.v1","spec_hash":"ace5017ef8413fa0fa8e80777387c9b1624cc80d91c52b557aeb21990e704ee6","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-24bfd8d28c271519be75","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v01\n- v03 --refutes--> v07\n- v06 --supports--> v00\n- v00 --refutes--> v03\n- v07 --refutes--> v04\n- v04 --supports--> v0a\n- v09 --refutes--> v06\n- v09 --supports--> v02\n- v03 --supports--> v05\n- v04 --supports--> v08\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8e8e12c82e6ed909c48b692f3fa0354f41e7a9b189ac6045726a44ace6efbdce","schema":"livegraph.boolean-propagation.v1","spec_hash":"24bfd8d28c271519be75e720518e32e8a0b972c2e960a127bc614cd1ead59307","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-5248aeeae22724e0964d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v09\n- v08 --refutes--> v07\n- v06 --supports--> v02\n- v01 --refutes--> v03\n- v00 --refutes--> v05\n- v05 --supports--> v06\n- v08 --supports--> v04\n- v01 --refutes--> v0a\n- v04 --refutes--> v00\n- v03 --refutes--> v08\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"168b782b84aee3d1204bbeb6e3bdb7b8d67d16c4eee9b140841f330df37f769a","schema":"livegraph.boolean-propagation.v1","spec_hash":"5248aeeae22724e0964d22e34012fda50d2edc2231425d235cae8de0a84e0aab","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":7,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-e49f77b7530f6fda8ab7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v04\n- v06 --refutes--> v00\n- v00 --refutes--> v08\n- v00 --refutes--> v0a\n- v07 --refutes--> v02\n- v05 --refutes--> v01\n- v0a --refutes--> v05\n- v09 --supports--> v07\n- v01 --supports--> v09\n- v07 --supports--> v03\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6893971ef2dc2ba2af5142069b5742a5bf608f35c4c97463b9e70c1eeee039f0","schema":"livegraph.boolean-propagation.v1","spec_hash":"e49f77b7530f6fda8ab786ef0bb7d18000bb7c13b23318a14e805e2d8873a256","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-f8b66d997bdcb124d73d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v03\n- v07 --refutes--> v0a\n- v06 --refutes--> v07\n- v09 --refutes--> v08\n- v09 --refutes--> v02\n- v04 --supports--> v01\n- v0a --supports--> v09\n- v05 --refutes--> v04\n- v05 --refutes--> v00\n- v01 --supports--> v06\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a102004ba3545116bc8750b8978f3677c5afb9780b5127ae09f04aff1702b55c","schema":"livegraph.boolean-propagation.v1","spec_hash":"f8b66d997bdcb124d73d003a0ea3f9f6ba21160bbf5e110694b79db08448a46c","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-6b07169c43c35a3c2637","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v01\n- v01 --supports--> v08\n- v04 --refutes--> v00\n- v02 --supports--> v07\n- v03 --supports--> v06\n- v07 --supports--> v04\n- v04 --refutes--> v05\n- v09 --refutes--> v02\n- v08 --refutes--> v09\n- v04 --refutes--> v0a\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c03c81b0d14b0284161328d042ea7719978acb301e160b4d24d2f494bb9b963d","schema":"livegraph.boolean-propagation.v1","spec_hash":"6b07169c43c35a3c2637fa1986587cf99668669a0950913fcbe0c677a273d512","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-ba73f594d20534e1379a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v09\n- v03 --supports--> v0a\n- v04 --supports--> v06\n- v07 --refutes--> v03\n- v03 --refutes--> v00\n- v07 --refutes--> v02\n- v06 --refutes--> v07\n- v05 --refutes--> v08\n- v06 --supports--> v01\n- v00 --supports--> v05\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3caaeff09a80f64d6ea01177259f14339127e932193edbfc629838a003639798","schema":"livegraph.boolean-propagation.v1","spec_hash":"ba73f594d20534e1379a8956c92648e1aaf71033240c8759a15084a1ddc8205c","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-7a1391cd55fc6adddcff","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v00\n- v09 --supports--> v07\n- v05 --supports--> v04\n- v0a --refutes--> v02\n- v06 --refutes--> v05\n- v01 --refutes--> v08\n- v05 --refutes--> v03\n- v04 --supports--> v09\n- v07 --supports--> v0a\n- v01 --refutes--> v06\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e198c0caf686c9b001f18c5d01030ad7154b6e49f0c645e244e4815c34f2b96b","schema":"livegraph.boolean-propagation.v1","spec_hash":"7a1391cd55fc6adddcff546bc3add1cf11e60e92bed20c90c7b36cb4cfd5014e","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-1a84fb4334166f6f76b1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v05\n- v00 --refutes--> v03\n- v0a --refutes--> v08\n- v09 --refutes--> v00\n- v08 --supports--> v01\n- v03 --refutes--> v06\n- v08 --refutes--> v04\n- v02 --supports--> v07\n- v03 --refutes--> v0a\n- v01 --supports--> v02\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2b069a0295f11d49766a1649e84156b4d85964080e0ab839ce3ca90c040c50f7","schema":"livegraph.boolean-propagation.v1","spec_hash":"1a84fb4334166f6f76b1050cf21b32816b48d221718b63ff5fd7737d9e38aa15","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-260f8a60540857a2116f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v07\n- v0a --refutes--> v02\n- v07 --refutes--> v09\n- v07 --supports--> v08\n- v05 --refutes--> v03\n- v0a --supports--> v04\n- v01 --supports--> v06\n- v06 --supports--> v0a\n- v06 --refutes--> v00\n- v02 --refutes--> v05\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c12da07a3e49f027b9aad2761fbf64cf0eb51b9dca25faafcef10417fae9d6d7","schema":"livegraph.boolean-propagation.v1","spec_hash":"260f8a60540857a2116fdc0513576540c4295297d8c617156feda9759726e2d5","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-012d01b515db97fb06ea","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v0a\n- v07 --supports--> v05\n- v09 --supports--> v06\n- v08 --supports--> v03\n- v0a --refutes--> v09\n- v01 --refutes--> v08\n- v06 --refutes--> v04\n- v04 --supports--> v01\n- v06 --refutes--> v02\n- v04 --refutes--> v00\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c31f0469d1fc2d53e3e9a9a8f300a0c53bcaabfe549d12cd1d8ddb8f1b48ca05","schema":"livegraph.boolean-propagation.v1","spec_hash":"012d01b515db97fb06ea074978570a428740ab25c834c795b9d8b9f92b3dd98a","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-1e2c97a45b040298f2be","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v00\n- v00 --supports--> v01\n- v00 --supports--> v02\n- v05 --refutes--> v08\n- v09 --refutes--> v0a\n- v03 --supports--> v07\n- v08 --supports--> v09\n- v0a --supports--> v03\n- v03 --supports--> v06\n- v03 --supports--> v04\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a14733a3a677fef977af71d67fe7c932c7ed1f91e78255e394bb72761cdf86e6","schema":"livegraph.boolean-propagation.v1","spec_hash":"1e2c97a45b040298f2be7c449b8e5ce7ea5f01c733a7801331724817a808e65a","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-2efcca8c6fbed0df12a4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v07\n- v07 --refutes--> v01\n- v04 --refutes--> v08\n- v02 --refutes--> v05\n- v0a --supports--> v02\n- v02 --supports--> v09\n- v03 --supports--> v0a\n- v06 --refutes--> v00\n- v04 --refutes--> v06\n- v07 --supports--> v04\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"778c547153e4d6eb8c76aa5e670543aab73e4757e3b6bc3057146505e5ce3267","schema":"livegraph.boolean-propagation.v1","spec_hash":"2efcca8c6fbed0df12a4ec065f158574649c3e167ac08964b4b54ba62b811ac2","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-19b8b57752ed6a620d94","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v04\n- v02 --supports--> v09\n- v0a --refutes--> v00\n- v00 --supports--> v06\n- v05 --supports--> v0a\n- v04 --refutes--> v08\n- v00 --supports--> v03\n- v04 --refutes--> v02\n- v09 --supports--> v07\n- v02 --supports--> v01\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b71e486474ec4d1d3e14053b3ca8407608ec9679a2380b9d3d7d7a36efc803d1","schema":"livegraph.boolean-propagation.v1","spec_hash":"19b8b57752ed6a620d94e5231c5c441b0d1e54401fa7e3242800a65537758582","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-d8239019874b5edf1cc4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v05\n- v01 --refutes--> v0a\n- v01 --refutes--> v00\n- v00 --supports--> v06\n- v05 --refutes--> v08\n- v05 --refutes--> v02\n- v06 --refutes--> v09\n- v05 --refutes--> v04\n- v02 --supports--> v07\n- v07 --supports--> v01\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2504301bf6a28aa4894899a17330397c23db4f493a31e1daf91da91361bc5939","schema":"livegraph.boolean-propagation.v1","spec_hash":"d8239019874b5edf1cc4c5773c99fd7e68516afedc98e29696722862b237d135","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":7,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-96243e6c79e515e69e53","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v09\n- v04 --refutes--> v07\n- v08 --supports--> v0a\n- v08 --refutes--> v02\n- v01 --refutes--> v04\n- v03 --refutes--> v05\n- v07 --refutes--> v08\n- v0a --refutes--> v00\n- v05 --refutes--> v01\n- v04 --supports--> v06\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"37831770c3387db1d88c1b10b77b8509fea43a9f4d0fcde3eef04dae1aabb1cd","schema":"livegraph.boolean-propagation.v1","spec_hash":"96243e6c79e515e69e5313e7b523ffd3f2a2b6d10b11fbbf9507d387809c7a07","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-2aa8eb005af99ff72949","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v05\n- v00 --refutes--> v0a\n- v09 --refutes--> v01\n- v08 --refutes--> v00\n- v09 --refutes--> v06\n- v04 --supports--> v09\n- v09 --supports--> v02\n- v07 --supports--> v04\n- v0a --refutes--> v07\n- v01 --refutes--> v03\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1fe8fda736a6d132729bd08109b282b6970ee2518728a976c6d53e43e506d3c9","schema":"livegraph.boolean-propagation.v1","spec_hash":"2aa8eb005af99ff7294987f1a0682cde62471dc566edb06ac04d7f3b5fc41a51","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-c44001f0647bb4b0443b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v09\n- v02 --refutes--> v07\n- v08 --supports--> v06\n- v07 --supports--> v03\n- v06 --supports--> v04\n- v02 --refutes--> v0a\n- v04 --refutes--> v01\n- v02 --refutes--> v00\n- v09 --refutes--> v02\n- v07 --supports--> v05\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b6da1064713253943cfa51ff06bf875e8cb0ae297ec596a7e9723ac28843cc8f","schema":"livegraph.boolean-propagation.v1","spec_hash":"c44001f0647bb4b0443b3ced3b454e87d816a66c50977db4f288ba74c5a95a76","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-9cbfa3490da307da7dbb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v05\n- v07 --refutes--> v09\n- v01 --refutes--> v08\n- v04 --refutes--> v07\n- v07 --supports--> v0a\n- v01 --refutes--> v06\n- v04 --refutes--> v02\n- v08 --supports--> v00\n- v09 --supports--> v01\n- v05 --refutes--> v03\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b9d93c1fc13c12ab44a6c44fdfed928adcb2a59cf5f55147420497379e142db3","schema":"livegraph.boolean-propagation.v1","spec_hash":"9cbfa3490da307da7dbbc435e5eab0fce17b077fc8f9f4981f0d6b3d8adf07cb","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-aafa1a6c7cb72c0e3211","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v07\n- v03 --supports--> v08\n- v09 --supports--> v00\n- v04 --supports--> v09\n- v08 --refutes--> v02\n- v06 --refutes--> v0a\n- v01 --supports--> v04\n- v09 --supports--> v05\n- v09 --supports--> v06\n- v0a --supports--> v03\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0129f1a9f12e441c9ab8250aa45f5ebb519531e60ebfa388d51b7c7c4a8e97f1","schema":"livegraph.boolean-propagation.v1","spec_hash":"aafa1a6c7cb72c0e32116f98bc7d5cb5e9feda111c9fa1041c1c21c0d16777c5","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-6ae18595061068a6297d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v08\n- v01 --refutes--> v05\n- v06 --refutes--> v00\n- v03 --refutes--> v0a\n- v02 --supports--> v06\n- v06 --refutes--> v07\n- v0a --refutes--> v04\n- v09 --supports--> v03\n- v07 --supports--> v09\n- v01 --supports--> v02\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1dbfe3a248c5b7546ac818a4f1f0a3021097ad59e25ca3fb2bb3c8b7991df2f4","schema":"livegraph.boolean-propagation.v1","spec_hash":"6ae18595061068a6297de966888b71d33f4551b23cc216a91fd9ed5dc68e24fc","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-e85e6dc50343f890fc95","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v08\n- v06 --supports--> v01\n- v05 --refutes--> v04\n- v0a --supports--> v09\n- v00 --supports--> v0a\n- v07 --refutes--> v02\n- v06 --supports--> v00\n- v08 --supports--> v03\n- v07 --refutes--> v05\n- v08 --refutes--> v06\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9b6e8ec9d7f98aa453a73ed796102b85a24e5fa789326384f42046f89c166f1c","schema":"livegraph.boolean-propagation.v1","spec_hash":"e85e6dc50343f890fc95d75ce1aa0740d531bef29cb3b24f77322558c518fa45","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-8a7a003524409cc1f04b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v06\n- v02 --refutes--> v03\n- v07 --supports--> v01\n- v04 --supports--> v09\n- v02 --refutes--> v0a\n- v08 --supports--> v00\n- v05 --refutes--> v02\n- v01 --supports--> v04\n- v04 --refutes--> v08\n- v00 --supports--> v05\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"322394edc64a4344d9980d2a221de815c0b57450e1a376d077bf65c3dab34a39","schema":"livegraph.boolean-propagation.v1","spec_hash":"8a7a003524409cc1f04b4550858e420d29ca3752422d9f815e774926b3a51d97","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":7,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-c0db0e320f28e0a16261","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v09\n- v04 --refutes--> v03\n- v02 --supports--> v07\n- v04 --supports--> v06\n- v03 --refutes--> v00\n- v00 --supports--> v0a\n- v04 --supports--> v01\n- v0a --refutes--> v05\n- v07 --refutes--> v04\n- v09 --refutes--> v08\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a37de7bc6a3b06ed24487e7c3881609dfd5d83fa7399936093ac5c2d1142e6fc","schema":"livegraph.boolean-propagation.v1","spec_hash":"c0db0e320f28e0a1626148000bb2ce6e27609ae8f311669619e8b76a21ea32ff","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-30d54b9442122a08f70c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v05\n- v07 --supports--> v06\n- v04 --supports--> v02\n- v07 --supports--> v03\n- v08 --supports--> v0a\n- v02 --refutes--> v08\n- v08 --supports--> v09\n- v0a --refutes--> v00\n- v00 --refutes--> v01\n- v01 --refutes--> v07\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a61c3f4b5f36a8624fd47c650c00ba59a915c98d44e528a8c4bf322d66d0b9e2","schema":"livegraph.boolean-propagation.v1","spec_hash":"30d54b9442122a08f70c87ac0e8ee3d373968f3b2b99cb4b9ac2255a166c5a60","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-7bf3b24607919f1fe910","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v02\n- v00 --refutes--> v04\n- v06 --refutes--> v00\n- v05 --refutes--> v01\n- v09 --supports--> v03\n- v09 --supports--> v0a\n- v02 --refutes--> v08\n- v02 --refutes--> v07\n- v0a --supports--> v05\n- v08 --refutes--> v06\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a5005a0ffb42eef5627054951b1799e9eca0b91ce7a2a2eea634273a3730370a","schema":"livegraph.boolean-propagation.v1","spec_hash":"7bf3b24607919f1fe9103719858a43a111204bb69fdab97f63d80f0e73bede4c","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-906f7cf2437ff8fd5a1c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v08\n- v0a --refutes--> v05\n- v08 --supports--> v06\n- v07 --supports--> v09\n- v06 --supports--> v00\n- v00 --refutes--> v01\n- v00 --refutes--> v03\n- v03 --refutes--> v04\n- v03 --supports--> v07\n- v02 --supports--> v0a\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8f2a9c9e6330a94875a39534b11b29519de649a8e72e0565917294bc7cb76f36","schema":"livegraph.boolean-propagation.v1","spec_hash":"906f7cf2437ff8fd5a1ca9c980e2dc1a9e9bf7506c37c5cb0698e22b8f75cb52","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-16e61e21d6566ba81f95","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v03\n- v09 --refutes--> v07\n- v08 --supports--> v02\n- v08 --refutes--> v06\n- v05 --refutes--> v0a\n- v03 --refutes--> v05\n- v02 --supports--> v00\n- v05 --supports--> v04\n- v09 --refutes--> v08\n- v01 --supports--> v09\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a91a797c131e9b060c90bf325c0aaf17e66e24cdd4ea49ff7ed35d9248f04d2b","schema":"livegraph.boolean-propagation.v1","spec_hash":"16e61e21d6566ba81f95c7e7c33daeeb2274c6f0c1a77dae7ca0305683a66b86","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-66ef5ada524e242cad49","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v03\n- v08 --supports--> v07\n- v03 --supports--> v00\n- v08 --supports--> v09\n- v06 --refutes--> v08\n- v08 --refutes--> v05\n- v07 --supports--> v01\n- v00 --refutes--> v02\n- v04 --refutes--> v06\n- v00 --refutes--> v04\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ccceb42ad7eac1deedaf018905074209bbd99182414e4d36cfd1add75b96f43e","schema":"livegraph.boolean-propagation.v1","spec_hash":"66ef5ada524e242cad49220ca5bc28d405100ca43871cb6666ec69216fc2b697","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-d0efb3dcfd2c1c1eb4ea","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v07\n- v01 --supports--> v03\n- v04 --refutes--> v0a\n- v0a --refutes--> v00\n- v04 --refutes--> v05\n- v09 --refutes--> v06\n- v04 --refutes--> v08\n- v02 --refutes--> v04\n- v09 --supports--> v02\n- v07 --supports--> v09\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6888f1c957966ffc7417dc7c607f6c159e1663f90179a0081e0e5c185950e9f7","schema":"livegraph.boolean-propagation.v1","spec_hash":"d0efb3dcfd2c1c1eb4eaa8add41972ae57ea14ccdda270fcebb728729bf52b04","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-954696c84c12eaff4643","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v00\n- v09 --refutes--> v04\n- v0a --supports--> v01\n- v01 --refutes--> v08\n- v06 --supports--> v02\n- v08 --refutes--> v03\n- v05 --refutes--> v09\n- v08 --refutes--> v07\n- v06 --refutes--> v05\n- v03 --supports--> v06\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"be0ba3864e8c130f65510abfa2927fec4c462e4b6620cdf4981cad536eaf15fd","schema":"livegraph.boolean-propagation.v1","spec_hash":"954696c84c12eaff46434d2e5fb199e09e6b9a0c20623e6e69129392a7f9ead5","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-281c676b48352b75ba7b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v01\n- v05 --refutes--> v03\n- v04 --supports--> v06\n- v01 --supports--> v0a\n- v00 --supports--> v02\n- v09 --supports--> v07\n- v08 --supports--> v09\n- v09 --refutes--> v04\n- v0a --supports--> v00\n- v00 --supports--> v05\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c0bba86eee09c72acd1edd7192a1661ebfcb2793fa8f207a2fd3852916a0271f","schema":"livegraph.boolean-propagation.v1","spec_hash":"281c676b48352b75ba7b568dba4702700e529b5aa2b77bc2709a5a2ab0d0d1f7","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-c7bd76b48ccc824ab900","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v04\n- v09 --supports--> v00\n- v01 --refutes--> v07\n- v01 --refutes--> v06\n- v02 --refutes--> v05\n- v00 --refutes--> v03\n- v07 --supports--> v02\n- v04 --supports--> v0a\n- v02 --supports--> v08\n- v04 --refutes--> v01\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2e6bef246d3c9244a2773239e5fa64d628d8f264a37885b6fdc30b34aa912887","schema":"livegraph.boolean-propagation.v1","spec_hash":"c7bd76b48ccc824ab900f6ab97f75f7fd4218f654fbec77887521c19629ad426","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-59a6b52af7d2f1212188","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v08\n- v02 --refutes--> v03\n- v03 --supports--> v06\n- v04 --supports--> v05\n- v08 --supports--> v02\n- v01 --refutes--> v00\n- v04 --supports--> v09\n- v06 --supports--> v01\n- v06 --refutes--> v07\n- v01 --supports--> v0a\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7bee4d6313aef7e05fc2af3f3ffbfb4883c27a578973402ac6dc565334cb7c11","schema":"livegraph.boolean-propagation.v1","spec_hash":"59a6b52af7d2f121218820a4ae2dc3955097297e124d36ffd90205f99b06f4e7","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-760bc46239226a9afd30","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v08\n- v03 --supports--> v02\n- v03 --supports--> v09\n- v02 --supports--> v0a\n- v02 --refutes--> v04\n- v07 --supports--> v03\n- v00 --refutes--> v07\n- v05 --supports--> v06\n- v01 --refutes--> v05\n- v06 --refutes--> v00\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fbe009fb044fb20d2ced7e108611bc4bf17cd3d36b2110995dfea467f7bbb9a5","schema":"livegraph.boolean-propagation.v1","spec_hash":"760bc46239226a9afd30a431827049ac911775b2df70fe4f669bbbeeb9410aab","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":7,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-c18d787a5c54564e76bc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v06\n- v0a --refutes--> v09\n- v06 --supports--> v02\n- v0a --refutes--> v08\n- v06 --supports--> v0a\n- v07 --refutes--> v01\n- v05 --refutes--> v00\n- v04 --refutes--> v05\n- v01 --refutes--> v03\n- v09 --supports--> v04\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2cd3f1341609bbacab5dbc4fb57cb4a4f55602121f730671239cd1bb60a24865","schema":"livegraph.boolean-propagation.v1","spec_hash":"c18d787a5c54564e76bce51d26048abb5396ef9afa1b8353723fe5dc9875fd02","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-30427f37a8b3849f44de","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v06\n- v04 --supports--> v08\n- v03 --supports--> v09\n- v07 --refutes--> v03\n- v05 --refutes--> v07\n- v09 --supports--> v02\n- v09 --refutes--> v00\n- v06 --refutes--> v01\n- v03 --supports--> v0a\n- v01 --refutes--> v05\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"79f66c02b16f2d5628dfbf23d57c98d73abfffb6563ad9c834b357b2a3ca1f3b","schema":"livegraph.boolean-propagation.v1","spec_hash":"30427f37a8b3849f44de847adc4b73c8adf68c5d35bbfa1ae969bb00b52b06b1","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-e2e7a2b8dc5a2e6dc3d4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v04\n- v03 --refutes--> v0a\n- v07 --refutes--> v09\n- v04 --supports--> v03\n- v00 --supports--> v01\n- v05 --refutes--> v02\n- v07 --supports--> v05\n- v02 --supports--> v00\n- v00 --refutes--> v08\n- v07 --supports--> v06\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"92ae3be6058199f5108e287e45397076329343ba80e2677536216faeafd67f10","schema":"livegraph.boolean-propagation.v1","spec_hash":"e2e7a2b8dc5a2e6dc3d440e217e34f9f7bce39b5d00836d4a4a92021a712e00e","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":7,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-2ac4a76f2377c54692e5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v02\n- v05 --refutes--> v04\n- v09 --supports--> v08\n- v02 --refutes--> v0a\n- v04 --supports--> v01\n- v09 --supports--> v05\n- v06 --supports--> v03\n- v03 --refutes--> v09\n- v09 --supports--> v07\n- v05 --refutes--> v00\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fd308d73a5245e3f009cf6beb3613fe19db8d31097fa4fa29b1806328a16f411","schema":"livegraph.boolean-propagation.v1","spec_hash":"2ac4a76f2377c54692e5ecb9f055ed3e326109027fcfe4744d34009484b0a32c","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-f8e03d685498ca839c14","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v09\n- v01 --refutes--> v08\n- v07 --supports--> v06\n- v00 --refutes--> v0a\n- v02 --supports--> v01\n- v04 --supports--> v02\n- v05 --refutes--> v00\n- v06 --supports--> v05\n- v05 --refutes--> v03\n- v01 --supports--> v07\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"aa4bd0f34dc2e6c66ca9c83683a6018e5f84989abeb60bc59b7a924550b8d9d3","schema":"livegraph.boolean-propagation.v1","spec_hash":"f8e03d685498ca839c1445d9b5379d6e551d11e33eccd54cbfbc4b134cb56ffb","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-9fc5b4bface52f949d34","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v0a\n- v0a --refutes--> v06\n- v00 --supports--> v02\n- v08 --refutes--> v00\n- v07 --refutes--> v03\n- v04 --refutes--> v08\n- v00 --supports--> v07\n- v07 --supports--> v05\n- v07 --refutes--> v01\n- v06 --supports--> v09\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0d08873ba69244f07ee83b4a934bd6e1a91f82137be337b832194c65a71b9097","schema":"livegraph.boolean-propagation.v1","spec_hash":"9fc5b4bface52f949d341fbfebbe39342702d113bab9af73ffc9bbc9409acdf0","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-3aa12ab99c65db58fd72","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v01\n- v02 --refutes--> v00\n- v03 --supports--> v05\n- v05 --supports--> v07\n- v0a --refutes--> v02\n- v02 --supports--> v06\n- v07 --refutes--> v08\n- v03 --refutes--> v04\n- v09 --supports--> v03\n- v08 --refutes--> v0a\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"30fd416c63dadc6126a34b8e47e78320df2e8c070cdb15d01fcb50c4c472b538","schema":"livegraph.boolean-propagation.v1","spec_hash":"3aa12ab99c65db58fd728b9137b633dcb99d562820b0859ef6f243026c027576","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-4d175a67562e4fc1d109","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v09\n- v06 --refutes--> v03\n- v0a --refutes--> v06\n- v09 --refutes--> v08\n- v07 --supports--> v0a\n- v03 --refutes--> v05\n- v01 --supports--> v02\n- v05 --refutes--> v04\n- v0a --supports--> v00\n- v05 --supports--> v01\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"853410180fd5239f6f65aea56f3429b015fa6f1d83e2766ef883047658448a53","schema":"livegraph.boolean-propagation.v1","spec_hash":"4d175a67562e4fc1d1091920f344c5417674ff0af46b3d7c5892434a1a8fb2c1","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-cbdab2812971aaec6bc4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v07\n- v07 --supports--> v03\n- v02 --supports--> v06\n- v00 --supports--> v0a\n- v06 --supports--> v08\n- v04 --refutes--> v02\n- v05 --refutes--> v09\n- v09 --refutes--> v00\n- v07 --refutes--> v04\n- v07 --refutes--> v01\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4d93b191797b3c774e8e70184f63bba886569965aadc57a4f7b2bc6370749c1e","schema":"livegraph.boolean-propagation.v1","spec_hash":"cbdab2812971aaec6bc4c957d2652ec0ca86ab0bd7fb76f654a24e2309721c0e","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-0698f691a99c59d79222","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v08\n- v06 --supports--> v02\n- v09 --refutes--> v04\n- v01 --refutes--> v0a\n- v09 --refutes--> v05\n- v06 --refutes--> v03\n- v08 --refutes--> v07\n- v06 --supports--> v00\n- v07 --refutes--> v09\n- v04 --supports--> v06\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"001beca4ddcaacf4a4e1fe4009e690947d26b6536bbb94d48044376de4381aff","schema":"livegraph.boolean-propagation.v1","spec_hash":"0698f691a99c59d7922202e458146b0ba8de7f1893ae9317a62be36cdc0970ea","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-c6660cc2a7b67d9bd87a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v03\n- v0a --refutes--> v00\n- v06 --refutes--> v04\n- v01 --supports--> v07\n- v08 --refutes--> v06\n- v02 --refutes--> v08\n- v01 --supports--> v0a\n- v00 --refutes--> v02\n- v06 --supports--> v09\n- v04 --refutes--> v05\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"094f76569b64422493f510651408b2e0796d04acfee35006b8d71d5efbf0e839","schema":"livegraph.boolean-propagation.v1","spec_hash":"c6660cc2a7b67d9bd87a5f526645de6813ccee0d573d278fa792017ac403ac7a","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-787036748120b2cd1959","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v06\n- v07 --refutes--> v00\n- v06 --refutes--> v07\n- v05 --refutes--> v0a\n- v05 --refutes--> v02\n- v05 --refutes--> v01\n- v07 --supports--> v08\n- v0a --supports--> v03\n- v03 --supports--> v04\n- v09 --refutes--> v05\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"98765ba283ea268175aa1fa759bcc1ecc3e9ceb36f5be6ccc3fdb417a6baa0eb","schema":"livegraph.boolean-propagation.v1","spec_hash":"787036748120b2cd19598951488679573e427f16f4981f95bdb3d9f9093d6337","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-c9de52ef543a4f154249","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v03\n- v06 --supports--> v0a\n- v07 --supports--> v06\n- v03 --supports--> v05\n- v0a --refutes--> v04\n- v06 --supports--> v08\n- v04 --supports--> v01\n- v07 --supports--> v09\n- v03 --supports--> v02\n- v00 --supports--> v07\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f1acc7d7e4e47afcf07c422246b2cb56daa94c6593cee0556e7964c02518c032","schema":"livegraph.boolean-propagation.v1","spec_hash":"c9de52ef543a4f1542494682bed7579519dc988ed99be7be16f6ce62508d96d1","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-617198f02700dae94479","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v08\n- v05 --supports--> v09\n- v05 --supports--> v0a\n- v05 --supports--> v02\n- v03 --refutes--> v05\n- v07 --refutes--> v03\n- v01 --refutes--> v07\n- v09 --refutes--> v06\n- v08 --refutes--> v04\n- v04 --refutes--> v00\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ee631c84bacb45675edb6e14d626996918737fc01d58186a164d6e4f6a33d985","schema":"livegraph.boolean-propagation.v1","spec_hash":"617198f02700dae94479eb936c7dfe22882a570fea3afd12be9d814144954788","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-d0b0dd4601a9f9b28dbc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v02\n- v06 --supports--> v03\n- v04 --supports--> v07\n- v05 --refutes--> v08\n- v06 --refutes--> v0a\n- v09 --supports--> v04\n- v08 --supports--> v01\n- v07 --refutes--> v05\n- v01 --supports--> v00\n- v01 --supports--> v06\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6f90001f91175500190e9d9fba3ec6087d55fbc0d832d59084e457c1679c5055","schema":"livegraph.boolean-propagation.v1","spec_hash":"d0b0dd4601a9f9b28dbcc91d900b19f9f8b77b907fcee74332463f81b2791e9e","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-670059f21895665ca91a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v02\n- v09 --refutes--> v07\n- v02 --supports--> v0a\n- v02 --supports--> v00\n- v0a --refutes--> v08\n- v07 --supports--> v05\n- v08 --refutes--> v09\n- v02 --refutes--> v01\n- v07 --refutes--> v03\n- v04 --supports--> v06\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e735b059527d42568c5f778cfb4c88cd62069b11a23fc2832976d8bd23070701","schema":"livegraph.boolean-propagation.v1","spec_hash":"670059f21895665ca91ac9b304ee9b36806580d646eca6421aa8beac33391afe","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-47b73452b33c8ce4d6f8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v0a\n- v05 --refutes--> v04\n- v0a --refutes--> v01\n- v08 --refutes--> v09\n- v07 --refutes--> v03\n- v01 --refutes--> v07\n- v0a --refutes--> v06\n- v09 --refutes--> v05\n- v07 --refutes--> v02\n- v01 --refutes--> v00\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f316e6af15e37c844027b72d6c1b063707cab6250571a2ea66e72991d08e159e","schema":"livegraph.boolean-propagation.v1","spec_hash":"47b73452b33c8ce4d6f828f7caf1685acafeaa50819dacfc50870ea1aa47fd90","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-51786e10b9a48e67cbf0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v04\n- v05 --supports--> v0a\n- v09 --supports--> v02\n- v00 --supports--> v06\n- v01 --refutes--> v07\n- v04 --refutes--> v03\n- v04 --supports--> v01\n- v01 --supports--> v05\n- v08 --refutes--> v09\n- v00 --refutes--> v08\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"aca4cfe3af08dd1aa53ea62d1dd87dd1e31f574048c8a9b7347ef940edd2a98f","schema":"livegraph.boolean-propagation.v1","spec_hash":"51786e10b9a48e67cbf0071b868b0dd4c1b7b93bb831de415be76524cebfb97a","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-b38db52b82bf0f085e40","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v02\n- v05 --refutes--> v04\n- v04 --supports--> v06\n- v02 --supports--> v01\n- v0a --supports--> v03\n- v05 --refutes--> v09\n- v0a --supports--> v05\n- v06 --supports--> v07\n- v07 --refutes--> v08\n- v08 --refutes--> v00\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"23341f1cf39cc34198f2fb66bd0d25c5e27f8877f485ccd30c7c1dc9fd3b49df","schema":"livegraph.boolean-propagation.v1","spec_hash":"b38db52b82bf0f085e40eb3ee95af65c74cb8e1d2da2abbad0bd0a29ee7c30c6","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-2b2819ef83e7815d0881","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v01\n- v06 --supports--> v00\n- v05 --refutes--> v06\n- v01 --supports--> v04\n- v05 --refutes--> v09\n- v00 --supports--> v03\n- v08 --supports--> v0a\n- v00 --refutes--> v07\n- v03 --refutes--> v08\n- v05 --supports--> v02\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6a06db8ba3f591a94bb28e002c459fd897dc9a904fb042f3fe6b1d6f2ed87de6","schema":"livegraph.boolean-propagation.v1","spec_hash":"2b2819ef83e7815d08810ae1ed93034ede62f6fbc455a8820162c22de35a63a5","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-ce7ed78e5ed36901ee7c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v09\n- v03 --supports--> v0a\n- v05 --supports--> v03\n- v06 --refutes--> v04\n- v07 --supports--> v06\n- v07 --refutes--> v00\n- v04 --refutes--> v02\n- v06 --supports--> v0b\n- v0c --refutes--> v07\n- v05 --supports--> v01\n- v02 --supports--> v08\n- v0a --refutes--> v0c\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cacb14718b5102b8d879169e02a7d2dccae61a758d1e8e2a16413fc12a8d749f","schema":"livegraph.boolean-propagation.v1","spec_hash":"ce7ed78e5ed36901ee7c970e35b2e8571407c2b474ec9f8d3bb70eede1d4484c","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-684965626e38964b26a8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v09\n- v04 --refutes--> v07\n- v09 --refutes--> v00\n- v07 --supports--> v0b\n- v02 --supports--> v04\n- v00 --refutes--> v0a\n- v09 --refutes--> v03\n- v01 --supports--> v06\n- v05 --supports--> v08\n- v05 --refutes--> v0c\n- v01 --supports--> v02\n- v00 --refutes--> v01\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e315def92dc10626a71dc34903bbd142b2e24c46f96fe8124a5a5696a4731af6","schema":"livegraph.boolean-propagation.v1","spec_hash":"684965626e38964b26a88bedf0c08b6d6a21c21e4d2c4286be392651661ae79d","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":8,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-94c67a2484dfcd264f37","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v02\n- v0c --supports--> v09\n- v0a --supports--> v05\n- v03 --refutes--> v0a\n- v08 --refutes--> v01\n- v05 --supports--> v0b\n- v0a --supports--> v06\n- v0b --refutes--> v04\n- v04 --supports--> v07\n- v08 --refutes--> v00\n- v0c --supports--> v08\n- v00 --refutes--> v03\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"848056ab18348ef1ed3458da4712846ebab2e48235c1a1eec962eecb8fa28d13","schema":"livegraph.boolean-propagation.v1","spec_hash":"94c67a2484dfcd264f37601b84dd3adea5fa42afe54f41357d140714d13f1ef2","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-883270eea3b53f9e7d76","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v05\n- v0b --supports--> v07\n- v0c --refutes--> v09\n- v0c --refutes--> v06\n- v03 --refutes--> v00\n- v00 --refutes--> v0a\n- v0a --refutes--> v0b\n- v06 --refutes--> v08\n- v08 --supports--> v04\n- v07 --supports--> v0c\n- v07 --refutes--> v02\n- v00 --supports--> v01\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6895e19eca9667edbf7af345d1b499b37763e960c481c2f817f1de7ce7fe352b","schema":"livegraph.boolean-propagation.v1","spec_hash":"883270eea3b53f9e7d76bc99319bc07e1295732e942212e6d06eebcbcef0c46a","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-613b10d5fedb077581a7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v04\n- v0a --supports--> v03\n- v06 --supports--> v05\n- v0b --refutes--> v0a\n- v04 --refutes--> v00\n- v0a --supports--> v01\n- v0b --supports--> v02\n- v07 --supports--> v0b\n- v00 --refutes--> v09\n- v0b --refutes--> v08\n- v09 --refutes--> v07\n- v07 --supports--> v0c\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"98870c8183484475043da6ca2e2980de2c6f848a95e038edf6f719e066ab36db","schema":"livegraph.boolean-propagation.v1","spec_hash":"613b10d5fedb077581a7a9e3323779c9ebb2bb141c6d50d7ea4267fc792f1ef3","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-67190a167e065d2b8a9f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v07\n- v0b --supports--> v06\n- v00 --refutes--> v05\n- v00 --refutes--> v01\n- v02 --supports--> v0a\n- v0b --refutes--> v00\n- v09 --refutes--> v08\n- v03 --refutes--> v0b\n- v08 --supports--> v03\n- v05 --refutes--> v02\n- v02 --supports--> v0c\n- v03 --refutes--> v04\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b2d6723b6d133862135086701169a5a2a039e72748ca3ebe0f628c3973ace90f","schema":"livegraph.boolean-propagation.v1","spec_hash":"67190a167e065d2b8a9fc7db9090ee9c308293decc2265aa42d57424488f90fd","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-14f4d65a3ba854b6e7c9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v0a\n- v03 --supports--> v01\n- v01 --supports--> v06\n- v00 --refutes--> v07\n- v07 --supports--> v02\n- v0c --supports--> v09\n- v00 --refutes--> v0b\n- v07 --refutes--> v04\n- v02 --supports--> v05\n- v08 --supports--> v0c\n- v01 --supports--> v08\n- v09 --supports--> v00\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"342df065d1761d7a7f88fe713304a46bd237ee629491c7d40aea337bfc4b2b9f","schema":"livegraph.boolean-propagation.v1","spec_hash":"14f4d65a3ba854b6e7c9402f36123946ab6690e8c1601615b089cc5ebb2c13e1","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-d2391827fe296dea12e0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v09\n- v09 --supports--> v04\n- v01 --supports--> v0b\n- v00 --supports--> v0a\n- v02 --supports--> v0c\n- v0c --refutes--> v01\n- v0c --refutes--> v03\n- v07 --refutes--> v02\n- v05 --refutes--> v00\n- v05 --supports--> v08\n- v09 --supports--> v05\n- v06 --refutes--> v07\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3308a35a3b78fe67d704a13a736921af7d2143c8abe0e9e1fc31901f95c580ac","schema":"livegraph.boolean-propagation.v1","spec_hash":"d2391827fe296dea12e06e4f00d9dbb394120a84790715d12c3030011d66b3d7","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-da03e6b81038708a7d00","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v0b\n- v07 --supports--> v02\n- v0c --refutes--> v03\n- v01 --refutes--> v06\n- v00 --refutes--> v09\n- v02 --refutes--> v00\n- v05 --refutes--> v04\n- v08 --refutes--> v05\n- v05 --refutes--> v0c\n- v09 --supports--> v01\n- v02 --refutes--> v0a\n- v0c --supports--> v07\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8778752525383cfab0181595993c9419a544c449d43ec473875a0e050388c9e4","schema":"livegraph.boolean-propagation.v1","spec_hash":"da03e6b81038708a7d00acbacb986cfe8bd79f8fdd7967f5a76f5794a974633a","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-097764fb43d54be92f21","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v00\n- v05 --refutes--> v03\n- v02 --refutes--> v09\n- v0a --refutes--> v05\n- v0c --supports--> v04\n- v00 --supports--> v08\n- v0b --supports--> v06\n- v05 --refutes--> v07\n- v08 --supports--> v0b\n- v05 --supports--> v01\n- v0c --supports--> v0a\n- v06 --refutes--> v0c\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"afeb68fdddf2b7da004a23a05cdd0e5a0a5dfc94abf9abeb3fbf8fb299e71397","schema":"livegraph.boolean-propagation.v1","spec_hash":"097764fb43d54be92f21665aba7e7d71e6d35f2e6e70fdaa11c899c9ebe6b9be","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-9c4e3a3dcc661fb7c34e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v01\n- v03 --supports--> v06\n- v04 --supports--> v0b\n- v05 --supports--> v00\n- v0a --refutes--> v03\n- v05 --refutes--> v08\n- v06 --refutes--> v05\n- v08 --refutes--> v0c\n- v0b --refutes--> v02\n- v04 --supports--> v09\n- v05 --refutes--> v07\n- v0c --refutes--> v04\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"628a87d53103a0ceb6720cd34a979eb6870c4eafebfc426873da6d4a3adcb957","schema":"livegraph.boolean-propagation.v1","spec_hash":"9c4e3a3dcc661fb7c34e508212236df146d3309416e42166d2934513937db479","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-fe8ff2f9168e427c4711","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v07\n- v07 --refutes--> v05\n- v05 --refutes--> v06\n- v04 --supports--> v0a\n- v05 --refutes--> v0b\n- v01 --refutes--> v08\n- v08 --supports--> v09\n- v08 --supports--> v00\n- v03 --refutes--> v04\n- v05 --refutes--> v02\n- v04 --supports--> v0c\n- v06 --supports--> v03\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ff5f80d8ab84828ab5d4ef100d7ac180ad41ddf6ca94adca7071fd0a5002c4c7","schema":"livegraph.boolean-propagation.v1","spec_hash":"fe8ff2f9168e427c47119d45c70a5dd149821f3a54a505d6c44cd00214b001a9","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":8,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-36d614f6109e1f509b34","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v0a\n- v07 --supports--> v09\n- v02 --refutes--> v00\n- v02 --supports--> v03\n- v05 --supports--> v06\n- v00 --refutes--> v0c\n- v09 --supports--> v01\n- v08 --supports--> v04\n- v0c --refutes--> v07\n- v09 --supports--> v08\n- v05 --supports--> v0b\n- v06 --refutes--> v02\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f2bbf3204fccc0025695aaf05f454504e8ea8ba0f9f4f7ded3c5701ac9562a2e","schema":"livegraph.boolean-propagation.v1","spec_hash":"36d614f6109e1f509b34975f11deb2c4d207a54144df1b9d9e462d9b44d5778e","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-a28ba7190b64ab53ecf2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v02\n- v07 --refutes--> v09\n- v02 --supports--> v05\n- v07 --supports--> v04\n- v0b --supports--> v06\n- v05 --refutes--> v01\n- v0a --refutes--> v0c\n- v07 --refutes--> v0b\n- v0c --supports--> v00\n- v01 --refutes--> v07\n- v0b --supports--> v0a\n- v07 --refutes--> v03\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b442020deded099bf999016591ad3629fd357b134885e2755ce5721b6d8d795b","schema":"livegraph.boolean-propagation.v1","spec_hash":"a28ba7190b64ab53ecf2ebb481fcacb9e3e053127b039b08e68085aac5a19b44","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-20b5d5b9931e44a891bb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v04\n- v08 --supports--> v01\n- v06 --supports--> v08\n- v0c --refutes--> v0a\n- v03 --supports--> v07\n- v0b --supports--> v0c\n- v01 --refutes--> v09\n- v02 --refutes--> v06\n- v02 --refutes--> v05\n- v07 --refutes--> v02\n- v02 --supports--> v00\n- v0c --supports--> v03\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b3643a0dedd07f7cfed72b7087f43e10833eebb0c551706a0c647b74dcef5526","schema":"livegraph.boolean-propagation.v1","spec_hash":"20b5d5b9931e44a891bb8bbdf3cb7671225eb4299625b100c75be480d4224406","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-aa6543d4a54404136089","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v07\n- v00 --supports--> v0c\n- v01 --refutes--> v03\n- v00 --supports--> v04\n- v0b --supports--> v08\n- v03 --refutes--> v0b\n- v08 --refutes--> v05\n- v05 --supports--> v06\n- v04 --refutes--> v0a\n- v01 --refutes--> v02\n- v0a --supports--> v01\n- v00 --refutes--> v09\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9076f3b2e1b69e513aa9681fd1b7c9dcc1b799c6c529ed8d00e3da0301ce8c80","schema":"livegraph.boolean-propagation.v1","spec_hash":"aa6543d4a5440413608947ffdf7e482e207022992dc5c76be176454c8f30df50","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-dc863b580d8eb3f54be3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v03\n- v09 --refutes--> v02\n- v01 --refutes--> v0c\n- v01 --supports--> v00\n- v09 --supports--> v0b\n- v03 --supports--> v09\n- v0b --refutes--> v04\n- v04 --supports--> v05\n- v03 --supports--> v06\n- v05 --supports--> v01\n- v0c --supports--> v07\n- v04 --refutes--> v08\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"48c0c3bb757cd4a36cecb5378addf693cc7509c9454224694f134114465ca8ca","schema":"livegraph.boolean-propagation.v1","spec_hash":"dc863b580d8eb3f54be337a47fa7752f0b22968cb62d2f7866a77efa0227fa11","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-161817e9961294b29b67","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v05\n- v00 --refutes--> v0b\n- v02 --supports--> v00\n- v03 --supports--> v07\n- v0b --supports--> v06\n- v09 --supports--> v03\n- v01 --refutes--> v02\n- v06 --refutes--> v08\n- v01 --supports--> v0c\n- v06 --supports--> v04\n- v02 --supports--> v0a\n- v08 --refutes--> v09\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c1e0dd43deaa8fddcb0f712a96001b183970256fe3d79310d79d88c40be9c194","schema":"livegraph.boolean-propagation.v1","spec_hash":"161817e9961294b29b6724ae2b99a3289ee3e5692fd900ad9f89059ce0981394","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-4cda140fbbc95e003403","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v07\n- v0b --supports--> v0c\n- v0c --supports--> v0a\n- v05 --supports--> v04\n- v04 --supports--> v03\n- v01 --refutes--> v09\n- v04 --refutes--> v00\n- v03 --supports--> v01\n- v05 --supports--> v02\n- v09 --refutes--> v0b\n- v03 --refutes--> v06\n- v0a --supports--> v08\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e2403a8c1ed5ef3b069f432e367ad427dbcd36d10eb8be7eb3eb79119fa96807","schema":"livegraph.boolean-propagation.v1","spec_hash":"4cda140fbbc95e003403777144a72c550f6594c8541c397fc476e33cbdc7cf7c","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-5da468296392746bdefd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v04\n- v02 --refutes--> v08\n- v0c --refutes--> v01\n- v0a --supports--> v06\n- v03 --refutes--> v0b\n- v08 --supports--> v03\n- v01 --supports--> v02\n- v0a --supports--> v05\n- v07 --supports--> v0a\n- v03 --supports--> v00\n- v07 --refutes--> v09\n- v0a --supports--> v0c\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"63083bd9e2144070d795ed971ac687f84e243ed6c9933edccb213bef4a05deb5","schema":"livegraph.boolean-propagation.v1","spec_hash":"5da468296392746bdefd8d3bd9bafe516ab617727a5b061d15e5562f670cf208","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-28fbda21ed51f181d7bf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v01\n- v0a --refutes--> v0b\n- v00 --supports--> v0a\n- v02 --supports--> v08\n- v08 --supports--> v04\n- v01 --refutes--> v06\n- v08 --supports--> v05\n- v01 --supports--> v0c\n- v06 --refutes--> v03\n- v0b --refutes--> v07\n- v06 --refutes--> v02\n- v09 --supports--> v00\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"72a0e0c46d5de620accea05fcae15902dd3e2ec30083c7ee494454c5e33414e4","schema":"livegraph.boolean-propagation.v1","spec_hash":"28fbda21ed51f181d7bf9dbf431756b256628770ad36ba2087454c897b2c4f67","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-fb5820849073a45999f7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v0b\n- v01 --refutes--> v0c\n- v02 --refutes--> v0a\n- v07 --supports--> v05\n- v0a --refutes--> v01\n- v08 --supports--> v03\n- v06 --refutes--> v04\n- v0b --refutes--> v08\n- v01 --supports--> v07\n- v06 --supports--> v09\n- v01 --supports--> v00\n- v08 --refutes--> v02\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"eaf8e44ebd8c6a8fc320cfae41ac5b8aaac6079fefe41a93272a485894bf7b1a","schema":"livegraph.boolean-propagation.v1","spec_hash":"fb5820849073a45999f7ffa37fe2e3a7cd942d72bec41015f9f7a90f34950531","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-95b06e60e366e7cd5240","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v06\n- v06 --refutes--> v01\n- v0b --supports--> v00\n- v02 --supports--> v08\n- v01 --supports--> v07\n- v07 --refutes--> v09\n- v00 --refutes--> v05\n- v04 --supports--> v02\n- v06 --refutes--> v03\n- v02 --supports--> v0c\n- v08 --refutes--> v0b\n- v07 --supports--> v0a\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6aa08bd94da30004374928b939963a85578c81b342be0124eaed1d5f789cb8fb","schema":"livegraph.boolean-propagation.v1","spec_hash":"95b06e60e366e7cd524044c21764cb5f43f49f1cc89b7b7c7bdade40c619e49e","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-3dacd6cceaec4c85fa32","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v05\n- v0c --supports--> v08\n- v0b --refutes--> v0a\n- v08 --supports--> v07\n- v0c --refutes--> v01\n- v07 --refutes--> v06\n- v07 --refutes--> v02\n- v05 --refutes--> v09\n- v06 --refutes--> v00\n- v04 --refutes--> v03\n- v05 --refutes--> v0b\n- v0b --supports--> v0c\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e3df8a4c526a752f80ab70d928584cef4be4e585a5c7853abcc8390c6292a196","schema":"livegraph.boolean-propagation.v1","spec_hash":"3dacd6cceaec4c85fa32037ba89cb539b64497456c57b4d4987475eb9461fef9","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-82aea5214d99e9fe14f6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v08\n- v00 --refutes--> v0a\n- v0a --refutes--> v07\n- v03 --refutes--> v05\n- v00 --refutes--> v02\n- v06 --refutes--> v09\n- v07 --supports--> v06\n- v0a --refutes--> v01\n- v03 --refutes--> v0c\n- v0c --refutes--> v0b\n- v0b --refutes--> v04\n- v04 --refutes--> v00\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bf4f5874f2df43eda0f813955bf64a26b36c57c91d5668e3aa218ea84544a1c0","schema":"livegraph.boolean-propagation.v1","spec_hash":"82aea5214d99e9fe14f6052629c2ecddf7687c826c5200fb301e4afe35490aa9","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-6b27a6f5e77a79cea998","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v06\n- v09 --supports--> v04\n- v0b --supports--> v02\n- v05 --supports--> v01\n- v02 --refutes--> v0a\n- v04 --refutes--> v08\n- v03 --refutes--> v0c\n- v05 --refutes--> v03\n- v0c --refutes--> v0b\n- v0b --supports--> v07\n- v0a --supports--> v09\n- v0b --refutes--> v00\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"10e2d852532909171ae82cdb8d6884109ff02ccd9fbe2ea0ffbe30f541bc9556","schema":"livegraph.boolean-propagation.v1","spec_hash":"6b27a6f5e77a79cea998be9d127838762f0118dcfc0334153a3a13cdfaba4a06","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-9906f252ec900f384e4a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v05\n- v03 --supports--> v08\n- v04 --supports--> v0a\n- v09 --refutes--> v01\n- v0c --supports--> v06\n- v0b --supports--> v03\n- v03 --supports--> v04\n- v07 --refutes--> v09\n- v00 --supports--> v0c\n- v05 --supports--> v02\n- v05 --supports--> v0b\n- v0c --refutes--> v07\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0a48d8e9796ea1f7de02fe77041c075fedbdb3672bc4e0d9508d8a0cc44677aa","schema":"livegraph.boolean-propagation.v1","spec_hash":"9906f252ec900f384e4af903e8fb246356030ef9198792426697184aef98ac58","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-ba7b582749f0f64c5710","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v01\n- v01 --supports--> v02\n- v00 --supports--> v07\n- v00 --refutes--> v0b\n- v0a --refutes--> v04\n- v0b --supports--> v06\n- v05 --refutes--> v0a\n- v02 --supports--> v05\n- v0c --supports--> v09\n- v0a --refutes--> v00\n- v01 --supports--> v03\n- v0c --supports--> v08\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8bbd6442a957f8ae862c670406d16a39fe398da55a4c335626f0fbb3a89cf8a6","schema":"livegraph.boolean-propagation.v1","spec_hash":"ba7b582749f0f64c5710aa7805aa17175ce5d2d435c19993b4fc567f610c7eea","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-f0a53fe0b835b5f1449b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v07\n- v07 --supports--> v0b\n- v0c --supports--> v08\n- v09 --supports--> v06\n- v07 --supports--> v05\n- v08 --refutes--> v04\n- v0c --refutes--> v00\n- v04 --refutes--> v02\n- v0c --supports--> v0a\n- v02 --refutes--> v09\n- v09 --refutes--> v01\n- v0b --supports--> v0c\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5bd92cea4243baafe8342de302f3eaa1c680437f0a68bac7d4e99b4f534a211b","schema":"livegraph.boolean-propagation.v1","spec_hash":"f0a53fe0b835b5f1449b03a780e47164b2a4e2417fbcec4a6df12b5578a40d17","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-1042d66366660a01f460","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v08\n- v09 --supports--> v0a\n- v06 --supports--> v01\n- v00 --refutes--> v07\n- v02 --supports--> v0c\n- v02 --supports--> v05\n- v06 --refutes--> v04\n- v07 --supports--> v06\n- v01 --refutes--> v09\n- v05 --refutes--> v03\n- v0a --refutes--> v02\n- v05 --refutes--> v0b\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e01c2202fd978f250a6773b8747c4b639b18b9bc3d2c5b90bddfa6ed6864a9d1","schema":"livegraph.boolean-propagation.v1","spec_hash":"1042d66366660a01f4603df93485718c0aac2ff381771dc98cb2e889dff52274","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":8,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-83d30c363df1a9ebb2bd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v02\n- v01 --supports--> v0c\n- v08 --supports--> v01\n- v06 --refutes--> v07\n- v08 --refutes--> v0b\n- v03 --supports--> v04\n- v0a --supports--> v03\n- v00 --supports--> v09\n- v01 --supports--> v0a\n- v02 --refutes--> v08\n- v03 --refutes--> v05\n- v05 --refutes--> v06\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"67c0e8b46ce5e1338508aee04bb70060e33a12a963a05d9d5e1c5a1f6b527444","schema":"livegraph.boolean-propagation.v1","spec_hash":"83d30c363df1a9ebb2bd9825ba3c1de1a9c364a1fe4124c07f7011ca67b7c2c7","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":8,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-88c56df2a26811c2f314","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v01\n- v00 --supports--> v0a\n- v01 --supports--> v0c\n- v03 --supports--> v07\n- v09 --supports--> v08\n- v0c --refutes--> v09\n- v03 --supports--> v06\n- v04 --refutes--> v02\n- v09 --supports--> v05\n- v0a --refutes--> v04\n- v09 --refutes--> v0b\n- v08 --refutes--> v03\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"65b2704f82b8e4611bd48c800f5e77697cb05e6aa469446776c8d0d46e068e59","schema":"livegraph.boolean-propagation.v1","spec_hash":"88c56df2a26811c2f314b0dd614d8384752d909d7431793a0f398c51c950ad69","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-3befee2c5f55a6ca197a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v03\n- v06 --supports--> v01\n- v0c --refutes--> v07\n- v06 --refutes--> v02\n- v02 --refutes--> v0c\n- v0c --refutes--> v00\n- v08 --supports--> v0b\n- v00 --refutes--> v08\n- v04 --refutes--> v0a\n- v05 --supports--> v06\n- v05 --refutes--> v09\n- v0b --supports--> v04\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2ba4e728759154ac5e56a1a373b38258ca0487ebdab2331748e77866f505d178","schema":"livegraph.boolean-propagation.v1","spec_hash":"3befee2c5f55a6ca197a49ef0435dd32fe0169459beb0bfe0307bfc1d264c782","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-207ff3de6c14fb7afc6e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v0c\n- v0c --refutes--> v0a\n- v07 --refutes--> v02\n- v00 --refutes--> v08\n- v03 --refutes--> v07\n- v07 --supports--> v01\n- v01 --refutes--> v04\n- v08 --refutes--> v09\n- v0b --supports--> v00\n- v08 --supports--> v03\n- v0c --supports--> v06\n- v05 --supports--> v0b\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3c5915d15d04ac0a128d46bbb0819608c118f5ac975f92d744c2721bd8074b9a","schema":"livegraph.boolean-propagation.v1","spec_hash":"207ff3de6c14fb7afc6ed0f51b044b009f0ae10aa8a90117e0d156d930d7149b","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-68b757254fd5e8385f6d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v01\n- v00 --refutes--> v0b\n- v0c --refutes--> v03\n- v09 --refutes--> v02\n- v05 --refutes--> v08\n- v0c --supports--> v06\n- v08 --refutes--> v07\n- v0a --refutes--> v05\n- v02 --supports--> v0c\n- v0c --supports--> v0a\n- v00 --supports--> v04\n- v07 --refutes--> v00\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6f250a7f75286cff48c835374675af4398b0fc695ddc9bbd8afbceefda6f5b79","schema":"livegraph.boolean-propagation.v1","spec_hash":"68b757254fd5e8385f6dc52d889d775588e561f48202531f6661b1579a903567","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-b979681f030cb7f96786","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v04\n- v02 --refutes--> v0b\n- v09 --supports--> v01\n- v01 --supports--> v0a\n- v0b --supports--> v08\n- v01 --supports--> v07\n- v0c --supports--> v00\n- v04 --refutes--> v09\n- v00 --supports--> v03\n- v09 --supports--> v05\n- v01 --refutes--> v06\n- v08 --refutes--> v0c\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c2bdd884db6e729be508471bc91c7512f29a223a9bbec240b3d48c97ce4b582c","schema":"livegraph.boolean-propagation.v1","spec_hash":"b979681f030cb7f9678697152879d5e4f66e9c66d0999de765d9218c6653aac0","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-3daa6077a39004982aad","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v00\n- v0c --refutes--> v04\n- v00 --refutes--> v06\n- v07 --supports--> v0a\n- v09 --refutes--> v01\n- v0b --refutes--> v09\n- v01 --supports--> v08\n- v06 --refutes--> v07\n- v09 --refutes--> v05\n- v04 --supports--> v03\n- v09 --refutes--> v02\n- v07 --refutes--> v0c\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"801ed7c5f5cb948d8410ec0c236fc6965fd1b4f458e1a669230176cf118ceda5","schema":"livegraph.boolean-propagation.v1","spec_hash":"3daa6077a39004982aadeb38e03a91d2adde0dce7c0d77519b74d77f74f8b75f","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-891831cfd8957c89d0c1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v03\n- v05 --refutes--> v06\n- v02 --refutes--> v04\n- v02 --refutes--> v0c\n- v06 --refutes--> v0b\n- v01 --supports--> v05\n- v0b --refutes--> v00\n- v00 --refutes--> v07\n- v09 --refutes--> v02\n- v00 --refutes--> v0a\n- v00 --refutes--> v08\n- v04 --refutes--> v01\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8a159a345b442e7a88b4c3ecec48225d4452886e1ef19427d021b8cb370ba066","schema":"livegraph.boolean-propagation.v1","spec_hash":"891831cfd8957c89d0c176138af52f478cd3a3b27e63d30cd545f67913987fad","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-c9cdcd3735d8fab6f7aa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v04\n- v00 --supports--> v01\n- v0a --supports--> v07\n- v00 --supports--> v08\n- v04 --supports--> v05\n- v0b --supports--> v06\n- v0a --refutes--> v09\n- v02 --refutes--> v0c\n- v03 --supports--> v0b\n- v09 --refutes--> v03\n- v02 --refutes--> v00\n- v04 --supports--> v02\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bae0b692c683095604186bc00537108889f6b51198557bc940806b8d7d06d08b","schema":"livegraph.boolean-propagation.v1","spec_hash":"c9cdcd3735d8fab6f7aaddc0e26d5fef74beb68b3232f94583359ef00455dee2","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-c7d8e7d49bee0c589a5a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v08\n- v08 --refutes--> v00\n- v07 --refutes--> v0a\n- v05 --supports--> v09\n- v05 --supports--> v02\n- v0a --supports--> v03\n- v01 --refutes--> v05\n- v02 --supports--> v06\n- v00 --supports--> v0c\n- v0c --supports--> v01\n- v02 --supports--> v04\n- v0c --refutes--> v0b\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2614d6aa82d0d354291e4b1f555f1c062705308c1e3d25a4e4018d72aa9d54ef","schema":"livegraph.boolean-propagation.v1","spec_hash":"c7d8e7d49bee0c589a5a7b8b18cac554ac1284af730c2f75e3c6bfeadb8eb0a3","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-56c0f5b82c664e752d44","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v03\n- v0c --refutes--> v0b\n- v07 --refutes--> v0a\n- v06 --supports--> v07\n- v0b --refutes--> v06\n- v05 --refutes--> v02\n- v0a --supports--> v00\n- v04 --refutes--> v0c\n- v05 --supports--> v09\n- v0a --supports--> v05\n- v0b --supports--> v08\n- v03 --supports--> v01\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"499fb0000f211c12162cf9533bef830731717cac6505b0782f9b1ffd8e9e43fd","schema":"livegraph.boolean-propagation.v1","spec_hash":"56c0f5b82c664e752d4496c4ebb5066e8842cb301d6f0de85542f4a9caa7bef2","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-82db2a7e87981c37aa34","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v0c\n- v01 --refutes--> v02\n- v09 --supports--> v08\n- v01 --refutes--> v00\n- v02 --refutes--> v0a\n- v04 --supports--> v09\n- v0b --supports--> v01\n- v0b --refutes--> v03\n- v06 --refutes--> v04\n- v02 --supports--> v05\n- v0c --refutes--> v06\n- v0c --refutes--> v07\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e87557275e4d6b15165dae8567b4a0cab4ee9710634cf9dedbfb1f416875143e","schema":"livegraph.boolean-propagation.v1","spec_hash":"82db2a7e87981c37aa34349c8b79e8881d562e9c537442fa40f38a7759b24ae3","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-e57e44132b537ac88bce","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v0c\n- v0c --refutes--> v05\n- v04 --supports--> v06\n- v06 --supports--> v03\n- v06 --refutes--> v08\n- v09 --refutes--> v04\n- v08 --refutes--> v00\n- v0a --supports--> v09\n- v0b --refutes--> v0a\n- v07 --supports--> v02\n- v06 --refutes--> v01\n- v07 --refutes--> v0b\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9775b16651f16c5b8fc66174d9b465149fd485ead85455f47989de6b40c25aff","schema":"livegraph.boolean-propagation.v1","spec_hash":"e57e44132b537ac88bced9f9e15aad582ddcf8cdb8da67e7c943962207c68c13","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-214af4c2d7bbf7beaa3d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v0c\n- v0c --supports--> v0b\n- v02 --refutes--> v05\n- v02 --supports--> v01\n- v06 --refutes--> v03\n- v05 --refutes--> v09\n- v06 --refutes--> v00\n- v00 --refutes--> v04\n- v0c --refutes--> v07\n- v04 --supports--> v02\n- v07 --refutes--> v08\n- v07 --refutes--> v0a\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"14ac8afc2ee36748a81343295c5604b57e6ae0f446f01a8bb3b731f8015c06d0","schema":"livegraph.boolean-propagation.v1","spec_hash":"214af4c2d7bbf7beaa3d7fe00bfeaf4bf36d6d87d94ac58cabefbc6f08f937b3","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-8ce5a50acecf65a39e01","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v05\n- v01 --refutes--> v03\n- v00 --supports--> v0b\n- v0c --refutes--> v04\n- v08 --refutes--> v09\n- v07 --refutes--> v0a\n- v08 --refutes--> v06\n- v0b --supports--> v08\n- v02 --supports--> v0c\n- v01 --supports--> v00\n- v0c --refutes--> v01\n- v0a --refutes--> v02\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e584cb79eb2e3b7d44b1a4415763c752616631c2501b8fafcc63e39667189300","schema":"livegraph.boolean-propagation.v1","spec_hash":"8ce5a50acecf65a39e01182851385011937e1b0bd2bd803914d70c72b163c682","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-79697b25eb9f55b970ad","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v03\n- v03 --supports--> v06\n- v01 --refutes--> v07\n- v05 --supports--> v0c\n- v05 --refutes--> v0b\n- v05 --supports--> v02\n- v08 --supports--> v04\n- v0a --supports--> v05\n- v0c --refutes--> v09\n- v0c --supports--> v01\n- v06 --refutes--> v00\n- v06 --supports--> v0a\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0aacf50960aeefc7df12dcb290feae5a3b3b0daf3767bbe78aea9987655936e4","schema":"livegraph.boolean-propagation.v1","spec_hash":"79697b25eb9f55b970ad532fdc2a5ea5ca47c73b85d500ded364c9baaa8dcc71","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-1a8bb3b757f867d6de3b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v08\n- v00 --supports--> v06\n- v04 --refutes--> v02\n- v08 --refutes--> v0a\n- v0c --supports--> v0b\n- v00 --supports--> v01\n- v01 --supports--> v09\n- v04 --refutes--> v03\n- v02 --supports--> v07\n- v0a --supports--> v00\n- v09 --refutes--> v04\n- v0c --refutes--> v05\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"48452fd5d3045387432f14d8762c1a7c82659e390200e78633125ef83042f10c","schema":"livegraph.boolean-propagation.v1","spec_hash":"1a8bb3b757f867d6de3b416c45b5ff5f8ece7c9f6f67bf65c1c46ca6e9cff6af","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-41ef40c5bc59009b3c43","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v08\n- v06 --refutes--> v03\n- v07 --refutes--> v0c\n- v09 --supports--> v0a\n- v0b --supports--> v09\n- v09 --supports--> v04\n- v01 --refutes--> v05\n- v03 --supports--> v01\n- v08 --refutes--> v06\n- v03 --refutes--> v00\n- v09 --refutes--> v02\n- v0c --refutes--> v0b\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b10add0bd12851fef0be0b2f3e483fc4016952247f35e5cc49b1b480ebc1714e","schema":"livegraph.boolean-propagation.v1","spec_hash":"41ef40c5bc59009b3c430154c3ef5bfc88a8cf2b922577744a5dc0e3b7ddb76d","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-ef8a0739c12cfe45eea3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v05\n- v04 --refutes--> v06\n- v0b --refutes--> v02\n- v04 --supports--> v0a\n- v05 --refutes--> v0c\n- v02 --refutes--> v04\n- v08 --supports--> v09\n- v08 --supports--> v00\n- v0c --refutes--> v01\n- v0b --refutes--> v03\n- v00 --supports--> v0b\n- v0c --refutes--> v07\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"17f14b9d3a861e48abed5306248d65a79d7c2d778adf4a2124e63b74b71bdb2c","schema":"livegraph.boolean-propagation.v1","spec_hash":"ef8a0739c12cfe45eea36f6d2b614f92d854e3790dda58f21ca25acb21090690","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-e917f184ff5502d2279d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v05\n- v05 --supports--> v0b\n- v06 --supports--> v09\n- v05 --supports--> v04\n- v02 --supports--> v0c\n- v0b --supports--> v08\n- v09 --refutes--> v01\n- v08 --supports--> v02\n- v08 --supports--> v00\n- v0c --supports--> v06\n- v05 --refutes--> v0a\n- v08 --supports--> v07\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6be428eca31ca7ffa7ef7db3fd27c776205534b3f2ab48aa56903841697727af","schema":"livegraph.boolean-propagation.v1","spec_hash":"e917f184ff5502d2279d07cb0d1f030d51d02fd541b95f0487eccd9f059d36bd","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-f7ddea027030d9880c80","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v06\n- v02 --refutes--> v0a\n- v06 --supports--> v00\n- v03 --refutes--> v08\n- v02 --supports--> v01\n- v00 --refutes--> v05\n- v04 --refutes--> v0b\n- v0b --refutes--> v02\n- v02 --refutes--> v09\n- v03 --supports--> v04\n- v05 --supports--> v03\n- v06 --supports--> v0c\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e3fd75625cfd01454d5544237bd7f3c31228c25ca7d91a9ddb54ba5a88084164","schema":"livegraph.boolean-propagation.v1","spec_hash":"f7ddea027030d9880c80cc61e622829f6162c2f248ae9a17acb2b0bdc054fefc","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-e8f98a5e970044833948","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v0a\n- v08 --refutes--> v09\n- v05 --refutes--> v07\n- v0b --refutes--> v08\n- v03 --refutes--> v0c\n- v03 --supports--> v06\n- v05 --supports--> v02\n- v04 --refutes--> v05\n- v01 --supports--> v03\n- v05 --refutes--> v01\n- v09 --refutes--> v04\n- v06 --supports--> v00\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4850f0022de97697f754fea3ad137f8640e433eaaabbabab113671b60e99acd1","schema":"livegraph.boolean-propagation.v1","spec_hash":"e8f98a5e970044833948e6f4c78b40a81e2eba1d37e3f76b28b2148def17a3c8","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-47537b577469a349542f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v01\n- v0c --refutes--> v0a\n- v01 --supports--> v08\n- v09 --supports--> v00\n- v0b --refutes--> v05\n- v04 --supports--> v0c\n- v08 --refutes--> v04\n- v0a --supports--> v0b\n- v0c --supports--> v03\n- v04 --supports--> v06\n- v09 --refutes--> v02\n- v09 --refutes--> v07\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"29c6e0d1dcd35e2f7b426c3c7c5d702c7fee402e39caf37b26a77e1b03afd284","schema":"livegraph.boolean-propagation.v1","spec_hash":"47537b577469a349542f4f9645528f564a7fe52572b393e13a91561760b3d8a0","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-0de8a50739caa4c69993","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v00\n- v07 --refutes--> v0b\n- v04 --supports--> v08\n- v0c --refutes--> v0a\n- v08 --refutes--> v02\n- v06 --refutes--> v05\n- v05 --refutes--> v03\n- v0a --supports--> v07\n- v08 --refutes--> v01\n- v01 --refutes--> v06\n- v05 --refutes--> v0c\n- v04 --refutes--> v09\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"69775ac4137b8ece969cdc7eb7a6e7949df20d8618ccf2f2cec23cd3d432ff74","schema":"livegraph.boolean-propagation.v1","spec_hash":"0de8a50739caa4c69993b4ab68fe459fc9a5703e709038e0da01c683d6d24bfb","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-4923c5496c7afef52d8f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v01\n- v00 --refutes--> v03\n- v05 --supports--> v07\n- v00 --refutes--> v0b\n- v08 --refutes--> v02\n- v06 --refutes--> v0c\n- v01 --supports--> v04\n- v06 --supports--> v08\n- v01 --refutes--> v0a\n- v02 --refutes--> v00\n- v01 --refutes--> v09\n- v09 --refutes--> v06\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c92a8d7a415602ad10e088ba9ac811f3a148c0afd52c68c27d40fe12d32116cd","schema":"livegraph.boolean-propagation.v1","spec_hash":"4923c5496c7afef52d8f085c98920ec99980535ee9d1492c441575ea0bdf8d9c","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":8,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-3b181c48116528e56a85","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v0a\n- v04 --refutes--> v05\n- v0a --refutes--> v0b\n- v06 --refutes--> v00\n- v01 --refutes--> v09\n- v0a --supports--> v08\n- v0a --supports--> v0c\n- v00 --refutes--> v03\n- v0c --supports--> v06\n- v01 --refutes--> v04\n- v05 --supports--> v07\n- v07 --supports--> v02\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d0635f9499a3d08c6c64ade18146e43d3c853e7fe3578b046622dec0f3064ca6","schema":"livegraph.boolean-propagation.v1","spec_hash":"3b181c48116528e56a8557a7fe81656fba3aa6e2fcd9df18addafd1011dd1b2e","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-cfa6715523ba232c5071","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v01\n- v01 --supports--> v0a\n- v02 --refutes--> v07\n- v04 --refutes--> v08\n- v08 --supports--> v05\n- v01 --refutes--> v02\n- v07 --refutes--> v06\n- v03 --supports--> v04\n- v04 --refutes--> v0c\n- v04 --refutes--> v09\n- v06 --supports--> v03\n- v02 --supports--> v0b\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7a4a253a9dff0a8a8120f01baf2dd8d6f1f798d356f48b9cbe113c1b2b782f70","schema":"livegraph.boolean-propagation.v1","spec_hash":"cfa6715523ba232c507130d67b5729c8b92bc610609eee0919f5903deac9afb8","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":8,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-091c25403629e9f2b4f1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v01\n- v0c --refutes--> v03\n- v0b --supports--> v07\n- v00 --refutes--> v04\n- v08 --supports--> v0c\n- v0b --supports--> v05\n- v02 --refutes--> v08\n- v05 --refutes--> v00\n- v06 --refutes--> v0b\n- v0c --refutes--> v09\n- v05 --refutes--> v0a\n- v04 --refutes--> v02\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"13a80c779f8df08b627e47064ba3bced501715febe33d58d578c0b522b10adb1","schema":"livegraph.boolean-propagation.v1","spec_hash":"091c25403629e9f2b4f1b53228bfa7c61c9406caceac2707466340e1688d30b8","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-c1086716a05019716747","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v07\n- v06 --supports--> v0a\n- v0b --supports--> v03\n- v07 --supports--> v05\n- v07 --supports--> v02\n- v05 --supports--> v0b\n- v08 --refutes--> v0c\n- v01 --refutes--> v06\n- v0b --refutes--> v01\n- v0c --refutes--> v09\n- v01 --supports--> v04\n- v0a --supports--> v00\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"684439799f345ed4fe912f190fe173414e684da5d35abd9f9bae4d6be41c8b0a","schema":"livegraph.boolean-propagation.v1","spec_hash":"c1086716a050197167476c97445d009cc200c6c8d18a0e327efc610397c58721","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-7a91bfe53606974a038f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v00\n- v0c --refutes--> v0a\n- v0a --supports--> v08\n- v06 --refutes--> v0c\n- v06 --supports--> v01\n- v0c --refutes--> v03\n- v02 --supports--> v04\n- v07 --refutes--> v0b\n- v07 --supports--> v09\n- v05 --supports--> v02\n- v07 --supports--> v05\n- v08 --refutes--> v07\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"757126278764e8e42c387782e4820de6cc460adeece155005973ab12e100ed15","schema":"livegraph.boolean-propagation.v1","spec_hash":"7a91bfe53606974a038ff44bc7313135a2d91b0998129929002a0431864f7a9e","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-b1c2e8658e9c1e9f9085","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v0c\n- v04 --refutes--> v08\n- v03 --refutes--> v09\n- v07 --supports--> v06\n- v02 --refutes--> v03\n- v06 --refutes--> v04\n- v05 --refutes--> v07\n- v03 --supports--> v0b\n- v04 --refutes--> v00\n- v0a --supports--> v05\n- v04 --refutes--> v02\n- v07 --supports--> v01\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"53e72735f6202647d6f5bb068c92a2276ae7340c98e4ae67995a086ed2a75a23","schema":"livegraph.boolean-propagation.v1","spec_hash":"b1c2e8658e9c1e9f90850b268790ab6a86900ecf0a40f5d63e26b46ad4be8358","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-7df9dc83da081607bc12","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v03\n- v03 --supports--> v00\n- v0b --refutes--> v01\n- v0c --refutes--> v08\n- v0b --supports--> v0a\n- v00 --refutes--> v0c\n- v02 --refutes--> v09\n- v06 --supports--> v0b\n- v00 --refutes--> v04\n- v0b --supports--> v05\n- v0c --supports--> v06\n- v09 --refutes--> v07\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b0c628b2e6b7eb57452f840e4c537de83a3a02bfa7f144eb68e76d079592990c","schema":"livegraph.boolean-propagation.v1","spec_hash":"7df9dc83da081607bc12f507b0291315f7a4b2ca0380d113b6044a08f406fd76","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-b15fea0d4ee8b150ddc6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v0b\n- v00 --supports--> v0a\n- v08 --refutes--> v03\n- v0c --supports--> v08\n- v09 --supports--> v02\n- v00 --supports--> v07\n- v0b --refutes--> v06\n- v03 --refutes--> v09\n- v0b --supports--> v05\n- v05 --supports--> v00\n- v00 --supports--> v0c\n- v08 --refutes--> v04\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"45909a7f5f516a2062952dd312178d8c9b8006d741023934ce9da1432296b0c3","schema":"livegraph.boolean-propagation.v1","spec_hash":"b15fea0d4ee8b150ddc61c1d06429ef18793ff20a88af34f1d2483a850db0296","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-1ae3d7100acd4eaff4bf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v00\n- v06 --refutes--> v0a\n- v00 --refutes--> v03\n- v05 --refutes--> v0c\n- v08 --supports--> v09\n- v09 --supports--> v0b\n- v04 --refutes--> v01\n- v01 --refutes--> v06\n- v0b --supports--> v05\n- v00 --supports--> v08\n- v09 --refutes--> v07\n- v04 --refutes--> v02\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4f53e3c3201119fe1dad228cb2817916d6fa999289363693f286bd438d6e758d","schema":"livegraph.boolean-propagation.v1","spec_hash":"1ae3d7100acd4eaff4bf21ed3186ecdae65d914cf2b96fc9bd20beb6a42695f3","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-af253f517aa755a29960","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v0c\n- v00 --refutes--> v05\n- v0a --supports--> v01\n- v08 --refutes--> v0a\n- v0c --refutes--> v00\n- v02 --supports--> v0b\n- v06 --supports--> v03\n- v00 --refutes--> v02\n- v06 --supports--> v07\n- v0a --refutes--> v09\n- v03 --refutes--> v04\n- v03 --refutes--> v08\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d2d0de456306eac8739bc5b11a69965ee2cadbc80f566d76fb403776e540bee6","schema":"livegraph.boolean-propagation.v1","spec_hash":"af253f517aa755a299609c0ca3539d90a83f68f3ae6ab321515ab6ff9ca01607","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-96935f5861780e7e1d0a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v04\n- v03 --refutes--> v00\n- v0c --refutes--> v01\n- v0c --supports--> v0a\n- v0a --refutes--> v07\n- v00 --supports--> v09\n- v02 --refutes--> v0b\n- v0b --supports--> v06\n- v05 --refutes--> v02\n- v09 --refutes--> v0c\n- v00 --refutes--> v08\n- v0b --refutes--> v03\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d527d678730898cd68b9c5e90d62c79fe86d3b2663d100a24b26694a0e3ff566","schema":"livegraph.boolean-propagation.v1","spec_hash":"96935f5861780e7e1d0a99644595aee21209ff851015003a803d4f7d27babeb8","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-7306e0c134bd873f96f4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v04\n- v0b --supports--> v02\n- v05 --refutes--> v00\n- v05 --refutes--> v08\n- v01 --refutes--> v0a\n- v0b --refutes--> v07\n- v01 --supports--> v03\n- v0c --refutes--> v01\n- v07 --supports--> v05\n- v06 --refutes--> v0c\n- v09 --refutes--> v0b\n- v08 --supports--> v06\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c3e1ec5240711162240c244e1d593fb9f1c544370a445cbf8896c3b3d9ea53b7","schema":"livegraph.boolean-propagation.v1","spec_hash":"7306e0c134bd873f96f4d6ea9d9857c5e72af3374148fa89fcb5eaf9071a55a0","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-9d57041ca9dd65a1f6e6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v0b\n- v02 --supports--> v09\n- v03 --supports--> v01\n- v08 --refutes--> v07\n- v0a --refutes--> v0c\n- v0b --supports--> v06\n- v08 --refutes--> v04\n- v03 --supports--> v00\n- v06 --refutes--> v03\n- v02 --refutes--> v05\n- v08 --supports--> v0a\n- v01 --refutes--> v08\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fd8c5526e22de00e1a2bec957cf22783857af4704cc4d25ad59c9f7e6bd9762e","schema":"livegraph.boolean-propagation.v1","spec_hash":"9d57041ca9dd65a1f6e6641bfac13f92e843c8749dc325d8cc048a4826a21817","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-0913331d8fa9e9970aeb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v00\n- v03 --supports--> v08\n- v06 --supports--> v04\n- v07 --supports--> v0b\n- v00 --supports--> v03\n- v02 --supports--> v05\n- v04 --refutes--> v02\n- v04 --refutes--> v0c\n- v08 --refutes--> v07\n- v08 --supports--> v0a\n- v09 --supports--> v06\n- v00 --supports--> v01\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"844d92dbef722d8302e558a8e2e88e05f40030a1671efe08c1ee10cbd0e89a3d","schema":"livegraph.boolean-propagation.v1","spec_hash":"0913331d8fa9e9970aeb67bc766d1bb5fcad1c215ac5d29c8dc2d6c8b34360fc","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":8,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-a441963541711d9bd1bd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v0c\n- v06 --refutes--> v07\n- v02 --refutes--> v06\n- v05 --supports--> v03\n- v04 --supports--> v00\n- v0a --refutes--> v05\n- v08 --supports--> v02\n- v0b --supports--> v04\n- v05 --supports--> v09\n- v03 --supports--> v08\n- v04 --refutes--> v01\n- v04 --refutes--> v0a\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"58e657ba439b81c91b988c4a60259ce621365c8475c660774f2a3e03745d411b","schema":"livegraph.boolean-propagation.v1","spec_hash":"a441963541711d9bd1bd6ee19f5da8b27bd92815af16357121876be81352dd48","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-715cf386534dc75d605b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v06\n- v09 --refutes--> v00\n- v08 --supports--> v05\n- v0b --supports--> v07\n- v04 --supports--> v0a\n- v08 --supports--> v01\n- v02 --supports--> v04\n- v04 --supports--> v0c\n- v09 --supports--> v03\n- v00 --refutes--> v08\n- v0a --supports--> v0b\n- v07 --supports--> v09\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"44d043c58776f9f5d736357d75073abcb0a774635b183b52b1b90abac62a9084","schema":"livegraph.boolean-propagation.v1","spec_hash":"715cf386534dc75d605bd0aa9ef891166223bff0e6fd6aea3cca5203bb6f78a1","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-7a52c1775b11ddf06f2f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v06\n- v05 --refutes--> v03\n- v07 --refutes--> v01\n- v0b --refutes--> v02\n- v06 --refutes--> v0c\n- v03 --refutes--> v07\n- v01 --supports--> v04\n- v00 --refutes--> v05\n- v0b --supports--> v09\n- v0c --refutes--> v0a\n- v01 --supports--> v0b\n- v05 --refutes--> v08\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"09926d68c505881522d500752b95dac84f6e74160dd538c0b6d8189b390ff046","schema":"livegraph.boolean-propagation.v1","spec_hash":"7a52c1775b11ddf06f2f08c2550205f5bb59828bde762ae08cf4ceda50414b9c","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-fc7f5149c1dcfd6cfa8b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v08\n- v01 --supports--> v0c\n- v0a --refutes--> v04\n- v0a --supports--> v03\n- v0b --supports--> v02\n- v0b --supports--> v05\n- v0a --supports--> v00\n- v0c --supports--> v07\n- v03 --supports--> v06\n- v05 --supports--> v09\n- v06 --refutes--> v01\n- v09 --refutes--> v0a\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8e2655ac9a88f073a12c441b7024f5211273fbef755155b8d048ee8571c9d662","schema":"livegraph.boolean-propagation.v1","spec_hash":"fc7f5149c1dcfd6cfa8b1b84c211e3cdb63d5d7c3dc42e2abe4fd523a02ec7b8","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-baa3adc168517f418cc2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v02\n- v05 --refutes--> v0b\n- v04 --supports--> v03\n- v08 --supports--> v01\n- v04 --refutes--> v0a\n- v02 --supports--> v09\n- v00 --supports--> v08\n- v0a --refutes--> v07\n- v0a --supports--> v00\n- v0b --supports--> v06\n- v05 --supports--> v0c\n- v06 --refutes--> v04\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"07e9bafde206be69e23a99b19fcbfa9bf2e7b7cffec2ce486d6e6d7c9d6d09da","schema":"livegraph.boolean-propagation.v1","spec_hash":"baa3adc168517f418cc255f8bf6e42973cf9e8a51814ce062bd5d9c6057125fc","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-2369e174774a1923307a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v06\n- v00 --supports--> v01\n- v06 --supports--> v09\n- v01 --refutes--> v05\n- v0c --refutes--> v03\n- v03 --supports--> v02\n- v07 --refutes--> v0b\n- v0a --supports--> v0c\n- v03 --supports--> v00\n- v03 --refutes--> v08\n- v0c --supports--> v04\n- v06 --supports--> v07\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"81ed7b13ab0fb3dd1a1cec19ae95009e0336fd151ce20d27f06db937c25eeb5c","schema":"livegraph.boolean-propagation.v1","spec_hash":"2369e174774a1923307a421b51f00659532f8c3bbf5127af2251ace94292c6b3","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-81cb456f640407ade463","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v00\n- v0a --supports--> v01\n- v03 --supports--> v02\n- v08 --supports--> v07\n- v00 --refutes--> v0c\n- v0b --supports--> v03\n- v01 --supports--> v05\n- v00 --supports--> v0a\n- v03 --supports--> v06\n- v08 --supports--> v0b\n- v05 --refutes--> v04\n- v05 --supports--> v08\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b59fc6c6e50c3c9fab79d9a71213061b7170a606aef597f81364ea95cebdf21d","schema":"livegraph.boolean-propagation.v1","spec_hash":"81cb456f640407ade4637a2c6467b20612931a1ba71224734bfa45b94d08fb9d","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-48d3bc89e37223addce2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v05\n- v05 --supports--> v0a\n- v01 --supports--> v08\n- v02 --refutes--> v09\n- v02 --supports--> v04\n- v02 --supports--> v03\n- v0a --refutes--> v0b\n- v08 --supports--> v02\n- v0a --supports--> v06\n- v0c --refutes--> v01\n- v03 --supports--> v07\n- v0a --refutes--> v00\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e1798d9e29ee3fc7300b99890771d18ac9787f9b766b7f319e42273741e0b84e","schema":"livegraph.boolean-propagation.v1","spec_hash":"48d3bc89e37223addce249bbb5aaf729075d1133a63fa2ea3162162f474db8da","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-00fdb98f682547a86f22","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v09\n- v05 --refutes--> v0c\n- v01 --refutes--> v0a\n- v03 --refutes--> v00\n- v09 --supports--> v06\n- v08 --supports--> v0b\n- v06 --refutes--> v04\n- v03 --refutes--> v07\n- v04 --supports--> v05\n- v02 --supports--> v08\n- v03 --refutes--> v02\n- v05 --refutes--> v03\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d55ff2c7fb228dcef839f7cfc3406c5636668bdc643e7cacaf47fd11ad861bea","schema":"livegraph.boolean-propagation.v1","spec_hash":"00fdb98f682547a86f2245371554093d9142c7279ac966eb87de62d2c5f3c64b","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":8,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-a424563adf6fdcef363f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v06\n- v03 --refutes--> v0c\n- v04 --supports--> v00\n- v05 --supports--> v04\n- v04 --supports--> v09\n- v04 --supports--> v07\n- v08 --supports--> v02\n- v08 --refutes--> v0a\n- v0a --supports--> v03\n- v08 --refutes--> v0b\n- v0c --refutes--> v05\n- v06 --supports--> v01\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"42d7af7d4e986804be33d3a9761487627d8ff06950ef278850b83674a3a47d1d","schema":"livegraph.boolean-propagation.v1","spec_hash":"a424563adf6fdcef363f2692e297cd3eb021b51bec02b4af4b19e96828c80330","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-693e9d351c87818edba7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v05\n- v07 --refutes--> v01\n- v0a --supports--> v02\n- v07 --supports--> v0c\n- v0b --refutes--> v03\n- v0c --refutes--> v0a\n- v06 --supports--> v09\n- v09 --supports--> v04\n- v09 --refutes--> v08\n- v08 --refutes--> v00\n- v05 --refutes--> v0b\n- v0a --supports--> v06\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d1e1daad350af270ac9607020299a8e47f751d36f40c53b34a617d479aa0cd79","schema":"livegraph.boolean-propagation.v1","spec_hash":"693e9d351c87818edba71d3b423196aace1f5b1fec40f0ffa45ddf7434331d08","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-cf02d2d8352a73cac6e0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v02\n- v04 --supports--> v0b\n- v05 --supports--> v00\n- v0a --supports--> v01\n- v02 --refutes--> v04\n- v05 --refutes--> v06\n- v06 --supports--> v09\n- v03 --supports--> v05\n- v0a --supports--> v08\n- v01 --refutes--> v0c\n- v01 --refutes--> v03\n- v09 --refutes--> v07\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a06fd12a21647591e1143980f812c8d8cdbec398632133adc2225e47301c6ba4","schema":"livegraph.boolean-propagation.v1","spec_hash":"cf02d2d8352a73cac6e046a25aed0069527b16e947f31fdc644cde1b376fc63e","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":8,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-b314ca58c6f7464c4f32","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v07\n- v03 --supports--> v0c\n- v08 --supports--> v06\n- v05 --refutes--> v0b\n- v07 --refutes--> v04\n- v03 --refutes--> v05\n- v0a --supports--> v00\n- v0b --supports--> v08\n- v00 --refutes--> v03\n- v05 --refutes--> v01\n- v03 --refutes--> v09\n- v07 --supports--> v0a\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2aaab0fc4b50745acaf14d762e4c38950282e765ff65e8203c94d8eb473d1d53","schema":"livegraph.boolean-propagation.v1","spec_hash":"b314ca58c6f7464c4f32c4d8a38b814f6513077cf1da0d624fd53abeafa3af92","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-2125ce463309d6eb9f05","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v01\n- v0b --refutes--> v0a\n- v04 --supports--> v02\n- v04 --refutes--> v05\n- v03 --refutes--> v06\n- v00 --supports--> v04\n- v06 --refutes--> v08\n- v02 --supports--> v0b\n- v06 --refutes--> v09\n- v0c --supports--> v03\n- v0a --supports--> v0c\n- v02 --refutes--> v07\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d0425fdda2c9ad67ee9e6d953f1421f01285d958838665ec72c81b1d631c8f33","schema":"livegraph.boolean-propagation.v1","spec_hash":"2125ce463309d6eb9f05bc5f0988b01b71affd33d10c8934af7098335d8a9802","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-664356ca093ced96ab19","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v00\n- v01 --supports--> v08\n- v00 --refutes--> v02\n- v00 --supports--> v01\n- v08 --supports--> v0b\n- v07 --refutes--> v05\n- v05 --supports--> v04\n- v0a --supports--> v03\n- v0b --supports--> v07\n- v0a --supports--> v06\n- v08 --refutes--> v0c\n- v0b --supports--> v09\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"69644e515f55255d3980a45ba249371e774764f4200e9648adf69272b03da652","schema":"livegraph.boolean-propagation.v1","spec_hash":"664356ca093ced96ab1972ad36ce80437bd6797edaf19b469f6597505e37b9b6","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-10c756ce31cc1fb7658c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v0b\n- v05 --supports--> v0c\n- v0b --supports--> v03\n- v0a --supports--> v06\n- v05 --refutes--> v09\n- v0a --refutes--> v02\n- v0b --refutes--> v00\n- v04 --supports--> v08\n- v03 --refutes--> v05\n- v06 --supports--> v04\n- v08 --supports--> v01\n- v07 --supports--> v0a\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6d76b01a3e85d369c77e368918d284170a945c44f76b1845563069eb92603a7c","schema":"livegraph.boolean-propagation.v1","spec_hash":"10c756ce31cc1fb7658c2bd7f67ce69787601b6eb0fa8af0dce70e039f58985e","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-d6366c799e2ab22f88ea","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v00\n- v05 --supports--> v06\n- v09 --supports--> v05\n- v01 --supports--> v09\n- v03 --refutes--> v07\n- v0c --refutes--> v0a\n- v02 --supports--> v08\n- v02 --supports--> v03\n- v0b --refutes--> v01\n- v03 --supports--> v04\n- v04 --supports--> v0c\n- v05 --supports--> v02\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ec341ed28e7e81c9fe8581bf0389bc274fe24ab8a5f1ea4221ac50f04c6fcce5","schema":"livegraph.boolean-propagation.v1","spec_hash":"d6366c799e2ab22f88ea69afc410af332bf35ee11c9f93b0101376ab7c68a76c","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-2702bf0c2afb3dc84392","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v05\n- v04 --supports--> v03\n- v03 --refutes--> v0b\n- v04 --refutes--> v06\n- v01 --refutes--> v04\n- v0b --refutes--> v0a\n- v01 --refutes--> v09\n- v0a --supports--> v00\n- v07 --refutes--> v02\n- v08 --refutes--> v01\n- v02 --refutes--> v08\n- v03 --refutes--> v0c\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ce431bfd9fb84b52d4063e44aca22fab7b57168a1153af0076e402ee2ddcf949","schema":"livegraph.boolean-propagation.v1","spec_hash":"2702bf0c2afb3dc8439225523491e00baafb6f11692fff4639199dde51ff927a","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-113e1896c0cafa0b8963","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v0a\n- v00 --refutes--> v08\n- v02 --refutes--> v00\n- v09 --refutes--> v07\n- v05 --supports--> v06\n- v0b --refutes--> v05\n- v03 --supports--> v0c\n- v09 --supports--> v04\n- v00 --refutes--> v03\n- v01 --supports--> v02\n- v05 --refutes--> v09\n- v0c --refutes--> v0b\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"59a995349ef27d478bd325611fa79ff6a315454c16ff23388733c593b91b75df","schema":"livegraph.boolean-propagation.v1","spec_hash":"113e1896c0cafa0b89633d6e2a01e54e23183aa28c84fff3972b07baee970446","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-cc16ead9e19bf6bf7c96","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v00\n- v08 --supports--> v0a\n- v0b --supports--> v01\n- v01 --refutes--> v0c\n- v01 --supports--> v02\n- v00 --refutes--> v0b\n- v02 --refutes--> v07\n- v02 --supports--> v09\n- v07 --refutes--> v05\n- v05 --supports--> v06\n- v05 --supports--> v08\n- v01 --refutes--> v04\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cad93bb2bbd76bc2688955df3285a06e48cca00ec573ebdb8dea848143ddee4a","schema":"livegraph.boolean-propagation.v1","spec_hash":"cc16ead9e19bf6bf7c96572f9d363c116c0538741563dcc96169a0e5caa2ec3a","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-3ec9a01d5106a5135519","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v00\n- v00 --supports--> v06\n- v02 --supports--> v05\n- v03 --supports--> v0a\n- v0a --supports--> v01\n- v00 --supports--> v03\n- v0a --supports--> v02\n- v02 --refutes--> v07\n- v02 --refutes--> v08\n- v04 --supports--> v0b\n- v08 --supports--> v04\n- v0c --supports--> v09\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b1e2081471de95219ab94493cdb9caf2f99ff43036aa78831ad6e88730582994","schema":"livegraph.boolean-propagation.v1","spec_hash":"3ec9a01d5106a51355199d11c1466003a65767f680dbcbedf59dfcfcce06c15a","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-0ffb02dac8d34e8709cb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v03\n- v04 --refutes--> v08\n- v07 --supports--> v02\n- v0a --supports--> v06\n- v0b --refutes--> v01\n- v01 --supports--> v0c\n- v01 --supports--> v07\n- v0a --supports--> v09\n- v08 --supports--> v0a\n- v00 --supports--> v05\n- v0a --refutes--> v0b\n- v05 --refutes--> v04\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d02fe57d857f31d239f00b6ffaa257eef1c49cb41e9514441ba84a56e5ea10e0","schema":"livegraph.boolean-propagation.v1","spec_hash":"0ffb02dac8d34e8709cb982bfa02bde3ddafc48536de0a81aa1f4cc5cd54d1d4","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-690214faed4002606bcc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v05\n- v07 --refutes--> v0a\n- v02 --supports--> v0b\n- v04 --supports--> v00\n- v0b --supports--> v0c\n- v01 --refutes--> v08\n- v0b --supports--> v06\n- v08 --refutes--> v04\n- v00 --refutes--> v02\n- v0c --supports--> v07\n- v04 --refutes--> v09\n- v04 --refutes--> v03\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fc7bb083d158a37976bfa8c155217d3c3e64b97d99e30082a5ae6864ba1f87d8","schema":"livegraph.boolean-propagation.v1","spec_hash":"690214faed4002606bcc1a94ded6f11b4104268552ec4d63ee71de605cf99fb1","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-b4d5961137205122f041","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v03\n- v0b --refutes--> v01\n- v02 --refutes--> v0a\n- v04 --supports--> v02\n- v00 --supports--> v04\n- v09 --supports--> v06\n- v0b --refutes--> v05\n- v02 --supports--> v07\n- v02 --refutes--> v0b\n- v03 --refutes--> v09\n- v0c --refutes--> v00\n- v09 --refutes--> v0c\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"be6136b27c717d20a7159091318140fc864f0d37f88ed168e665e6de51c963d7","schema":"livegraph.boolean-propagation.v1","spec_hash":"b4d5961137205122f0419fe83bc999a28063bd9e2901a44139b3449f4ab94adc","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-fe7551d21e7efa4ca507","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v03\n- v04 --refutes--> v0c\n- v07 --supports--> v08\n- v05 --refutes--> v0b\n- v08 --supports--> v00\n- v03 --supports--> v07\n- v08 --supports--> v04\n- v0b --refutes--> v02\n- v02 --refutes--> v06\n- v0b --refutes--> v01\n- v02 --supports--> v0a\n- v03 --refutes--> v09\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"71f16055d4d5e677bbf1e7e6916620a402fcb25cca2bd02250c1cd496fbdba9f","schema":"livegraph.boolean-propagation.v1","spec_hash":"fe7551d21e7efa4ca5079c5dce25d17f8e5f25abc0aff038a911aeaa679ff503","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-5d4a019e64a267842ea0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v0c\n- v08 --supports--> v05\n- v09 --refutes--> v0b\n- v01 --refutes--> v06\n- v09 --refutes--> v04\n- v04 --refutes--> v0a\n- v00 --supports--> v08\n- v0a --supports--> v00\n- v05 --refutes--> v01\n- v01 --supports--> v07\n- v07 --supports--> v02\n- v04 --supports--> v03\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5cb6ffcda19fc4b265f2b48d002f0816977ef07e6e225deab44264441a7d019b","schema":"livegraph.boolean-propagation.v1","spec_hash":"5d4a019e64a267842ea028980df52b18128308df001f7cb6aa21e499baa276b6","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":8,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-90b73d1e5a9eacdee18b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v05\n- v0c --supports--> v03\n- v04 --refutes--> v0b\n- v08 --supports--> v02\n- v00 --refutes--> v04\n- v05 --refutes--> v00\n- v01 --supports--> v06\n- v0b --refutes--> v07\n- v04 --refutes--> v0a\n- v01 --supports--> v09\n- v0b --refutes--> v0c\n- v01 --refutes--> v08\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b16ba6fa25b2fe51c90f6227488f1297b80d911860dd28b88ddb4ec78eb8d5a0","schema":"livegraph.boolean-propagation.v1","spec_hash":"90b73d1e5a9eacdee18bcb0a69b79a3c853129b24872abdf624fe302b19ed05e","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-10616fbcfe0d87eba711","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v08\n- v03 --refutes--> v01\n- v04 --supports--> v07\n- v00 --refutes--> v09\n- v05 --refutes--> v0c\n- v0c --refutes--> v06\n- v03 --supports--> v02\n- v01 --refutes--> v0b\n- v0c --supports--> v00\n- v09 --supports--> v0a\n- v09 --refutes--> v03\n- v04 --supports--> v05\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e535ca7f44ec130738f80f21659578dfe0fa6b6557b37c82c7ce986d082acb79","schema":"livegraph.boolean-propagation.v1","spec_hash":"10616fbcfe0d87eba711a788ad04053f0f6ff173105a9cefc1ae05993f9a96be","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-fc42c89ef67217ba0015","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v02\n- v0c --supports--> v06\n- v01 --refutes--> v05\n- v04 --supports--> v07\n- v06 --supports--> v0a\n- v04 --refutes--> v01\n- v05 --refutes--> v03\n- v04 --supports--> v08\n- v01 --supports--> v00\n- v02 --refutes--> v04\n- v05 --supports--> v09\n- v03 --supports--> v0b\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a40a47680f3c0a45a1b8088afd3424fd002da7555beb2a71a409b7dc44ef0646","schema":"livegraph.boolean-propagation.v1","spec_hash":"fc42c89ef67217ba0015b26df8d86ebf475498dd876844859afececd41edb5b3","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-c8ee9a66f9c8e6a71b42","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v05\n- v0c --refutes--> v00\n- v08 --refutes--> v06\n- v04 --refutes--> v02\n- v07 --supports--> v09\n- v01 --refutes--> v0c\n- v06 --refutes--> v04\n- v0a --refutes--> v0b\n- v0b --refutes--> v03\n- v0b --refutes--> v07\n- v07 --supports--> v01\n- v04 --supports--> v0a\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"83300f052c977fced55a10bbfdd23999f0891fafa27c632fb3457db0de5e7301","schema":"livegraph.boolean-propagation.v1","spec_hash":"c8ee9a66f9c8e6a71b421e621ae07d0835b507be65d1e07f371438d2d6a88f0e","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-0ab39ff4274ab3279dd0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v01\n- v03 --supports--> v08\n- v0c --refutes--> v07\n- v0b --supports--> v03\n- v08 --supports--> v05\n- v03 --supports--> v04\n- v09 --supports--> v0b\n- v06 --supports--> v00\n- v05 --supports--> v0c\n- v06 --supports--> v02\n- v06 --refutes--> v09\n- v09 --supports--> v0a\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"603ec5da8ec46ba2d10f4c585c13e3bd637bf1d5262a7d687507853782533aab","schema":"livegraph.boolean-propagation.v1","spec_hash":"0ab39ff4274ab3279dd0ecaa25eb9cde7fc243379d3d923448d537ebebe9e535","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-e385ce6d90f2d82f24b1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v09\n- v0c --refutes--> v0a\n- v05 --supports--> v01\n- v05 --refutes--> v0b\n- v09 --refutes--> v06\n- v09 --supports--> v02\n- v0a --supports--> v08\n- v02 --supports--> v07\n- v07 --refutes--> v03\n- v04 --refutes--> v00\n- v07 --refutes--> v05\n- v01 --refutes--> v04\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3f9a82a5ed4bf79cf7a256165faeeabce1a65a54fed76e9474dab05cf67f795c","schema":"livegraph.boolean-propagation.v1","spec_hash":"e385ce6d90f2d82f24b197968d0e0b186f0cedc9c043dcc413c2be8b1954820b","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-18a45b2ff65c59c5d888","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v08\n- v08 --supports--> v00\n- v0a --supports--> v05\n- v02 --supports--> v0c\n- v07 --refutes--> v04\n- v08 --supports--> v07\n- v05 --supports--> v09\n- v08 --supports--> v06\n- v02 --supports--> v03\n- v0c --refutes--> v01\n- v04 --refutes--> v0a\n- v0c --supports--> v0b\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5e019d930ac63837b952ffd1a64c1c6e72ce7f831c3b3220981cb57857cfdcbc","schema":"livegraph.boolean-propagation.v1","spec_hash":"18a45b2ff65c59c5d888691962baad374702a758ba880ece08dccf50c5490ece","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-a6b13c61987fdad847de","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v04\n- v04 --refutes--> v09\n- v0c --supports--> v00\n- v09 --refutes--> v0a\n- v02 --supports--> v03\n- v05 --refutes--> v01\n- v08 --refutes--> v05\n- v09 --supports--> v08\n- v02 --supports--> v07\n- v03 --refutes--> v06\n- v03 --supports--> v0b\n- v01 --refutes--> v0c\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b685d4c15905fab7e0971769efcb60e3588e4739de60343845bbca38f34f52ea","schema":"livegraph.boolean-propagation.v1","spec_hash":"a6b13c61987fdad847dee9ea8d474026b075eae54dd10ca28d05342ab7943e52","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-354899201b9a1593f822","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v0b\n- v00 --refutes--> v01\n- v07 --supports--> v05\n- v0c --refutes--> v0a\n- v0a --supports--> v09\n- v04 --supports--> v06\n- v0b --supports--> v03\n- v00 --supports--> v08\n- v0c --supports--> v02\n- v04 --refutes--> v00\n- v07 --supports--> v0c\n- v01 --supports--> v07\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"108782b6be04936fb08a751c5e5b4894c10adcc61df38289e4aa54e6554cc48a","schema":"livegraph.boolean-propagation.v1","spec_hash":"354899201b9a1593f82259872775abcc7db4f812ae22f137c6e820eea64cfa34","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-0469c329eae9c7057931","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v06\n- v08 --refutes--> v03\n- v06 --refutes--> v01\n- v02 --supports--> v0a\n- v06 --supports--> v05\n- v05 --refutes--> v09\n- v09 --supports--> v08\n- v07 --supports--> v0b\n- v09 --refutes--> v0c\n- v02 --refutes--> v00\n- v02 --refutes--> v07\n- v03 --refutes--> v04\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7dbfe57bcf07defd8d0e74c1246c0c89512e36e24569ee03b3c36b7ecf7caf6c","schema":"livegraph.boolean-propagation.v1","spec_hash":"0469c329eae9c7057931e5171d6c860115641636fe6517af02a93de472698a11","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-459a48447b1f616b7763","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v05\n- v09 --refutes--> v0a\n- v01 --refutes--> v02\n- v00 --supports--> v09\n- v0a --refutes--> v07\n- v07 --refutes--> v01\n- v02 --refutes--> v0b\n- v08 --supports--> v03\n- v08 --supports--> v04\n- v04 --supports--> v00\n- v0a --refutes--> v0c\n- v07 --refutes--> v06\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b1f39cb635beb51a3bc94cff9c2fe67fdb8ace0b177229b96c5a1a6098c1883d","schema":"livegraph.boolean-propagation.v1","spec_hash":"459a48447b1f616b77637297f05be40ca778ea788b64c0e4560ff3a98809e2e2","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":8,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-0d4962c295a4b360bc8f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v09\n- v00 --supports--> v03\n- v08 --refutes--> v01\n- v0c --refutes--> v04\n- v01 --supports--> v05\n- v09 --refutes--> v0a\n- v0c --refutes--> v06\n- v0b --refutes--> v00\n- v00 --refutes--> v08\n- v02 --supports--> v07\n- v09 --refutes--> v0c\n- v05 --refutes--> v02\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cce8a2d5b39e1390e4f96eb6cb881a3135510007d35800a8fe54b01b9bfe43cc","schema":"livegraph.boolean-propagation.v1","spec_hash":"0d4962c295a4b360bc8f2490717b0952a57f0a61d1af8c97495a8195d2779822","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-324d11e5c0c593234f91","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v0c\n- v01 --refutes--> v04\n- v08 --refutes--> v02\n- v07 --supports--> v06\n- v0a --refutes--> v03\n- v08 --refutes--> v05\n- v05 --refutes--> v09\n- v0a --refutes--> v0b\n- v00 --supports--> v08\n- v0b --refutes--> v01\n- v06 --refutes--> v00\n- v05 --refutes--> v0a\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e324f1a26e4ea57efd4dd4a6aa756df4c1a752cc498f8b5aa9afc987ea782150","schema":"livegraph.boolean-propagation.v1","spec_hash":"324d11e5c0c593234f91ccf9a64020e3cb7746c34dc766bcec14fd93bdc3265f","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-e38113c4e030649698fe","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v0c\n- v06 --supports--> v00\n- v05 --supports--> v08\n- v08 --refutes--> v06\n- v07 --refutes--> v05\n- v09 --refutes--> v04\n- v01 --supports--> v09\n- v07 --refutes--> v02\n- v06 --refutes--> v03\n- v09 --refutes--> v0a\n- v03 --refutes--> v0b\n- v0a --refutes--> v07\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a93616b504378b6f2f2000f0a4b28120b86fdae25bdc30b80d5ce396505895f2","schema":"livegraph.boolean-propagation.v1","spec_hash":"e38113c4e030649698fecce0d0478ea923e0f111242eb0a478e5608ed55cc869","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":8,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-1ae05e38bb8cd97a3286","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v01\n- v07 --refutes--> v00\n- v0a --supports--> v0b\n- v07 --supports--> v0c\n- v04 --supports--> v07\n- v05 --supports--> v0a\n- v0b --refutes--> v09\n- v06 --refutes--> v08\n- v05 --supports--> v03\n- v09 --refutes--> v06\n- v08 --refutes--> v04\n- v04 --refutes--> v02\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"66b628bbbf7529b44d556a46986778480f9f0f9ef37cb4691a41194e1deb4756","schema":"livegraph.boolean-propagation.v1","spec_hash":"1ae05e38bb8cd97a32866f8899a59c4f7035ae2d112ffeeda3809911b14286cb","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-5e60775ff2b140f51aae","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v01\n- v06 --supports--> v09\n- v07 --refutes--> v08\n- v03 --supports--> v00\n- v02 --supports--> v06\n- v08 --refutes--> v03\n- v0a --refutes--> v02\n- v08 --refutes--> v0c\n- v01 --refutes--> v05\n- v01 --refutes--> v0b\n- v03 --supports--> v04\n- v06 --refutes--> v07\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c4022ffc947259f67477c25b110fb142e85aef0df2e2cd863f7c715a0e63d59c","schema":"livegraph.boolean-propagation.v1","spec_hash":"5e60775ff2b140f51aae0b1a911dd7a10830ffe0066b3b3c390e3fd425e6c211","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":8,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-93c97292de17229fcee4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v09\n- v05 --supports--> v04\n- v07 --refutes--> v05\n- v01 --supports--> v08\n- v01 --refutes--> v0c\n- v01 --supports--> v0a\n- v06 --supports--> v03\n- v03 --refutes--> v02\n- v07 --supports--> v00\n- v04 --supports--> v01\n- v0c --refutes--> v0b\n- v03 --supports--> v07\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e419b577847a67e391acff2d615b66bad04ae93624beb90a3cd327e272b170e5","schema":"livegraph.boolean-propagation.v1","spec_hash":"93c97292de17229fcee4ba289e266939e6ded5cc9e4d0d5dfe5ba8d56ca75933","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-c81d56c5760f37316a20","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v0b\n- v0b --supports--> v0a\n- v08 --supports--> v02\n- v05 --supports--> v06\n- v02 --supports--> v04\n- v07 --refutes--> v08\n- v03 --refutes--> v0c\n- v09 --supports--> v05\n- v0c --supports--> v09\n- v0b --refutes--> v01\n- v0c --supports--> v00\n- v02 --supports--> v03\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"588008b4b6a8d920175fea96b3eea2ef7a10f94057c88ff58391a16b481beaf5","schema":"livegraph.boolean-propagation.v1","spec_hash":"c81d56c5760f37316a20a917222bf11836e6aeff43db22d38375a9a53a5421e2","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-58c2c4dacbb40a002844","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v0c\n- v03 --supports--> v04\n- v04 --refutes--> v01\n- v0c --refutes--> v07\n- v01 --supports--> v00\n- v0a --supports--> v05\n- v05 --supports--> v09\n- v0c --supports--> v0a\n- v02 --refutes--> v03\n- v04 --supports--> v06\n- v0a --refutes--> v08\n- v01 --refutes--> v0b\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bc1dc0069d4ce703c3a60a21c1b9da02367d2e21cbad679c7390b79ad1c97be1","schema":"livegraph.boolean-propagation.v1","spec_hash":"58c2c4dacbb40a0028443bd7ba237bebb2d0af64c9a2981c3c192a9c0ee7b4f0","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-bf75c30512fb68666412","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v02\n- v03 --refutes--> v09\n- v03 --refutes--> v01\n- v04 --supports--> v03\n- v0a --refutes--> v06\n- v00 --refutes--> v08\n- v08 --refutes--> v0b\n- v02 --supports--> v04\n- v05 --refutes--> v0c\n- v08 --supports--> v0a\n- v09 --supports--> v05\n- v03 --supports--> v07\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0e1dc34a13808a382aaa8a30158e7e13472912eb784837d34e64adc69ab265de","schema":"livegraph.boolean-propagation.v1","spec_hash":"bf75c30512fb686664123e104ba0015bb7e6b5bb2f752d0ab1eb5360b7ff6df2","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":8,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-d11bc8a1e4c791074302","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v09\n- v09 --refutes--> v06\n- v0b --supports--> v00\n- v04 --supports--> v0a\n- v06 --refutes--> v05\n- v0a --supports--> v08\n- v0a --refutes--> v01\n- v0a --refutes--> v03\n- v06 --supports--> v0c\n- v07 --refutes--> v04\n- v00 --supports--> v02\n- v0c --supports--> v07\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"69e47b06e4f5e5498bbd59ed61dff11b1ee728bfa907dc60246f321de93da420","schema":"livegraph.boolean-propagation.v1","spec_hash":"d11bc8a1e4c791074302d6dea13d7e9e4517c0f6243a5f475cee940800352d70","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-e3c78a9b3ba6e574dbc3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v01\n- v09 --supports--> v0c\n- v02 --supports--> v04\n- v05 --refutes--> v07\n- v04 --supports--> v0a\n- v01 --supports--> v0b\n- v0b --refutes--> v09\n- v0b --supports--> v00\n- v09 --supports--> v05\n- v05 --supports--> v03\n- v06 --supports--> v02\n- v02 --refutes--> v08\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"119c4bf4dc1a073b4b50bcb791da2b2d5b26a1d7f9b7a7bafc34460f7bcafa46","schema":"livegraph.boolean-propagation.v1","spec_hash":"e3c78a9b3ba6e574dbc3c28a92ee431a9078e07ebec7dc86c08c6320e538a3b7","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-683af0c1c66bcd4a9733","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v07\n- v0a --supports--> v02\n- v05 --supports--> v04\n- v08 --supports--> v03\n- v07 --refutes--> v05\n- v04 --refutes--> v0a\n- v0b --refutes--> v0c\n- v05 --refutes--> v01\n- v03 --supports--> v0b\n- v07 --supports--> v00\n- v09 --refutes--> v08\n- v08 --supports--> v06\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"df4619d3d9429b403bf697acf74d05490dd2f7f3b2ac81ec3b6d87f4a33c2b4e","schema":"livegraph.boolean-propagation.v1","spec_hash":"683af0c1c66bcd4a97337bd7cb855832cd6e21f367f1e4108cdf38de4e2711ff","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-901b6992f2719cec6bc7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v02\n- v06 --refutes--> v09\n- v00 --supports--> v04\n- v09 --refutes--> v08\n- v04 --supports--> v0a\n- v04 --supports--> v0c\n- v02 --supports--> v07\n- v05 --refutes--> v00\n- v09 --supports--> v0b\n- v01 --refutes--> v06\n- v09 --supports--> v05\n- v0a --refutes--> v03\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6d6ebdfbe1357177c6990ecdd1c8fa2d9a72e082ab88589d494857010a39babc","schema":"livegraph.boolean-propagation.v1","spec_hash":"901b6992f2719cec6bc79a958a65e8919c9941f1778fec92ddefdc5f04ce2407","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":8,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-6f3c58ab4106d04deceb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v02\n- v08 --supports--> v0c\n- v07 --refutes--> v04\n- v02 --refutes--> v06\n- v01 --supports--> v08\n- v02 --supports--> v0b\n- v09 --refutes--> v05\n- v07 --supports--> v01\n- v08 --supports--> v00\n- v05 --supports--> v0a\n- v0b --supports--> v07\n- v0c --refutes--> v03\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"283a7009ba3528367259a2a6e6a8337f412db9b789810cdd38579fa0bd552a24","schema":"livegraph.boolean-propagation.v1","spec_hash":"6f3c58ab4106d04decebeb32adf0f49bb40da946755226d35388a65b59158ce6","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-2c659b65bfb981fd6825","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v08\n- v0c --supports--> v0a\n- v09 --refutes--> v0b\n- v08 --refutes--> v05\n- v09 --supports--> v01\n- v07 --refutes--> v09\n- v02 --supports--> v0c\n- v09 --supports--> v03\n- v09 --supports--> v06\n- v08 --supports--> v02\n- v0a --supports--> v04\n- v04 --refutes--> v07\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6c3106a67e7f15e1f4edc41eb62204b6e5c0fa460d07f8257fcdee33238e31ce","schema":"livegraph.boolean-propagation.v1","spec_hash":"2c659b65bfb981fd6825452db435d3c741a343df166d4529d961fa29fbf80b48","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-afe214c052f92d2c1746","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v03\n- v08 --supports--> v05\n- v00 --supports--> v08\n- v03 --supports--> v04\n- v0a --supports--> v01\n- v09 --supports--> v0b\n- v03 --supports--> v0c\n- v00 --supports--> v06\n- v08 --refutes--> v09\n- v01 --refutes--> v02\n- v04 --supports--> v0a\n- v08 --supports--> v07\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d063197b02d142743bfa09ecad7f6f741a542bce32b278bbcf6535bc6fac8733","schema":"livegraph.boolean-propagation.v1","spec_hash":"afe214c052f92d2c174637d7f9a19e01af1cccbc931476cb7059303e379f60ae","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-c18a3faf25d20923e3c9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v0c\n- v02 --refutes--> v01\n- v05 --supports--> v09\n- v0a --refutes--> v06\n- v07 --refutes--> v08\n- v06 --supports--> v03\n- v02 --supports--> v07\n- v02 --supports--> v04\n- v07 --supports--> v0a\n- v09 --supports--> v02\n- v05 --supports--> v00\n- v0c --supports--> v0b\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"50e47fa2bc9909396c48cc82a745ac4c3be8aad92072c3ca4d55f7ffbecc12c3","schema":"livegraph.boolean-propagation.v1","spec_hash":"c18a3faf25d20923e3c929124f1aa62795a02722f6a7d01c38e1171d529abc5d","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-6aefe356a5c954361d54","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v02\n- v01 --refutes--> v04\n- v04 --refutes--> v0a\n- v02 --refutes--> v05\n- v07 --refutes--> v06\n- v07 --supports--> v0b\n- v00 --refutes--> v08\n- v06 --refutes--> v0c\n- v02 --refutes--> v00\n- v0a --supports--> v07\n- v06 --refutes--> v03\n- v06 --supports--> v09\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f8dfca7c4db925b93380c1c2b891a1814c098d1d54d78be86201354e76465bed","schema":"livegraph.boolean-propagation.v1","spec_hash":"6aefe356a5c954361d544463c06662c3276a19dde9c07d218dceb8b60504ea28","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":8,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-c7a2c16393576e7e8283","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v0a\n- v09 --supports--> v07\n- v03 --refutes--> v0c\n- v02 --supports--> v06\n- v05 --refutes--> v08\n- v00 --supports--> v04\n- v01 --supports--> v02\n- v07 --refutes--> v05\n- v05 --refutes--> v01\n- v03 --refutes--> v00\n- v02 --supports--> v0b\n- v04 --supports--> v09\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0928a0976aa51370e1df76a6d75b772b8f89c9ecff65c79fa969b0f44181d3b1","schema":"livegraph.boolean-propagation.v1","spec_hash":"c7a2c16393576e7e82834f6c9d210d65ec023cad188f13b7b00dba9b58b137e7","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":8,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-fe3bcda5e183462a9f9e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v0c\n- v06 --refutes--> v0a\n- v09 --supports--> v02\n- v02 --supports--> v01\n- v09 --refutes--> v04\n- v06 --refutes--> v08\n- v05 --supports--> v09\n- v02 --supports--> v00\n- v08 --supports--> v07\n- v07 --supports--> v03\n- v03 --supports--> v0b\n- v0b --supports--> v05\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"49a2c7fa2b696b7f3c1967ff6d9ab054d87a1c3230066fd8c9f3df5b539880c8","schema":"livegraph.boolean-propagation.v1","spec_hash":"fe3bcda5e183462a9f9ec20d21e54dae1d61e67deaf361a43218df89d61d5413","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-39e846a3b9c876306785","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v02\n- v03 --refutes--> v00\n- v0c --refutes--> v05\n- v01 --supports--> v0a\n- v05 --refutes--> v08\n- v0b --supports--> v03\n- v03 --refutes--> v07\n- v07 --refutes--> v06\n- v0a --supports--> v0c\n- v0b --refutes--> v04\n- v05 --refutes--> v0b\n- v06 --supports--> v09\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"08d7d9cc28af8a3ffffbe1132cfac451e0d535766709cbcb2e65ca2536a366d4","schema":"livegraph.boolean-propagation.v1","spec_hash":"39e846a3b9c87630678574882707513c93eb1b5dbc0740a15f51f990799b284c","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-f3e7056ad14eccf01d74","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v02\n- v03 --refutes--> v0a\n- v0c --refutes--> v05\n- v05 --refutes--> v06\n- v07 --refutes--> v01\n- v06 --refutes--> v08\n- v05 --refutes--> v0b\n- v0a --refutes--> v00\n- v09 --refutes--> v03\n- v04 --supports--> v07\n- v06 --refutes--> v04\n- v0a --supports--> v0c\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2598c932867578c0f0f03d7d95feb8183e94ecb59b7d14594639f10ed09638a1","schema":"livegraph.boolean-propagation.v1","spec_hash":"f3e7056ad14eccf01d745abb37905a96069ce792ad0382ef55b757789b43f928","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-acbe001c58042fb79625","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v04\n- v00 --supports--> v05\n- v09 --refutes--> v0b\n- v04 --refutes--> v09\n- v0c --supports--> v07\n- v01 --supports--> v06\n- v0c --refutes--> v08\n- v0a --refutes--> v0c\n- v08 --supports--> v01\n- v09 --supports--> v02\n- v01 --refutes--> v00\n- v03 --refutes--> v0a\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0522b5ec54de13357720031c272b24687f308ac9d19929aefde28ce2196269f8","schema":"livegraph.boolean-propagation.v1","spec_hash":"acbe001c58042fb79625b32f2308b2c12c56f2113d21378d140306d01245e55c","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-8b16f01fec76c6cc6ce4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v09\n- v02 --refutes--> v08\n- v09 --supports--> v02\n- v0b --refutes--> v06\n- v04 --supports--> v0a\n- v02 --supports--> v07\n- v04 --refutes--> v0b\n- v05 --supports--> v01\n- v05 --supports--> v0c\n- v07 --supports--> v05\n- v0c --supports--> v04\n- v04 --refutes--> v00\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f48d336e4692203f211fef7acee79a1d0a5e5d516fd18094382f52e2b3456040","schema":"livegraph.boolean-propagation.v1","spec_hash":"8b16f01fec76c6cc6ce48cdf58fb7c25689726c5c61e023e2361a236d33158f1","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-8fb3d83ed4bac401475c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v0c\n- v09 --refutes--> v00\n- v0c --refutes--> v09\n- v0a --supports--> v05\n- v02 --supports--> v08\n- v05 --supports--> v02\n- v05 --refutes--> v04\n- v0c --supports--> v07\n- v06 --refutes--> v0b\n- v08 --refutes--> v03\n- v00 --refutes--> v06\n- v09 --refutes--> v01\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4899bb4e5f744095d38028accc8b93c8d7447da76515961e484ddd854d12a300","schema":"livegraph.boolean-propagation.v1","spec_hash":"8fb3d83ed4bac401475cfad14ccbbb3fe00b16617b2607d66dffff1073e860e0","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-56b6f45916bce695f7d4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v07\n- v02 --supports--> v0a\n- v0b --refutes--> v01\n- v09 --refutes--> v03\n- v02 --supports--> v04\n- v01 --supports--> v06\n- v00 --supports--> v08\n- v09 --refutes--> v05\n- v0c --refutes--> v02\n- v08 --refutes--> v0c\n- v01 --supports--> v00\n- v03 --supports--> v0b\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c837122397b4a6890e56945d914517ca02221d9d2e7b45825854604d78e97f0d","schema":"livegraph.boolean-propagation.v1","spec_hash":"56b6f45916bce695f7d48cbfaddfa518f172218a670648e04c0fe8842f0f2317","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-c84b6ed9f5486b7e4810","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v09\n- v04 --supports--> v00\n- v0a --supports--> v01\n- v02 --supports--> v03\n- v02 --refutes--> v06\n- v0a --supports--> v0b\n- v06 --refutes--> v0c\n- v06 --supports--> v07\n- v0b --supports--> v08\n- v09 --supports--> v0a\n- v08 --supports--> v05\n- v00 --supports--> v02\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"58d0eb1c3dc8acc8d5875ec3c7c97c2b2eae08f80108863c32bf90d3cfe10ec2","schema":"livegraph.boolean-propagation.v1","spec_hash":"c84b6ed9f5486b7e48108e7188e777177ed4990dc6c9a61708f7135a356b24f9","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-5507f66b605679ce050c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v00\n- v0c --refutes--> v01\n- v01 --supports--> v03\n- v03 --supports--> v09\n- v03 --supports--> v05\n- v05 --supports--> v07\n- v06 --supports--> v0a\n- v04 --refutes--> v0c\n- v03 --supports--> v0b\n- v0a --refutes--> v04\n- v07 --supports--> v02\n- v07 --refutes--> v08\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0d3c8793165478f4790c44e8e9597cdda61fe0df031e5924a311fd0c3c646c3e","schema":"livegraph.boolean-propagation.v1","spec_hash":"5507f66b605679ce050cc664fda951dcbd94f3ab4e5548390a440607d74d42dd","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-10c9c8f45924187bdacc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v02\n- v02 --supports--> v0c\n- v03 --refutes--> v06\n- v02 --refutes--> v09\n- v04 --supports--> v08\n- v0a --supports--> v00\n- v07 --supports--> v01\n- v08 --supports--> v07\n- v0b --supports--> v03\n- v07 --supports--> v05\n- v02 --supports--> v04\n- v00 --refutes--> v0b\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"771ed6d775d5eb8c8ebf1aadaa63331f32abf5b110716b47946d48aaae301650","schema":"livegraph.boolean-propagation.v1","spec_hash":"10c9c8f45924187bdacc0b05c764137db3d97c129f32334658270503a9d19bbb","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-7625faaf039d6c3b236c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v04\n- v00 --refutes--> v0a\n- v0a --refutes--> v0b\n- v09 --refutes--> v06\n- v08 --supports--> v0c\n- v0a --supports--> v02\n- v04 --refutes--> v09\n- v00 --supports--> v07\n- v05 --supports--> v00\n- v06 --refutes--> v01\n- v08 --supports--> v05\n- v04 --refutes--> v03\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a24ee48b38149f285f8285b091ae96cf08f675108781f6737a50e2e5101435bf","schema":"livegraph.boolean-propagation.v1","spec_hash":"7625faaf039d6c3b236c862d552d786d340e9889068696e503fd08cd17844577","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":8,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-d418f1df260d21c7f6b2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v03\n- v03 --supports--> v0a\n- v02 --refutes--> v05\n- v06 --supports--> v01\n- v05 --supports--> v07\n- v06 --refutes--> v02\n- v09 --supports--> v04\n- v09 --supports--> v0c\n- v00 --supports--> v08\n- v07 --supports--> v00\n- v03 --refutes--> v0b\n- v09 --refutes--> v06\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7d9b2e22c654c06673d418ebcfeb087c4eff9cb7c64638f2843c0afa998f3f47","schema":"livegraph.boolean-propagation.v1","spec_hash":"d418f1df260d21c7f6b25d11c73f54f94beb35848cced97c28c7fb998fdb230d","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":8,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-9ce97a496072c8c9a329","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v02\n- v06 --refutes--> v00\n- v04 --supports--> v05\n- v06 --refutes--> v0b\n- v0c --refutes--> v07\n- v0b --refutes--> v0a\n- v0c --supports--> v04\n- v08 --refutes--> v0c\n- v08 --refutes--> v01\n- v03 --supports--> v09\n- v09 --refutes--> v08\n- v05 --refutes--> v06\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e73d802f79db719bd5c8bdf30de7946f94aef0af1a0e4ab1746745ea9d88f57f","schema":"livegraph.boolean-propagation.v1","spec_hash":"9ce97a496072c8c9a329ca1f5f3c8c02e0693db09747821134d4b66df5f44f2c","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-916d8c942b53a9bfe556","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v04\n- v00 --refutes--> v05\n- v02 --refutes--> v00\n- v05 --supports--> v0a\n- v0b --refutes--> v03\n- v06 --supports--> v0b\n- v0b --supports--> v01\n- v03 --refutes--> v07\n- v07 --refutes--> v02\n- v09 --refutes--> v0c\n- v09 --supports--> v06\n- v05 --refutes--> v08\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a6acc78c825e6ba138396b7150410e3cf3683d5c3fbb71f99e742acb7fafa90f","schema":"livegraph.boolean-propagation.v1","spec_hash":"916d8c942b53a9bfe55640d77a790d47177d5e88186fdbdbce4b5466fde49e45","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-6a14edd0f9762aa34136","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v02\n- v0c --refutes--> v09\n- v0a --supports--> v03\n- v00 --refutes--> v06\n- v05 --refutes--> v00\n- v03 --supports--> v0b\n- v00 --refutes--> v0c\n- v05 --refutes--> v08\n- v09 --supports--> v07\n- v04 --refutes--> v05\n- v01 --refutes--> v0a\n- v03 --refutes--> v04\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"da2ddc6b09a2ce78e0211606badd7db8e1e9480f83af219b745e9f0692169d4e","schema":"livegraph.boolean-propagation.v1","spec_hash":"6a14edd0f9762aa34136572a2b18d3888c33220020c11fd1508a561b720a2dfb","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-66241c0fcba1cc60c136","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v07\n- v00 --refutes--> v06\n- v0b --refutes--> v08\n- v08 --refutes--> v01\n- v04 --supports--> v0c\n- v0b --supports--> v0a\n- v05 --supports--> v00\n- v04 --supports--> v05\n- v0b --supports--> v02\n- v0b --supports--> v09\n- v03 --supports--> v0b\n- v07 --refutes--> v03\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8cad2e66566c50e80383400b5b01242124159532a0d4b9065b8f1a23d5bc1ad1","schema":"livegraph.boolean-propagation.v1","spec_hash":"66241c0fcba1cc60c1363bdfed408c602ed5bfb63f67c5ba444832414d66fe10","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-16c399ffbec5437ddf33","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v04\n- v06 --refutes--> v05\n- v0c --refutes--> v09\n- v01 --refutes--> v03\n- v06 --refutes--> v0c\n- v00 --refutes--> v02\n- v00 --refutes--> v06\n- v09 --supports--> v07\n- v04 --supports--> v01\n- v04 --refutes--> v0a\n- v09 --refutes--> v08\n- v0b --supports--> v00\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"100554f2ecabf9772296033a96fdaf35714163d811ba17eab5724d69751fda85","schema":"livegraph.boolean-propagation.v1","spec_hash":"16c399ffbec5437ddf33e9ff4200fa37151ac807338ffb205c511cd811394cfc","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-f4c730bff668b40af3a4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v08\n- v03 --refutes--> v06\n- v0c --refutes--> v0b\n- v05 --refutes--> v03\n- v01 --supports--> v0a\n- v08 --supports--> v02\n- v0a --supports--> v04\n- v03 --refutes--> v01\n- v05 --supports--> v00\n- v05 --supports--> v07\n- v09 --refutes--> v05\n- v0a --supports--> v0c\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0a410487b3c0d65eab5b805aab0521697d95c93aed0a7b0ec93c2c8f359ab1e5","schema":"livegraph.boolean-propagation.v1","spec_hash":"f4c730bff668b40af3a453dce5fc3581647cb2a89e3c91d2dd67cb697b125b31","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-1114067d5f77ad3c218e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v02\n- v02 --supports--> v03\n- v05 --supports--> v0a\n- v09 --refutes--> v06\n- v00 --refutes--> v08\n- v0b --refutes--> v01\n- v00 --refutes--> v0b\n- v01 --refutes--> v04\n- v03 --refutes--> v00\n- v01 --refutes--> v05\n- v0b --refutes--> v07\n- v0b --supports--> v0c\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"440515403b564599cb4a14e262d28f3a6510a880c0e3aa4ca5a19cba75a87001","schema":"livegraph.boolean-propagation.v1","spec_hash":"1114067d5f77ad3c218ec31edb3d1f7ece4af79c5b346bc5c3f0e9007bc8798c","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-71584f3b59874f7db2e5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v04\n- v0a --supports--> v00\n- v06 --refutes--> v0c\n- v04 --refutes--> v03\n- v0b --supports--> v0a\n- v04 --refutes--> v06\n- v09 --refutes--> v07\n- v01 --supports--> v05\n- v08 --supports--> v09\n- v0c --refutes--> v0b\n- v09 --refutes--> v01\n- v01 --refutes--> v02\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"31b56b90cf8961a89eeb3f817eadf6d9bd9868175f43f2dc986ca5c890a5d8f7","schema":"livegraph.boolean-propagation.v1","spec_hash":"71584f3b59874f7db2e58dccee09ebeb6f16f7e94e515b962c5bd5645f2d7ca6","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-df29789b88c3bc866258","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v08\n- v02 --refutes--> v0b\n- v0a --refutes--> v0c\n- v00 --refutes--> v03\n- v01 --supports--> v02\n- v07 --refutes--> v05\n- v07 --supports--> v09\n- v04 --supports--> v01\n- v0c --refutes--> v04\n- v02 --supports--> v06\n- v07 --refutes--> v00\n- v03 --refutes--> v0a\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3dae651185d7b2bf401fc39797ae3fade9f85873b5a6cb854df05fec48463536","schema":"livegraph.boolean-propagation.v1","spec_hash":"df29789b88c3bc866258d6bb4b97e1c3b0278bf8e7daa4f81aa56377c3b68268","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-ad19525423a6e12cd926","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v08\n- v01 --supports--> v0c\n- v0c --refutes--> v02\n- v01 --refutes--> v04\n- v09 --refutes--> v0a\n- v0c --refutes--> v03\n- v0a --refutes--> v00\n- v02 --refutes--> v06\n- v08 --refutes--> v01\n- v02 --supports--> v07\n- v06 --refutes--> v09\n- v01 --refutes--> v05\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b3e56b90c6c6f15d2b83679f739ed85c2c4614809c4087deb4af6a9346819bdf","schema":"livegraph.boolean-propagation.v1","spec_hash":"ad19525423a6e12cd9265c2b6fd6d2ace577e8e7f3c1fa3f6e96a990ccc141a1","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-2df6887d196c041b82f1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v08\n- v00 --supports--> v0c\n- v02 --refutes--> v01\n- v07 --refutes--> v05\n- v03 --refutes--> v06\n- v09 --supports--> v00\n- v0b --refutes--> v09\n- v06 --refutes--> v0b\n- v06 --refutes--> v04\n- v02 --supports--> v03\n- v05 --supports--> v02\n- v07 --supports--> v0a\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1a797a0004c3ab525bd4c80fb72f7222b535efdb1cbb327be667e79aa886eae3","schema":"livegraph.boolean-propagation.v1","spec_hash":"2df6887d196c041b82f126562134ac76bcb4877a43c8be7f47e8660edd05a466","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-715b3057f8cb2295eb39","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v09\n- v07 --supports--> v08\n- v0b --refutes--> v03\n- v0a --supports--> v04\n- v05 --refutes--> v06\n- v08 --supports--> v0b\n- v03 --refutes--> v05\n- v07 --supports--> v0c\n- v01 --refutes--> v00\n- v0a --refutes--> v02\n- v01 --refutes--> v0a\n- v06 --refutes--> v01\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7e8546b74d8de0dc7fe9d898e312f61f5a43406a56268cb462c48287ef0f44a6","schema":"livegraph.boolean-propagation.v1","spec_hash":"715b3057f8cb2295eb393023dfa0d14275ddff7eafe1e7f341a9dbd24e3b3dd8","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-d350c2ad5a21e88a4564","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v08\n- v0a --refutes--> v09\n- v09 --refutes--> v0b\n- v00 --refutes--> v05\n- v0a --refutes--> v01\n- v05 --supports--> v07\n- v02 --supports--> v04\n- v09 --supports--> v02\n- v04 --supports--> v00\n- v03 --refutes--> v06\n- v06 --supports--> v0a\n- v03 --refutes--> v0c\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9e2cbfd11e2d04bc0c2a1484681ec8e9f9ce8e4b198b21e9ddc87b4892188aeb","schema":"livegraph.boolean-propagation.v1","spec_hash":"d350c2ad5a21e88a4564d8b9b5b25601461fc163e882d65f75830278e722eedd","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":8,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-c5f9d6561c6d5055f524","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v06\n- v0b --refutes--> v05\n- v0c --refutes--> v09\n- v09 --refutes--> v0a\n- v0a --refutes--> v02\n- v04 --refutes--> v0b\n- v03 --supports--> v04\n- v05 --supports--> v00\n- v01 --supports--> v03\n- v0c --supports--> v07\n- v0a --supports--> v01\n- v09 --supports--> v08\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8b65da7025483bc91e43080d8e5c27bec2d76f442c4433c6272b630e8856132e","schema":"livegraph.boolean-propagation.v1","spec_hash":"c5f9d6561c6d5055f5248f7c1289e3f874f4d2f0d2a7cf0fd3ab3f32841bc0e5","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-2d772d74ea6e20b1b361","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v0a\n- v02 --supports--> v08\n- v09 --supports--> v0b\n- v03 --supports--> v04\n- v01 --supports--> v00\n- v03 --supports--> v02\n- v02 --refutes--> v07\n- v03 --supports--> v06\n- v00 --supports--> v05\n- v05 --refutes--> v0c\n- v09 --supports--> v01\n- v0a --supports--> v09\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b1155dfccc53c4b0cafd013db991ad505f1105ffb940945d22dab0a43cb3a56d","schema":"livegraph.boolean-propagation.v1","spec_hash":"2d772d74ea6e20b1b361fe5b1a5ef302c75d90a7080ca9110d45caeb525894cd","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":8,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-83f63aee24f556f8aa86","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v0b\n- v01 --supports--> v08\n- v0a --supports--> v01\n- v05 --refutes--> v09\n- v08 --refutes--> v04\n- v00 --refutes--> v05\n- v06 --supports--> v02\n- v06 --supports--> v0a\n- v09 --supports--> v0c\n- v00 --supports--> v07\n- v04 --refutes--> v00\n- v08 --refutes--> v03\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d0e326b83b6f83bf437f8c346284e740342a1b6efb23cfec6ba933a615d07618","schema":"livegraph.boolean-propagation.v1","spec_hash":"83f63aee24f556f8aa862278726b36069f3294973119c30de46fb756c10bb806","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":8,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-f4bcb06320376538dbb6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v05\n- v05 --refutes--> v02\n- v06 --supports--> v01\n- v02 --supports--> v03\n- v07 --refutes--> v09\n- v03 --refutes--> v04\n- v0c --refutes--> v0b\n- v09 --supports--> v06\n- v0a --supports--> v00\n- v0a --refutes--> v08\n- v03 --refutes--> v0c\n- v0a --supports--> v07\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c58424c6eab031f17fb2f98943c174dac02c2baf3a4c5a05e4e03aa161f626d2","schema":"livegraph.boolean-propagation.v1","spec_hash":"f4bcb06320376538dbb6678c8435aeebac010103e98c6f38f3a78916b47ea77b","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-aa8cd78d70c197b75747","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v03\n- v01 --refutes--> v09\n- v0b --refutes--> v04\n- v08 --supports--> v0c\n- v00 --supports--> v01\n- v09 --supports--> v0b\n- v0b --supports--> v0a\n- v01 --refutes--> v02\n- v01 --refutes--> v06\n- v0a --supports--> v07\n- v0c --refutes--> v05\n- v05 --supports--> v00\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"46c5522931b5fec9f52f10ceec4df6762372ecc3f7138f70d2dbd4b2b135834a","schema":"livegraph.boolean-propagation.v1","spec_hash":"aa8cd78d70c197b757476fd1f7219762c5f1a3c681afccd1ded8256a5621772d","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-d9b1d3e94b5c4a635628","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v05\n- v07 --supports--> v0b\n- v06 --refutes--> v03\n- v01 --refutes--> v08\n- v08 --supports--> v09\n- v05 --refutes--> v06\n- v07 --supports--> v02\n- v0b --supports--> v0c\n- v08 --refutes--> v04\n- v04 --supports--> v0a\n- v08 --supports--> v00\n- v0a --supports--> v07\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cb6962e8948f25170c8a6c5ba6b9d874d27e487202614bfeafc938bddd609b55","schema":"livegraph.boolean-propagation.v1","spec_hash":"d9b1d3e94b5c4a635628a9b5bdd69138ac3359b89dba4d10f2187442a006c0f4","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-fba729e3d5be29f91345","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v0b\n- v07 --supports--> v09\n- v03 --refutes--> v0c\n- v0b --supports--> v02\n- v00 --refutes--> v08\n- v0b --supports--> v07\n- v00 --supports--> v06\n- v04 --refutes--> v03\n- v06 --supports--> v01\n- v00 --refutes--> v05\n- v03 --supports--> v00\n- v0a --supports--> v04\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d5594c46ba36c237800d020437159cf5f40ea5e7590bf20ac82650355a8c22f4","schema":"livegraph.boolean-propagation.v1","spec_hash":"fba729e3d5be29f9134597fab0b4e5d4f3cae0cfab0b2a9bd70a29b19323a67b","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-c95b567736b13b1e3c52","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v04\n- v05 --supports--> v06\n- v07 --refutes--> v0a\n- v02 --refutes--> v0b\n- v09 --refutes--> v05\n- v09 --refutes--> v01\n- v04 --supports--> v00\n- v00 --refutes--> v03\n- v04 --supports--> v08\n- v07 --refutes--> v09\n- v05 --refutes--> v0c\n- v0c --refutes--> v02\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f1bfd9933ce7c2f91b82ac0e689f0d04261fb119ce3afa8af7e16b14c2660b43","schema":"livegraph.boolean-propagation.v1","spec_hash":"c95b567736b13b1e3c52a0f445eded8d8aec4a67954a59e6cfb0aaf2a8450ff5","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-2add1df4534ac8490a24","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v01\n- v07 --supports--> v02\n- v02 --supports--> v0c\n- v0c --supports--> v03\n- v00 --supports--> v07\n- v07 --refutes--> v0a\n- v01 --supports--> v06\n- v03 --refutes--> v04\n- v02 --refutes--> v09\n- v04 --refutes--> v0b\n- v02 --supports--> v08\n- v01 --supports--> v00\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"be09b3ad2da5faff56e7e799a10a3440e130d70fdaccc7c5bed8ced168eaa455","schema":"livegraph.boolean-propagation.v1","spec_hash":"2add1df4534ac8490a24edc3dd925bb6b38d7bc2f5c5acaabe0aa526ae533826","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-3c6aef5809158f5c1aad","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v08\n- v0a --supports--> v00\n- v01 --refutes--> v0c\n- v00 --supports--> v04\n- v0c --supports--> v02\n- v0a --refutes--> v03\n- v05 --refutes--> v09\n- v06 --refutes--> v0b\n- v02 --refutes--> v05\n- v00 --refutes--> v01\n- v06 --supports--> v0a\n- v09 --supports--> v07\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d1fd8a5e4fd150aeaa611f5baa443c80dadeb46031ce7c6973793c81a46fd175","schema":"livegraph.boolean-propagation.v1","spec_hash":"3c6aef5809158f5c1aad3ad9644081b812ac74b798c06cca30020e7fc03dc76f","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-de05eb4e45b89de55cef","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v00\n- v04 --refutes--> v05\n- v02 --refutes--> v0b\n- v04 --supports--> v0c\n- v0a --supports--> v08\n- v09 --refutes--> v07\n- v08 --supports--> v01\n- v07 --supports--> v04\n- v08 --supports--> v02\n- v0c --refutes--> v06\n- v0c --supports--> v0a\n- v0b --supports--> v03\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"51b195e9dc58fe4530c23d560b270a75f73eac72f5e14f844460d47c8cda2190","schema":"livegraph.boolean-propagation.v1","spec_hash":"de05eb4e45b89de55cef536680853302da865b40dc158aa9b63e2516446014cf","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-4f603ff072ec72688edf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v01\n- v0c --refutes--> v03\n- v04 --refutes--> v06\n- v00 --refutes--> v04\n- v06 --supports--> v02\n- v08 --supports--> v0a\n- v02 --refutes--> v0b\n- v03 --supports--> v05\n- v03 --supports--> v07\n- v08 --refutes--> v00\n- v0b --supports--> v09\n- v0b --supports--> v0c\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5ff979d2554e0654ecaffc2f6e62b57ba747a43c690bf5fa181568a59fd1a1ba","schema":"livegraph.boolean-propagation.v1","spec_hash":"4f603ff072ec72688edfc19432f37c1a58690d58994b74dda76420d4ba71713e","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-f5b0b18114812c2826a2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v09\n- v07 --supports--> v0c\n- v0b --supports--> v02\n- v00 --refutes--> v03\n- v03 --refutes--> v01\n- v03 --supports--> v0a\n- v04 --supports--> v08\n- v0c --supports--> v00\n- v04 --refutes--> v05\n- v02 --supports--> v07\n- v04 --supports--> v0b\n- v07 --supports--> v06\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"508bb0bc9ed6298f0777358369900900fe2692a663eb03d06b5e1a08abedf6bc","schema":"livegraph.boolean-propagation.v1","spec_hash":"f5b0b18114812c2826a27d87669d9835e32c3f0960bac3bbfbc81d54c2130e28","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-fc7ec86a825eb29ad822","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v0c\n- v01 --supports--> v07\n- v06 --supports--> v01\n- v02 --supports--> v09\n- v08 --supports--> v0a\n- v0b --refutes--> v08\n- v06 --refutes--> v03\n- v07 --refutes--> v00\n- v09 --supports--> v06\n- v0b --refutes--> v05\n- v07 --refutes--> v04\n- v04 --supports--> v0b\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ebf72c2acb8cc062d047fcba04408518dc3be009421f6f367b152b0f82ecd615","schema":"livegraph.boolean-propagation.v1","spec_hash":"fc7ec86a825eb29ad822dba9d31daca2e428a827c027d2fb7739170e523fe061","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-4dd7a727ad7abf8ebf44","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v09\n- v0b --supports--> v01\n- v07 --supports--> v0a\n- v08 --refutes--> v0c\n- v05 --supports--> v02\n- v05 --supports--> v08\n- v04 --supports--> v07\n- v08 --supports--> v00\n- v0a --refutes--> v0b\n- v08 --refutes--> v04\n- v09 --supports--> v06\n- v0a --supports--> v03\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"123f3ef5326b2514274dae088763f0dc250456fcef1e3bf27e2edc54a0d04232","schema":"livegraph.boolean-propagation.v1","spec_hash":"4dd7a727ad7abf8ebf447e83081bcf8aa36f54cf2c456e226ae3f6937a400de1","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-cc8ae7c59e918224bd77","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v0b\n- v00 --supports--> v0a\n- v03 --refutes--> v06\n- v08 --supports--> v05\n- v01 --supports--> v00\n- v03 --supports--> v04\n- v07 --supports--> v0c\n- v03 --refutes--> v07\n- v01 --refutes--> v09\n- v0b --supports--> v03\n- v0a --refutes--> v08\n- v02 --supports--> v01\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"64e2088e85fa9fce7dd45404ae4aa26d2e886df72e15c552270e273fc9395512","schema":"livegraph.boolean-propagation.v1","spec_hash":"cc8ae7c59e918224bd777c51c578396ebcc773de20ddfa546bc7754bdbdf0124","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-28a7b0ea495817be63d8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v01\n- v08 --refutes--> v03\n- v07 --refutes--> v0b\n- v09 --supports--> v0c\n- v09 --supports--> v05\n- v04 --refutes--> v08\n- v03 --supports--> v06\n- v00 --refutes--> v09\n- v0a --refutes--> v00\n- v07 --supports--> v0a\n- v09 --supports--> v04\n- v02 --supports--> v07\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5167ee764feae8df16edaccfd8cdddc02734ac8cdf54fa655bf93abfd4a7c814","schema":"livegraph.boolean-propagation.v1","spec_hash":"28a7b0ea495817be63d81227fe27718bb975a3717e74484886fd3ebf69e8d9c2","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":8,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-d0e16bbacf824ba6668a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v0a\n- v01 --refutes--> v0b\n- v07 --refutes--> v01\n- v00 --supports--> v0c\n- v01 --supports--> v00\n- v08 --refutes--> v03\n- v07 --refutes--> v04\n- v0c --supports--> v09\n- v0c --refutes--> v06\n- v06 --refutes--> v08\n- v0a --refutes--> v07\n- v00 --supports--> v02\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cccbcc5fca1aba5dcc69f7e21abae3998292d66c1c23884eccb635ffdd2abcbf","schema":"livegraph.boolean-propagation.v1","spec_hash":"d0e16bbacf824ba6668ae9490af5b1f553321700dda025a9e8e6c4b12ddf5609","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-1830018817cc8e912b16","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v05\n- v00 --supports--> v09\n- v09 --refutes--> v0c\n- v08 --refutes--> v02\n- v09 --refutes--> v0a\n- v07 --refutes--> v03\n- v08 --supports--> v04\n- v07 --refutes--> v06\n- v0c --supports--> v01\n- v06 --supports--> v08\n- v02 --refutes--> v0b\n- v01 --supports--> v07\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"78021be50c7a5ec7fa4885d319b75f0367e47a5d8a7ec88a8726da03609b80df","schema":"livegraph.boolean-propagation.v1","spec_hash":"1830018817cc8e912b16c6f92b2761ef7d79f620d060ffd980ad10b6767e6bd4","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-e74a70a02111c644d281","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v08\n- v09 --refutes--> v05\n- v03 --supports--> v04\n- v0a --supports--> v09\n- v05 --supports--> v02\n- v0a --refutes--> v0b\n- v01 --refutes--> v07\n- v08 --refutes--> v03\n- v09 --refutes--> v00\n- v08 --supports--> v06\n- v02 --refutes--> v0c\n- v04 --supports--> v0a\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"71be509721a5020fa472970d1d27be2cc6b2fe449f52d31ef225f66cf87ba552","schema":"livegraph.boolean-propagation.v1","spec_hash":"e74a70a02111c644d281b7163ff39a1df3c99ace740019ee91ace1705c9075e4","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":8,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-9b5748c6c366eabc038c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v02\n- v04 --refutes--> v08\n- v00 --refutes--> v0c\n- v04 --supports--> v03\n- v05 --supports--> v01\n- v0a --refutes--> v06\n- v06 --supports--> v0b\n- v02 --supports--> v05\n- v00 --refutes--> v04\n- v05 --supports--> v09\n- v0b --supports--> v07\n- v08 --refutes--> v0a\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7e2635c3c7d48359d616375a53f1e084daf3daed3d73dddcfac24b1c9b7ba1e6","schema":"livegraph.boolean-propagation.v1","spec_hash":"9b5748c6c366eabc038c9dd11ff208a09ca91dfaf8be10fbae6d538ade40b52b","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-bb716ab7748a31a3636e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v03\n- v09 --refutes--> v0c\n- v0b --supports--> v02\n- v01 --refutes--> v05\n- v01 --refutes--> v06\n- v02 --supports--> v08\n- v04 --supports--> v0a\n- v04 --supports--> v0b\n- v08 --refutes--> v01\n- v07 --refutes--> v04\n- v0c --supports--> v07\n- v0c --supports--> v00\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"20502d7293b2e805040f00c33d04985919bf30dbf09985d00dfe12706aa825f3","schema":"livegraph.boolean-propagation.v1","spec_hash":"bb716ab7748a31a3636e6a7ab00ea3a405934bfc4ebd84c399fb9ae35ec756ec","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-7b94b85f2e0287461115","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v01\n- v0c --refutes--> v0a\n- v07 --refutes--> v00\n- v05 --refutes--> v02\n- v04 --refutes--> v07\n- v09 --refutes--> v05\n- v02 --refutes--> v03\n- v00 --supports--> v0b\n- v0a --refutes--> v08\n- v0a --supports--> v09\n- v0a --refutes--> v06\n- v00 --refutes--> v0c\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9d1595a62c58e2cf57baf99133eed4b877bd2196d22347f5f49efedc35e87e03","schema":"livegraph.boolean-propagation.v1","spec_hash":"7b94b85f2e028746111598471f4bd4ef712a6e6fdd7a642c59cdce5d98757983","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-01e160c81c07055e3c35","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v08\n- v07 --supports--> v01\n- v01 --refutes--> v0c\n- v0b --supports--> v02\n- v01 --supports--> v09\n- v03 --supports--> v06\n- v09 --refutes--> v04\n- v0a --supports--> v07\n- v09 --refutes--> v0b\n- v08 --supports--> v05\n- v0b --supports--> v00\n- v00 --refutes--> v03\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"470e18fbce8ad25d10659c8d32ea5117a7b1882395b11cf6bb0e1102bb310aff","schema":"livegraph.boolean-propagation.v1","spec_hash":"01e160c81c07055e3c35950a0607401b2ed1bf1585e93ffced6c054848c11618","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-1449bcc1b1da00c0fae4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v05\n- v0a --supports--> v07\n- v03 --refutes--> v0a\n- v08 --refutes--> v0c\n- v01 --supports--> v08\n- v01 --supports--> v02\n- v0c --refutes--> v00\n- v00 --refutes--> v03\n- v0a --refutes--> v06\n- v08 --supports--> v0b\n- v0a --refutes--> v09\n- v04 --refutes--> v01\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"eb67ee5f1ecd7fd0981e5fac5fb580045f0e69d08836e389899157bf6191705b","schema":"livegraph.boolean-propagation.v1","spec_hash":"1449bcc1b1da00c0fae45a04aa71d52b294794148fa31c9276783afd7d1d26e3","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-e0e5b4f8e29fb5eb25a7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v02\n- v07 --supports--> v09\n- v0b --supports--> v0c\n- v0b --refutes--> v01\n- v0a --supports--> v07\n- v07 --refutes--> v06\n- v0b --refutes--> v00\n- v01 --refutes--> v04\n- v0a --refutes--> v08\n- v05 --refutes--> v0b\n- v03 --refutes--> v05\n- v04 --refutes--> v0a\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e0584af3355004275e1c4876c3bbf4ffd2a75d7ebe323c81fbee4ea7cf00d4e2","schema":"livegraph.boolean-propagation.v1","spec_hash":"e0e5b4f8e29fb5eb25a76608910f3c4ed74b832fd4ab47033994801ef1636eb2","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-a4cc0b983207eb06de48","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v02\n- v04 --supports--> v03\n- v03 --refutes--> v08\n- v05 --supports--> v07\n- v09 --refutes--> v00\n- v0a --supports--> v01\n- v03 --supports--> v0c\n- v0a --refutes--> v06\n- v09 --supports--> v0b\n- v03 --refutes--> v09\n- v07 --refutes--> v04\n- v01 --supports--> v05\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c305a7b64b1000054524a6679abceeb27566bdd7208cb1702eb3034c1d4ad844","schema":"livegraph.boolean-propagation.v1","spec_hash":"a4cc0b983207eb06de483dba44e705b842d61e7f55abec75c3b50b31c0945dfd","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":8,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-6670312a2814f8648e07","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v05\n- v09 --refutes--> v02\n- v02 --supports--> v00\n- v03 --refutes--> v0b\n- v0b --supports--> v08\n- v06 --supports--> v03\n- v01 --supports--> v04\n- v09 --supports--> v0c\n- v00 --supports--> v06\n- v01 --refutes--> v0a\n- v02 --refutes--> v07\n- v04 --refutes--> v09\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"31a168c10699985a271d4b9d0651bb743fa21152c0b4215d66d4dc0c8ccf0bc7","schema":"livegraph.boolean-propagation.v1","spec_hash":"6670312a2814f8648e07ac2371f656fddf4bbfc65899800a95d1f3d7b3db1ebf","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-31f664c649c3dce84152","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v01\n- v00 --refutes--> v07\n- v09 --supports--> v00\n- v04 --supports--> v08\n- v06 --refutes--> v09\n- v00 --refutes--> v04\n- v01 --refutes--> v05\n- v02 --supports--> v0a\n- v05 --refutes--> v06\n- v05 --refutes--> v03\n- v05 --refutes--> v0c\n- v02 --supports--> v0b\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8d101a8d9b23e7309433d1ff8e84e2a55403689e97ec5fd1b38340979f4b8872","schema":"livegraph.boolean-propagation.v1","spec_hash":"31f664c649c3dce84152b236f8dc19c40636d6cf6272ffe299283f50a92c3ef3","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":8,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-8727802d4eaa27c3df99","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v02\n- v0b --refutes--> v01\n- v01 --supports--> v06\n- v0a --refutes--> v07\n- v07 --supports--> v00\n- v00 --supports--> v04\n- v00 --refutes--> v03\n- v01 --supports--> v05\n- v08 --supports--> v0a\n- v09 --refutes--> v0b\n- v00 --refutes--> v0c\n- v04 --supports--> v09\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1d6d2ab58af2552057ce942036deaca51ad67c2764ed8df0412bb1965b62cc53","schema":"livegraph.boolean-propagation.v1","spec_hash":"8727802d4eaa27c3df99119cd584925d9fc4d8a6208a5c90c8693e8ed229a61f","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-24990b72981c49118bee","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v09\n- v02 --supports--> v0c\n- v05 --supports--> v0a\n- v05 --supports--> v0b\n- v09 --refutes--> v06\n- v0c --refutes--> v00\n- v05 --refutes--> v08\n- v00 --supports--> v04\n- v03 --refutes--> v07\n- v05 --supports--> v03\n- v04 --supports--> v05\n- v01 --supports--> v02\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5e6870f704867b5702168e34c35728aa1d24d57d03a302b7111791047c4b5815","schema":"livegraph.boolean-propagation.v1","spec_hash":"24990b72981c49118bee656be9b60af07b6e3534ec71c225918fd25565639734","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-2c81653625c3fe95967b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v01\n- v00 --supports--> v05\n- v00 --refutes--> v07\n- v02 --refutes--> v08\n- v01 --refutes--> v00\n- v0a --supports--> v04\n- v02 --refutes--> v0c\n- v0b --refutes--> v0a\n- v06 --refutes--> v0b\n- v06 --refutes--> v03\n- v04 --refutes--> v09\n- v00 --refutes--> v02\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"19b1d963a17a53bc6324810616630789be3982fff12c2063bf0bffd2848a70d0","schema":"livegraph.boolean-propagation.v1","spec_hash":"2c81653625c3fe95967b5394b69437b6c9ea05f2f27e5d755e4836e464d16e63","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-95a7da8a9f014be4dfb2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v06\n- v0c --supports--> v08\n- v06 --refutes--> v0c\n- v0a --supports--> v04\n- v07 --refutes--> v02\n- v08 --supports--> v0b\n- v08 --supports--> v09\n- v0a --refutes--> v01\n- v03 --refutes--> v00\n- v07 --supports--> v03\n- v04 --refutes--> v07\n- v03 --supports--> v05\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5065b08b715c0161d7d2d6e6f0bf095435dd6d17b82573db05fbc6339c47724f","schema":"livegraph.boolean-propagation.v1","spec_hash":"95a7da8a9f014be4dfb2ce2f98fb661ff8f5650eeaf6a04cb907f06dcd39e5d2","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-29487a3251d451997b78","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v07\n- v05 --supports--> v00\n- v01 --refutes--> v08\n- v0c --supports--> v02\n- v00 --supports--> v01\n- v08 --supports--> v06\n- v01 --supports--> v03\n- v05 --refutes--> v04\n- v06 --supports--> v0c\n- v0a --supports--> v05\n- v0a --refutes--> v0b\n- v09 --refutes--> v0a\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"01ab59aacd65d87666bf53cb251da9599e5afeb084aeefd28a241cb27ed51fad","schema":"livegraph.boolean-propagation.v1","spec_hash":"29487a3251d451997b789d1cda6c1188a3c3b4a97bc362a0fcebc2be1bdbe8b5","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-250802641428b0b87b56","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v02\n- v0a --supports--> v04\n- v07 --refutes--> v0b\n- v07 --refutes--> v03\n- v04 --supports--> v0c\n- v09 --refutes--> v08\n- v09 --refutes--> v05\n- v01 --refutes--> v09\n- v07 --supports--> v06\n- v08 --refutes--> v00\n- v00 --supports--> v0a\n- v0b --supports--> v01\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"195fad0a44fa1a74519ac6b1926de45cd3a77a8ba41798f77e7ddea6a6a14f76","schema":"livegraph.boolean-propagation.v1","spec_hash":"250802641428b0b87b5695d4412005eec61ab25a2c364ad0c8230db7616af4ae","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-ac4f33b9fe31df7c2cac","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v09\n- v05 --refutes--> v06\n- v02 --supports--> v0a\n- v06 --refutes--> v00\n- v01 --refutes--> v02\n- v02 --supports--> v04\n- v04 --supports--> v07\n- v03 --refutes--> v0b\n- v04 --refutes--> v03\n- v05 --refutes--> v08\n- v0c --supports--> v01\n- v0b --refutes--> v05\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f2ceb8b837313f3a8bdd654e3f205e7362910d7cb2d70096a21abb8a8380daef","schema":"livegraph.boolean-propagation.v1","spec_hash":"ac4f33b9fe31df7c2cac0863aae3af7eb7e63975f434f88644e4608c8d4f39cd","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-5476fa6faa0eb5d5476e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v08\n- v02 --refutes--> v09\n- v07 --refutes--> v0a\n- v0a --supports--> v00\n- v06 --refutes--> v0b\n- v06 --supports--> v0c\n- v08 --refutes--> v03\n- v0a --supports--> v04\n- v00 --refutes--> v06\n- v01 --refutes--> v02\n- v03 --refutes--> v05\n- v03 --refutes--> v01\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f1dbacc849fdcb833e3da52eb0e008b688c0a996d8021668cdb989826624d923","schema":"livegraph.boolean-propagation.v1","spec_hash":"5476fa6faa0eb5d5476e0acc29f4e44e980f9729406ce9a90618023c8fc385d5","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":8,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-207d056ac617e02620b2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v08\n- v0c --supports--> v00\n- v06 --refutes--> v09\n- v0b --supports--> v04\n- v03 --refutes--> v0a\n- v03 --supports--> v02\n- v00 --supports--> v03\n- v05 --supports--> v07\n- v0c --refutes--> v01\n- v04 --supports--> v06\n- v06 --supports--> v05\n- v02 --refutes--> v0b\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"03a3ad09800812b7303262905e7f0785441a6437ab227af900222868796b4a18","schema":"livegraph.boolean-propagation.v1","spec_hash":"207d056ac617e02620b2c3b3798c61cb43f1074575d6981328468fa7e09b7908","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-f2edfd03c3dd64282010","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v06\n- v05 --refutes--> v04\n- v04 --supports--> v00\n- v08 --supports--> v0a\n- v03 --refutes--> v09\n- v08 --supports--> v07\n- v08 --supports--> v01\n- v06 --supports--> v0b\n- v00 --refutes--> v08\n- v0b --supports--> v05\n- v04 --refutes--> v02\n- v05 --supports--> v0c\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8a973c2c41f8042625eb5b499c66226f56ddcedfdf96b8136055aa875703fbbe","schema":"livegraph.boolean-propagation.v1","spec_hash":"f2edfd03c3dd64282010214848b21012de59f1a4b55155203a8d171f4c794a5e","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-4e0b24bdb0fc42192e5f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v00\n- v0c --supports--> v09\n- v08 --supports--> v0a\n- v04 --refutes--> v06\n- v01 --supports--> v07\n- v0b --refutes--> v03\n- v0a --refutes--> v0c\n- v0c --supports--> v05\n- v04 --refutes--> v01\n- v0b --supports--> v04\n- v00 --supports--> v08\n- v02 --refutes--> v0b\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"efb6376bdbc15b686535ea205965e234d7f4a23b530286d0d73a9ece63b62c49","schema":"livegraph.boolean-propagation.v1","spec_hash":"4e0b24bdb0fc42192e5f5cd8565fab6ebc7cac7530a7d96074e15c24f3f23b56","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-aa773f4b14181305a184","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v03\n- v03 --refutes--> v09\n- v0c --refutes--> v07\n- v02 --supports--> v00\n- v0a --refutes--> v08\n- v04 --refutes--> v0a\n- v03 --refutes--> v06\n- v03 --supports--> v02\n- v0a --supports--> v0b\n- v00 --supports--> v01\n- v01 --refutes--> v0c\n- v0c --supports--> v05\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e8fee5890eb50128ee6304671b8cb0691c3bfe5a12cfa0c40c50db21caca6841","schema":"livegraph.boolean-propagation.v1","spec_hash":"aa773f4b14181305a184c3b6b40889417efb50a00012e83bffeba7b180895392","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-8ee9061d5658690a3bbd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v08\n- v03 --refutes--> v05\n- v0c --refutes--> v03\n- v04 --refutes--> v00\n- v02 --refutes--> v0c\n- v02 --refutes--> v01\n- v06 --supports--> v09\n- v09 --supports--> v0b\n- v06 --refutes--> v0a\n- v09 --refutes--> v07\n- v03 --refutes--> v04\n- v08 --refutes--> v06\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"aee46d0f6f56eda64ba29d3ee83b988a28f10680fee43945f6a32f1741567a81","schema":"livegraph.boolean-propagation.v1","spec_hash":"8ee9061d5658690a3bbd4c5a4e8a7c537a4ce3fcada234ca5512ff38827e533d","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-88bbf1a1fb9e15f8d778","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v04\n- v03 --refutes--> v00\n- v0a --refutes--> v0b\n- v0c --supports--> v06\n- v07 --refutes--> v01\n- v01 --refutes--> v03\n- v08 --supports--> v09\n- v08 --supports--> v05\n- v05 --supports--> v0c\n- v0c --refutes--> v07\n- v07 --refutes--> v02\n- v0b --refutes--> v08\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0cd28cce5e84d96e43cbe448e6f9f2a452352a5b212d9c3b5dbd832818379f46","schema":"livegraph.boolean-propagation.v1","spec_hash":"88bbf1a1fb9e15f8d7786ca47165a5c41cc6c9d27eeec1fa9ecfe87beed29cca","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-4badf35e8195f21b8dc1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v00\n- v08 --refutes--> v0b\n- v01 --supports--> v0c\n- v05 --supports--> v03\n- v06 --refutes--> v0a\n- v01 --supports--> v04\n- v04 --supports--> v08\n- v0a --supports--> v09\n- v0a --supports--> v07\n- v02 --supports--> v01\n- v03 --refutes--> v06\n- v08 --refutes--> v05\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e5bd6fa01f88ead193a2d1ecdb70a18af8df3886328315ebfbab13882e489bfe","schema":"livegraph.boolean-propagation.v1","spec_hash":"4badf35e8195f21b8dc1e8d05efd96ab8b904efee20b294dc46643aabb59170c","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":8,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-932c7f1aab5c52bf0cfa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v08\n- v06 --refutes--> v04\n- v08 --supports--> v06\n- v0c --refutes--> v05\n- v02 --supports--> v0b\n- v0a --refutes--> v07\n- v01 --refutes--> v09\n- v02 --refutes--> v0c\n- v06 --supports--> v02\n- v05 --refutes--> v03\n- v05 --supports--> v0a\n- v08 --supports--> v00\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2b4e1b8f5cf5cbaf9b7d765c304b5eeb0b97885f5a25435c245686d501dfc36f","schema":"livegraph.boolean-propagation.v1","spec_hash":"932c7f1aab5c52bf0cfabb943dc01f59f9e5551dd75f8ef45ec0ab13c10baa02","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-df7e228518148f733cdd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v03\n- v07 --refutes--> v02\n- v07 --supports--> v00\n- v08 --refutes--> v06\n- v04 --supports--> v0b\n- v06 --supports--> v04\n- v01 --supports--> v05\n- v00 --supports--> v0a\n- v01 --supports--> v07\n- v0c --refutes--> v08\n- v04 --supports--> v09\n- v09 --refutes--> v01\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7edfbdc6faa4f88d4d572d1578ecdfd64261796c4f6157588460a9426142f2de","schema":"livegraph.boolean-propagation.v1","spec_hash":"df7e228518148f733cddbc4534c473362b8721b1fbf46ab15154f1bca82ad27b","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-9ea13eb510c1c1c9d161","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v04\n- v04 --supports--> v02\n- v0a --refutes--> v00\n- v08 --supports--> v09\n- v05 --supports--> v01\n- v01 --supports--> v08\n- v05 --supports--> v0b\n- v00 --supports--> v07\n- v08 --refutes--> v06\n- v04 --supports--> v0a\n- v07 --supports--> v03\n- v00 --refutes--> v0c\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2e1eddfb7b862b7f058e66cbbb77289e9f0bd84ce3c9ccf7f6c89ea36d8bb43e","schema":"livegraph.boolean-propagation.v1","spec_hash":"9ea13eb510c1c1c9d1612da391114273f3cc1124c162c872a4174240fb998e25","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":8,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-b9615952e9f89c24fca4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v05\n- v06 --refutes--> v0c\n- v00 --refutes--> v0a\n- v03 --supports--> v06\n- v03 --refutes--> v01\n- v00 --supports--> v02\n- v09 --refutes--> v00\n- v05 --supports--> v03\n- v00 --supports--> v08\n- v04 --refutes--> v07\n- v0c --supports--> v04\n- v0c --refutes--> v0b\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e1926f174f76412b5d3479b7df27e683faa729ee61c05cd26e8fb7793dfce0ec","schema":"livegraph.boolean-propagation.v1","spec_hash":"b9615952e9f89c24fca49c31105f9c11bc344601a5d56c5f33f33749904051b7","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-57334684c04f2d0747dd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v05\n- v03 --refutes--> v08\n- v08 --supports--> v09\n- v07 --refutes--> v00\n- v06 --refutes--> v0a\n- v07 --refutes--> v0c\n- v04 --supports--> v03\n- v08 --supports--> v0b\n- v0a --refutes--> v07\n- v02 --supports--> v01\n- v08 --supports--> v02\n- v01 --refutes--> v06\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f0fa718a4e83f05ee1e5dc9696f4ecf3eec9f4ea950e6b3a067ae913e0d1af60","schema":"livegraph.boolean-propagation.v1","spec_hash":"57334684c04f2d0747dd2c3b823362dfb33bf888a6c4c525fd249a447063876f","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":8,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-a5cc904f4023c4d2b07d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v0a\n- v04 --supports--> v05\n- v07 --refutes--> v01\n- v04 --refutes--> v08\n- v06 --refutes--> v03\n- v0c --refutes--> v06\n- v0c --refutes--> v0b\n- v04 --supports--> v0c\n- v07 --refutes--> v09\n- v00 --refutes--> v04\n- v0a --supports--> v00\n- v03 --refutes--> v02\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7fcb0b5fafa0068b5062d8be86b15ee898be10e9ed2024b2f4b26c33e35a8227","schema":"livegraph.boolean-propagation.v1","spec_hash":"a5cc904f4023c4d2b07d1f0f95aee489b880c76f642c8dc8acdee8d122f03daa","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-9b560c011078c1d588ad","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v09\n- v06 --refutes--> v04\n- v09 --supports--> v05\n- v06 --supports--> v07\n- v01 --supports--> v06\n- v02 --refutes--> v08\n- v04 --supports--> v0a\n- v01 --supports--> v03\n- v09 --supports--> v02\n- v0b --refutes--> v01\n- v04 --refutes--> v00\n- v0a --supports--> v0c\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"436b3f0c3e4decd7cd161fbd014dbe97f884141ae1984b2339b2f50b902a9088","schema":"livegraph.boolean-propagation.v1","spec_hash":"9b560c011078c1d588ad611bc8ea54ccafafc68004dc415beb968d73292ca244","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-65b941716c986aadebcf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v02\n- v06 --supports--> v0c\n- v06 --refutes--> v05\n- v0b --refutes--> v06\n- v0b --refutes--> v04\n- v07 --refutes--> v01\n- v05 --supports--> v03\n- v06 --refutes--> v00\n- v02 --refutes--> v0a\n- v03 --refutes--> v08\n- v08 --supports--> v07\n- v02 --refutes--> v09\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"245c8d8dcf0bd9320bdf3b192d802bcccbce5010b257f7f356c81eec92e3d955","schema":"livegraph.boolean-propagation.v1","spec_hash":"65b941716c986aadebcf8e168ef80b8a2b0979f27d6c9654b0df3e3926c36756","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-bb600b7200ee5c28853e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v09\n- v09 --supports--> v05\n- v03 --refutes--> v04\n- v05 --supports--> v07\n- v0b --supports--> v00\n- v09 --supports--> v02\n- v0a --refutes--> v0b\n- v03 --refutes--> v0a\n- v00 --supports--> v01\n- v05 --supports--> v08\n- v08 --supports--> v03\n- v09 --supports--> v0c\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b0b791da84d95f84044e488a09945f4bddabef13e13532951d69fbe492bdca54","schema":"livegraph.boolean-propagation.v1","spec_hash":"bb600b7200ee5c28853ecc0e73ea3b02cbacfa93ee98d6b2907944582aafe305","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-9c41093aa17bba9d46e8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v07\n- v05 --supports--> v0c\n- v0c --supports--> v01\n- v0c --supports--> v09\n- v0a --supports--> v02\n- v0c --supports--> v06\n- v05 --refutes--> v0b\n- v06 --refutes--> v04\n- v04 --supports--> v08\n- v00 --refutes--> v05\n- v0a --refutes--> v00\n- v07 --refutes--> v03\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e411c951b3c488a521e9a4ef9aa02f4622cb74251f18e6aeebea6399c4122974","schema":"livegraph.boolean-propagation.v1","spec_hash":"9c41093aa17bba9d46e832650270b14dcf31d60af9ae2913df72f66e56c08d7c","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-529bc6170721c470dd3e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v03\n- v08 --refutes--> v07\n- v0c --refutes--> v04\n- v03 --refutes--> v06\n- v08 --supports--> v00\n- v01 --refutes--> v08\n- v05 --supports--> v0b\n- v0a --supports--> v05\n- v07 --refutes--> v09\n- v04 --refutes--> v0a\n- v0c --refutes--> v02\n- v03 --refutes--> v01\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"792ce4f18a313cedb5b1b7a7ce68c63e220d95e6c018df6654c75270da990816","schema":"livegraph.boolean-propagation.v1","spec_hash":"529bc6170721c470dd3ef46c5eef70e227f162b2daaf637fe20119985313b5d2","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-b05c3ebc1d419fb8f8f5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v07\n- v08 --refutes--> v09\n- v05 --supports--> v02\n- v03 --refutes--> v0b\n- v05 --supports--> v0a\n- v0c --refutes--> v05\n- v09 --refutes--> v03\n- v00 --supports--> v08\n- v09 --supports--> v04\n- v07 --supports--> v06\n- v09 --refutes--> v01\n- v07 --supports--> v00\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d98234931912e5c10fa270ef8c7c74048f05958c54929d2c6d1451b039e3188e","schema":"livegraph.boolean-propagation.v1","spec_hash":"b05c3ebc1d419fb8f8f537527f02c5f739c603e83224a512411313279870fdf2","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-64ac38962de2fc29bcac","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v04\n- v09 --refutes--> v0a\n- v04 --supports--> v00\n- v08 --refutes--> v06\n- v0c --supports--> v03\n- v04 --supports--> v05\n- v06 --refutes--> v01\n- v06 --refutes--> v0b\n- v02 --supports--> v09\n- v09 --refutes--> v08\n- v06 --supports--> v07\n- v00 --supports--> v02\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"be9914b1549e61f92cf4d65a251138f8eeb2c9cf8726ccc82b62fffa5c5032d7","schema":"livegraph.boolean-propagation.v1","spec_hash":"64ac38962de2fc29bcac8d1f9214154af64f5248e1ab8be7d3f3bd67ff3baa01","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-8b6ba1290be15db8f87c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v05\n- v0a --supports--> v09\n- v05 --supports--> v0a\n- v0c --supports--> v01\n- v0c --supports--> v06\n- v0b --refutes--> v0c\n- v00 --refutes--> v0b\n- v0b --supports--> v03\n- v06 --refutes--> v02\n- v04 --refutes--> v00\n- v0c --refutes--> v08\n- v0b --refutes--> v07\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ceff19dde294e8f9dd6f470a71cdb639c4a0c26961a9500ef71cefed95cabd28","schema":"livegraph.boolean-propagation.v1","spec_hash":"8b6ba1290be15db8f87c17f8534a778cf27c7233ab61196bb8cff010a58d16a0","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-d4e0eed2e84894a4471c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v0b\n- v0b --refutes--> v02\n- v08 --refutes--> v0a\n- v06 --refutes--> v05\n- v0b --supports--> v03\n- v05 --refutes--> v08\n- v0a --refutes--> v01\n- v01 --supports--> v00\n- v01 --refutes--> v09\n- v0a --refutes--> v0c\n- v0b --refutes--> v04\n- v07 --refutes--> v06\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"af535554125e67245fa5c87a3ec3a414961c574dd144444d4021b4f16b4972e5","schema":"livegraph.boolean-propagation.v1","spec_hash":"d4e0eed2e84894a4471c009e59084497d6aee238a70a43d6e8c39c802556ebb2","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-cd81cb2beaeadbf60564","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v0a\n- v05 --refutes--> v00\n- v0c --refutes--> v05\n- v0b --supports--> v01\n- v04 --supports--> v08\n- v04 --supports--> v09\n- v04 --refutes--> v03\n- v01 --refutes--> v06\n- v03 --supports--> v0c\n- v05 --supports--> v02\n- v00 --refutes--> v0b\n- v04 --refutes--> v07\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a8ba18e1b472fd28c7a915ef9383768b92f7a416cdfde0d916dc5584036f7de9","schema":"livegraph.boolean-propagation.v1","spec_hash":"cd81cb2beaeadbf60564d84332712e929d1d3f8e497ddfc3fbe6cbe0a2a4e66a","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-27e51662ddc26ce73344","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v0c\n- v02 --refutes--> v00\n- v02 --supports--> v07\n- v06 --supports--> v08\n- v08 --supports--> v01\n- v00 --refutes--> v0a\n- v01 --supports--> v02\n- v0a --refutes--> v09\n- v06 --refutes--> v04\n- v0a --refutes--> v05\n- v09 --refutes--> v0b\n- v0b --refutes--> v03\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b2acd746e3b8c820523db292b189f0dc10afe8f6c4bef40670ede6d50f1bf5ca","schema":"livegraph.boolean-propagation.v1","spec_hash":"27e51662ddc26ce7334461bdb01e03b3ac8cc95fd7fb3b63aa28a7b195160605","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-6584e88f2980ab6dd719","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v06\n- v06 --refutes--> v05\n- v00 --supports--> v08\n- v02 --supports--> v03\n- v03 --supports--> v00\n- v00 --refutes--> v07\n- v00 --supports--> v04\n- v01 --refutes--> v02\n- v06 --supports--> v09\n- v03 --supports--> v0b\n- v0a --refutes--> v01\n- v09 --refutes--> v0a\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"349b53eabb000797c59d97d3487a51085fabc9a2ad788018ca966162f9c3f2e9","schema":"livegraph.boolean-propagation.v1","spec_hash":"6584e88f2980ab6dd71945846d05de2931c4083f373164c4850ed89c38ed8426","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-c156c4ea2223868e227a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v03\n- v08 --supports--> v06\n- v04 --refutes--> v09\n- v0a --refutes--> v05\n- v04 --refutes--> v0a\n- v0a --refutes--> v02\n- v03 --supports--> v04\n- v0c --refutes--> v00\n- v07 --supports--> v0c\n- v0a --supports--> v08\n- v06 --refutes--> v01\n- v06 --refutes--> v0b\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9e6a696074a59c4095037fea9bc780b333ed9c5112cf85e2c64dd512d86ca3af","schema":"livegraph.boolean-propagation.v1","spec_hash":"c156c4ea2223868e227a622e5a28ee2ef3e2b49d06c738db34d3fc9187029f11","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-b1189c60104821bc3f20","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v09\n- v07 --refutes--> v05\n- v01 --supports--> v0b\n- v07 --refutes--> v0c\n- v00 --refutes--> v04\n- v02 --supports--> v0a\n- v0c --refutes--> v01\n- v04 --refutes--> v02\n- v00 --supports--> v03\n- v09 --supports--> v07\n- v08 --refutes--> v00\n- v0c --refutes--> v06\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5f677d7922e99b736e523ba112e8863fd96ceeb44c01968f79cac73311247324","schema":"livegraph.boolean-propagation.v1","spec_hash":"b1189c60104821bc3f206b44eee359822c5e12016f81106aa8c71b6d4edfca95","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-e1b99700c1d051ee16f1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v03\n- v0a --supports--> v0c\n- v03 --refutes--> v07\n- v00 --refutes--> v06\n- v00 --supports--> v01\n- v0a --supports--> v09\n- v0b --refutes--> v08\n- v08 --supports--> v00\n- v0a --refutes--> v02\n- v07 --supports--> v0a\n- v01 --refutes--> v04\n- v04 --supports--> v05\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"97504eb48180b6928766005bc69da33be89aec1c270a8b5011970da7b5f67bb1","schema":"livegraph.boolean-propagation.v1","spec_hash":"e1b99700c1d051ee16f1b6ce657bb968094dc8ab8dff8373eee86488c15c5c5e","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-264d1132d2b3d9757035","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v09\n- v04 --supports--> v03\n- v01 --supports--> v06\n- v04 --supports--> v0b\n- v0b --supports--> v0a\n- v08 --supports--> v05\n- v05 --supports--> v04\n- v08 --supports--> v07\n- v0a --supports--> v0c\n- v09 --refutes--> v02\n- v02 --supports--> v01\n- v0a --supports--> v00\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ee92870761d89f74103af6c1ab5eb96913cba830d852f1077ba8287a39e1b96a","schema":"livegraph.boolean-propagation.v1","spec_hash":"264d1132d2b3d975703509309b4c84a56c585f7126dff2757f5eeb43beab20b7","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-476afd19ada9ade1903e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v05\n- v02 --refutes--> v04\n- v04 --supports--> v0b\n- v08 --supports--> v00\n- v01 --refutes--> v03\n- v04 --supports--> v0c\n- v04 --supports--> v07\n- v06 --supports--> v09\n- v09 --refutes--> v01\n- v0c --supports--> v06\n- v03 --refutes--> v08\n- v03 --refutes--> v0a\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b70a0a2f9b8373bbedef6bb24f71c91e834430fe01c89962674adc60a276ef19","schema":"livegraph.boolean-propagation.v1","spec_hash":"476afd19ada9ade1903eaced98e4b60986a48e211532ac267de0a1d27c9562e8","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-6bf84b7211436b0c1b40","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v07\n- v00 --supports--> v0a\n- v0a --refutes--> v01\n- v09 --supports--> v0b\n- v09 --supports--> v00\n- v03 --refutes--> v09\n- v09 --supports--> v04\n- v0c --refutes--> v08\n- v02 --refutes--> v03\n- v0c --supports--> v02\n- v07 --supports--> v05\n- v07 --supports--> v06\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5202c460c163dfedc961abcb6107bd151f5b2ae0af0c8c73457efc6d63b35714","schema":"livegraph.boolean-propagation.v1","spec_hash":"6bf84b7211436b0c1b4039f4374ba603bb65361ea7eb4e2d07bf58a9f685adcb","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-116456dc1ccb75e9418d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v00\n- v08 --refutes--> v0a\n- v02 --refutes--> v03\n- v02 --supports--> v0b\n- v00 --refutes--> v0c\n- v07 --supports--> v02\n- v04 --supports--> v06\n- v0c --supports--> v01\n- v07 --refutes--> v05\n- v08 --supports--> v04\n- v04 --supports--> v07\n- v03 --refutes--> v09\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"94fba3d11a8fce401365861d2a85b8041149947722b1e925b78bfbb7428b0395","schema":"livegraph.boolean-propagation.v1","spec_hash":"116456dc1ccb75e9418d15e0391fbef80f2b94304111f813af86765396fa96a4","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-fe0df08bc5eab66c9150","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v04\n- v0b --refutes--> v05\n- v02 --supports--> v06\n- v09 --supports--> v01\n- v09 --supports--> v07\n- v09 --refutes--> v0c\n- v04 --supports--> v03\n- v08 --refutes--> v0a\n- v09 --refutes--> v02\n- v06 --refutes--> v0b\n- v00 --supports--> v08\n- v03 --refutes--> v00\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0371d9e15aa4da9e3c8f32c3d08ca5fadf9b0cdbd7b931ca8ffcf052318f5ec4","schema":"livegraph.boolean-propagation.v1","spec_hash":"fe0df08bc5eab66c9150e4d899abeaa54b1043790c42e2994629b0fdb4c2a0b8","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-04238b7d96c8ae341c78","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v07\n- v04 --supports--> v01\n- v02 --supports--> v09\n- v08 --refutes--> v0c\n- v0a --supports--> v00\n- v05 --refutes--> v02\n- v0a --supports--> v05\n- v06 --supports--> v0a\n- v07 --refutes--> v06\n- v08 --refutes--> v03\n- v04 --supports--> v08\n- v03 --refutes--> v0b\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d1f1e8937e709adf953a2ba1131781e9b9751375c043adb59d76d09d8e66799b","schema":"livegraph.boolean-propagation.v1","spec_hash":"04238b7d96c8ae341c78b8f1fc942e67a9da1c204e65c30a8505c165ab6a5cbb","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":8,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-c9bac46f36ffd2200170","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v08\n- v02 --supports--> v09\n- v09 --refutes--> v07\n- v07 --supports--> v04\n- v06 --supports--> v02\n- v06 --supports--> v0a\n- v09 --supports--> v03\n- v04 --supports--> v0b\n- v08 --refutes--> v05\n- v02 --refutes--> v0c\n- v05 --supports--> v00\n- v06 --refutes--> v01\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f912aacdf4d61115874dafee75829e37c775bff7020bc07222cee9a94266e9e5","schema":"livegraph.boolean-propagation.v1","spec_hash":"c9bac46f36ffd2200170d6f0272165df409568f08b87b3425c5fbea1723d1751","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-439e9e3ff6b2c8bbf3c7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v09\n- v05 --supports--> v00\n- v0a --supports--> v0c\n- v02 --supports--> v0b\n- v05 --refutes--> v03\n- v0c --refutes--> v02\n- v02 --refutes--> v06\n- v07 --supports--> v0a\n- v02 --supports--> v01\n- v00 --refutes--> v07\n- v09 --refutes--> v04\n- v07 --refutes--> v08\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8df6e9713590af0b04b88dd7f8746be6b5df260b92bac59dbb592cfd55ae149a","schema":"livegraph.boolean-propagation.v1","spec_hash":"439e9e3ff6b2c8bbf3c7b992a073165606986d5715df6bcd7542e885ce04b2df","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-36e240aff3afdccec6d2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v03\n- v01 --refutes--> v02\n- v01 --supports--> v07\n- v03 --refutes--> v06\n- v02 --refutes--> v08\n- v0b --supports--> v01\n- v00 --supports--> v0b\n- v06 --refutes--> v00\n- v0a --refutes--> v0c\n- v09 --refutes--> v05\n- v03 --supports--> v04\n- v02 --supports--> v0a\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8e2a67dc20480274b4bfbcb798c99a110341999ccbd6d5683069bd306f0bb102","schema":"livegraph.boolean-propagation.v1","spec_hash":"36e240aff3afdccec6d2e1a819eca5d91de175aa97643df27fce838885e61546","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-e802dbabd579f6d1e6d4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v00\n- v0b --supports--> v02\n- v02 --refutes--> v04\n- v01 --supports--> v05\n- v03 --refutes--> v01\n- v07 --supports--> v08\n- v04 --refutes--> v07\n- v00 --supports--> v0a\n- v01 --refutes--> v09\n- v0a --supports--> v03\n- v01 --supports--> v0c\n- v0b --refutes--> v06\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"06c44ab6540fdaf234133867823e16ab3307f13489c4e393eae3b0969d6b9125","schema":"livegraph.boolean-propagation.v1","spec_hash":"e802dbabd579f6d1e6d40420a5eb693d550be536fa42a95c2ab07cd89a6b7d3c","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-8287a935bbcad85a9c18","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v0b\n- v04 --supports--> v07\n- v00 --refutes--> v05\n- v05 --supports--> v02\n- v09 --refutes--> v06\n- v0a --refutes--> v03\n- v0b --supports--> v00\n- v05 --refutes--> v04\n- v06 --refutes--> v0a\n- v04 --refutes--> v0c\n- v04 --supports--> v08\n- v04 --supports--> v01\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"86579b637e9ecf8d91ae5b66a92bd4c6a3915a2b8c5ade2734beecf4abab677d","schema":"livegraph.boolean-propagation.v1","spec_hash":"8287a935bbcad85a9c187719733954ed8451935452a0c18744e78f9f7daed457","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-6a2fcf2076f1b350cdf4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v02\n- v0c --refutes--> v03\n- v0a --supports--> v07\n- v0a --supports--> v04\n- v08 --refutes--> v0a\n- v01 --refutes--> v08\n- v06 --refutes--> v05\n- v0b --supports--> v0c\n- v08 --supports--> v09\n- v0c --refutes--> v06\n- v02 --refutes--> v0b\n- v06 --supports--> v01\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3f45e74eaf1ad8d42d1f31d327897986caba9da5018b689e4341f61b9c5a62e7","schema":"livegraph.boolean-propagation.v1","spec_hash":"6a2fcf2076f1b350cdf46937f963067cfcf5d79f45cfffa434a1b7360a937d54","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-9f427f7c53e177a2a917","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v0c\n- v05 --supports--> v07\n- v08 --refutes--> v01\n- v01 --supports--> v05\n- v08 --supports--> v03\n- v0b --supports--> v00\n- v05 --refutes--> v09\n- v02 --supports--> v06\n- v09 --refutes--> v02\n- v0a --supports--> v04\n- v0a --supports--> v0b\n- v06 --supports--> v0a\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"78601337aa4eaa506d419b5f5c7a5f37613c7219b747aafb53c7ffe6554a5a2a","schema":"livegraph.boolean-propagation.v1","spec_hash":"9f427f7c53e177a2a9178fbbf10d6d15b95911a926d385bbd47a8afeade5698a","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-6bca85a1b6fb9d0d61f3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v02\n- v08 --refutes--> v09\n- v02 --refutes--> v01\n- v04 --supports--> v08\n- v07 --refutes--> v04\n- v08 --refutes--> v06\n- v02 --supports--> v0b\n- v03 --supports--> v05\n- v06 --refutes--> v00\n- v0c --supports--> v07\n- v02 --refutes--> v0c\n- v0a --supports--> v03\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"388c6a4d559be085b5fe207c1458ec5e9b18f4bc71c6669946bddb3743b36612","schema":"livegraph.boolean-propagation.v1","spec_hash":"6bca85a1b6fb9d0d61f304267e295e3ad3ece777814eb671325814f9ec14ae5c","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-4d293995856d626e360f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v08\n- v0c --refutes--> v0b\n- v00 --supports--> v0c\n- v09 --supports--> v07\n- v02 --refutes--> v06\n- v0c --refutes--> v05\n- v06 --refutes--> v01\n- v05 --supports--> v04\n- v09 --supports--> v0a\n- v03 --supports--> v09\n- v06 --refutes--> v00\n- v07 --supports--> v02\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7c63a3fec0f757921afd4c6c958ee46cb1818ba343c0a82c3c9e6463ec9a538c","schema":"livegraph.boolean-propagation.v1","spec_hash":"4d293995856d626e360fb8dc682be4bbe11ff7332570356912d0ac060fad6850","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-bdf380776f653a9ab4e1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v06\n- v07 --supports--> v0b\n- v05 --supports--> v00\n- v0a --refutes--> v05\n- v05 --refutes--> v07\n- v04 --supports--> v0c\n- v0c --refutes--> v08\n- v0c --refutes--> v09\n- v0b --refutes--> v01\n- v03 --supports--> v02\n- v03 --refutes--> v04\n- v01 --supports--> v03\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7fc391c4acd12c83447593f004c69b91e83787b9968fdac19d1d45e832fa4b93","schema":"livegraph.boolean-propagation.v1","spec_hash":"bdf380776f653a9ab4e1d095e9897b333bae28902c85eb92da04c7e23840d375","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-b15628132c44615481fc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v05\n- v08 --supports--> v0a\n- v0a --supports--> v0c\n- v0b --supports--> v00\n- v07 --supports--> v09\n- v0b --supports--> v06\n- v09 --supports--> v04\n- v05 --refutes--> v08\n- v04 --refutes--> v0b\n- v08 --supports--> v02\n- v0b --refutes--> v01\n- v05 --refutes--> v03\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3fa35b8e78adc54e239a2e5df5a4b8e869ca3ca85629f99a7f474e9dab5235f8","schema":"livegraph.boolean-propagation.v1","spec_hash":"b15628132c44615481fc02650b6b12068ac9830551c572bea76c5b8cf4165a8a","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-c47998ea50c4e9ab1dd4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v04\n- v05 --supports--> v09\n- v06 --refutes--> v07\n- v08 --refutes--> v0c\n- v0a --supports--> v0b\n- v08 --refutes--> v02\n- v09 --supports--> v0a\n- v0b --supports--> v08\n- v09 --supports--> v00\n- v09 --supports--> v03\n- v02 --refutes--> v01\n- v07 --supports--> v05\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8a24f24cb18b8c52ffe9ec189982159d15a7fa989429f37a8e00ef07d2b784e8","schema":"livegraph.boolean-propagation.v1","spec_hash":"c47998ea50c4e9ab1dd447f59e1074734b058b8a3ede263c40f237d8e5774968","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-431cb41d47a5eab0360b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v0b\n- v08 --supports--> v09\n- v05 --refutes--> v0a\n- v06 --refutes--> v01\n- v00 --supports--> v0c\n- v01 --supports--> v08\n- v06 --supports--> v04\n- v0a --refutes--> v00\n- v09 --refutes--> v02\n- v00 --refutes--> v03\n- v08 --refutes--> v07\n- v0c --refutes--> v06\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"30be8d25a356de3df3b4edb23a722c0c937b2461e048005525d9e532abaff86b","schema":"livegraph.boolean-propagation.v1","spec_hash":"431cb41d47a5eab0360b43f7b6fb01c897facdb0a010612ea285cda992025738","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-2f7d10c15e7981547bee","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v0b\n- v03 --refutes--> v09\n- v06 --refutes--> v04\n- v0a --supports--> v07\n- v08 --refutes--> v0c\n- v09 --supports--> v08\n- v0a --supports--> v00\n- v0c --refutes--> v02\n- v06 --refutes--> v03\n- v02 --supports--> v01\n- v0a --supports--> v05\n- v01 --refutes--> v0a\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"592f00706acf8ccbae64e6056287f216e1546fcd50bb9bca7bce68b9c37c2596","schema":"livegraph.boolean-propagation.v1","spec_hash":"2f7d10c15e7981547bee5f75e76338b0ebfa09cfba7b723e8b562059bffd794a","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-910c83061a5bfea58060","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v0c\n- v03 --supports--> v07\n- v07 --supports--> v02\n- v07 --refutes--> v0a\n- v0c --supports--> v08\n- v08 --refutes--> v01\n- v0a --refutes--> v04\n- v06 --refutes--> v05\n- v09 --supports--> v03\n- v04 --refutes--> v06\n- v07 --refutes--> v0b\n- v04 --supports--> v00\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"51fca5589b7a0eb69994ec4aa0dd4e56919f125de4c2d0200bc1f0f8102a9dd3","schema":"livegraph.boolean-propagation.v1","spec_hash":"910c83061a5bfea5806032acc78c583c2191b57a67ec4f6f1061fd6eaf49e1d2","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-25cee5ea8ed2b141f0f8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v0b\n- v03 --refutes--> v07\n- v00 --refutes--> v06\n- v0b --refutes--> v0c\n- v0b --supports--> v00\n- v06 --supports--> v01\n- v00 --supports--> v08\n- v06 --refutes--> v04\n- v07 --refutes--> v02\n- v09 --refutes--> v03\n- v05 --refutes--> v09\n- v02 --refutes--> v0a\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"68d415229a0e0028670eadf0eda5cca3849e2cc0de9039c1c305740dd15368e8","schema":"livegraph.boolean-propagation.v1","spec_hash":"25cee5ea8ed2b141f0f817a390edbd96fd82206b64267ab66328e5b2419f5451","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-b2bff50b6365045b315b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v06\n- v0a --supports--> v00\n- v02 --refutes--> v01\n- v01 --supports--> v05\n- v01 --supports--> v0c\n- v07 --supports--> v0b\n- v08 --supports--> v0a\n- v01 --supports--> v03\n- v06 --supports--> v08\n- v07 --supports--> v04\n- v00 --refutes--> v07\n- v07 --supports--> v02\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"80590e52ba97cb111989ef3755cacacb8282b8d1bea3934f89b94d86abbf6a18","schema":"livegraph.boolean-propagation.v1","spec_hash":"b2bff50b6365045b315b40076f7993d9029d2965b9866417ed3d09e2a3192a9a","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":8,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-1067183c77ae05db7d57","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v03\n- v02 --refutes--> v09\n- v0b --supports--> v00\n- v02 --supports--> v07\n- v04 --refutes--> v08\n- v01 --refutes--> v0b\n- v04 --supports--> v06\n- v07 --supports--> v05\n- v02 --refutes--> v0c\n- v0a --refutes--> v01\n- v07 --refutes--> v04\n- v03 --supports--> v0a\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bf8cb83afeaac7051e443dc646e804990d9a53c3be4cc53cdd1ff49721c2a6f1","schema":"livegraph.boolean-propagation.v1","spec_hash":"1067183c77ae05db7d57f71de4a09fbcab913ea917ba03d589cf5e09aa0b89e4","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-ac2b565eee257d587e2e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v01\n- v02 --supports--> v05\n- v07 --supports--> v08\n- v0b --refutes--> v03\n- v00 --refutes--> v02\n- v07 --supports--> v04\n- v03 --supports--> v09\n- v0c --supports--> v00\n- v04 --supports--> v06\n- v0a --supports--> v0c\n- v02 --refutes--> v07\n- v03 --refutes--> v0a\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bee95e6ef05dd57a3126381ee786ac7e3950d53348a5a3bbd3a5675fe24c445a","schema":"livegraph.boolean-propagation.v1","spec_hash":"ac2b565eee257d587e2eea12eaf995d6f471f49caaab15591b35ad52511e6b80","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-6259ba80c58b5c8b91fa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v07\n- v00 --refutes--> v0b\n- v03 --supports--> v0c\n- v0a --refutes--> v01\n- v08 --supports--> v09\n- v03 --supports--> v06\n- v01 --refutes--> v02\n- v08 --supports--> v05\n- v06 --refutes--> v0a\n- v04 --supports--> v08\n- v02 --refutes--> v00\n- v00 --supports--> v04\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"33a60c855f25301acdcecf9c6f49818f343bbd79fe080c177574d0e0806c1bc9","schema":"livegraph.boolean-propagation.v1","spec_hash":"6259ba80c58b5c8b91facac5407c79a6b1bdd8579740da52ef5e6d30911ebed7","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":8,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-50a269ca3df14eb076f7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v04\n- v00 --refutes--> v07\n- v07 --supports--> v0a\n- v06 --supports--> v00\n- v08 --supports--> v09\n- v08 --supports--> v06\n- v05 --refutes--> v03\n- v0b --refutes--> v02\n- v0b --refutes--> v0c\n- v05 --supports--> v08\n- v04 --supports--> v0b\n- v01 --supports--> v05\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ef95d10c24bc5135c5c519451e129cb0a7eeb663b8bacf6ae878c69f4e145f41","schema":"livegraph.boolean-propagation.v1","spec_hash":"50a269ca3df14eb076f70bb9777871a0fcf11d209e7a092500d66e29f6d7f501","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-919ac1e57ee27a71b8a8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v09\n- v04 --supports--> v0b\n- v09 --supports--> v05\n- v04 --refutes--> v02\n- v03 --refutes--> v04\n- v01 --refutes--> v0c\n- v03 --supports--> v08\n- v00 --refutes--> v06\n- v0b --supports--> v00\n- v00 --refutes--> v01\n- v00 --supports--> v07\n- v0a --refutes--> v03\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"21cbbacabce497dc94e5d8d4f7b8dd0501f1a177e2156260e55571ed3c4b9d97","schema":"livegraph.boolean-propagation.v1","spec_hash":"919ac1e57ee27a71b8a833ba59598c3cddc7a525efdfc4a8d99b3635b0a7af5d","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-86b9ca271e72014621bf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v07\n- v0c --refutes--> v04\n- v01 --supports--> v02\n- v0b --supports--> v06\n- v0c --supports--> v05\n- v00 --refutes--> v0a\n- v0b --supports--> v03\n- v08 --refutes--> v00\n- v06 --supports--> v0c\n- v02 --supports--> v0b\n- v00 --refutes--> v09\n- v04 --supports--> v08\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"54b0aeb04c6ca66556fa8d180f7b35e682cd11e6c8ccc6d53c54c25d5a4fda9b","schema":"livegraph.boolean-propagation.v1","spec_hash":"86b9ca271e72014621bf9c1373aa768e4384827e4e8ce7d9d7956c1ad33f507c","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-6040f6da083a84be9daa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v01\n- v07 --supports--> v00\n- v0a --refutes--> v07\n- v06 --supports--> v0c\n- v09 --supports--> v02\n- v09 --supports--> v04\n- v08 --supports--> v05\n- v0b --supports--> v09\n- v09 --supports--> v03\n- v06 --supports--> v08\n- v00 --supports--> v0b\n- v05 --refutes--> v0a\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"56a788fcee3b37a65d06b853d73e7a397e192163b5ff6f2f0e41fc3a752891cd","schema":"livegraph.boolean-propagation.v1","spec_hash":"6040f6da083a84be9daae1348ada39ae5c2b234afdbea6c3a182e09473062b76","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-2d02493b3560f6e4025a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v0b\n- v00 --supports--> v07\n- v04 --supports--> v0c\n- v0c --supports--> v06\n- v02 --supports--> v01\n- v09 --supports--> v05\n- v05 --refutes--> v03\n- v0a --supports--> v08\n- v05 --refutes--> v00\n- v02 --refutes--> v09\n- v08 --refutes--> v04\n- v0c --refutes--> v02\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"43e8ce2ad1a52e84da80d95cdbe1a644d24cc02c16bbc6ac38a514fe0518df0a","schema":"livegraph.boolean-propagation.v1","spec_hash":"2d02493b3560f6e4025a9cd018695c90599ff4d9e331f023a70b9a2ce88daf46","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-7de16adae7ee77aaf2b6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v0b\n- v02 --supports--> v0c\n- v0a --supports--> v08\n- v0c --refutes--> v06\n- v06 --refutes--> v01\n- v04 --supports--> v07\n- v06 --refutes--> v00\n- v03 --supports--> v09\n- v08 --refutes--> v03\n- v01 --refutes--> v0a\n- v03 --refutes--> v05\n- v04 --refutes--> v02\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c3b15aeda9f2033ea62ee527fb142826bd2e3480543a5d8d0b40510a915d24f7","schema":"livegraph.boolean-propagation.v1","spec_hash":"7de16adae7ee77aaf2b6990fe72e8ec44aad21738ee4b4d9fda45882ab283b0b","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-2353178537e3ada54313","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v05\n- v05 --supports--> v03\n- v09 --refutes--> v04\n- v06 --supports--> v0c\n- v00 --supports--> v0b\n- v06 --refutes--> v07\n- v07 --supports--> v00\n- v09 --refutes--> v06\n- v02 --refutes--> v08\n- v03 --refutes--> v0a\n- v06 --supports--> v01\n- v03 --supports--> v09\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5b38c9861dd3ab37859e9ed09c5273e37d9dbc6e82fd02c9d22182271a3623c3","schema":"livegraph.boolean-propagation.v1","spec_hash":"2353178537e3ada5431334f29b4a5e33924da51a097f331a89f28ed1a88f113e","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":8,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-cf9550c5a46c6da7b22f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v05\n- v03 --refutes--> v04\n- v02 --refutes--> v03\n- v06 --refutes--> v0b\n- v02 --refutes--> v0c\n- v09 --refutes--> v00\n- v01 --supports--> v06\n- v07 --refutes--> v02\n- v0a --supports--> v07\n- v07 --supports--> v08\n- v04 --refutes--> v09\n- v09 --supports--> v01\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"92f28594e144053b9a934a6275255b04a1e989d40ca9f00e99275efaca87a781","schema":"livegraph.boolean-propagation.v1","spec_hash":"cf9550c5a46c6da7b22f17d4db1b618a60c4e550049a3c7de51d2f0bfe2421a8","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":8,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-8bfe3f23b51b7b1272f1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v03\n- v02 --refutes--> v00\n- v05 --refutes--> v08\n- v03 --refutes--> v0c\n- v04 --supports--> v06\n- v04 --refutes--> v09\n- v08 --refutes--> v01\n- v03 --supports--> v05\n- v01 --supports--> v0b\n- v01 --supports--> v07\n- v04 --supports--> v02\n- v0b --refutes--> v04\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dbbfb54867d73506116cd2e3a2125db2b1b093920cd5f86e65a433ac52112166","schema":"livegraph.boolean-propagation.v1","spec_hash":"8bfe3f23b51b7b1272f107256126410163c2d783bbf22cb0ecf298880195a48d","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-d4caede283e0f7b0d929","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v05\n- v09 --refutes--> v02\n- v01 --supports--> v09\n- v0c --refutes--> v01\n- v01 --refutes--> v00\n- v04 --refutes--> v0c\n- v07 --supports--> v08\n- v06 --refutes--> v07\n- v06 --refutes--> v03\n- v07 --supports--> v0a\n- v09 --supports--> v06\n- v05 --refutes--> v0b\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"efc7f5768a3faf98351feb1d4d86b4a601a9ad3f6f575abf11ed8719a3fb59af","schema":"livegraph.boolean-propagation.v1","spec_hash":"d4caede283e0f7b0d92940a6bc707a19c535dad676a6735438f9fb9afd2e8438","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-55863054d4896d80ba34","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v04\n- v08 --supports--> v0c\n- v08 --supports--> v0b\n- v07 --supports--> v05\n- v0b --supports--> v09\n- v0a --supports--> v03\n- v04 --supports--> v08\n- v01 --supports--> v07\n- v03 --refutes--> v06\n- v01 --refutes--> v00\n- v03 --refutes--> v01\n- v03 --supports--> v02\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"25465dac0601f1a90777c3df16b918fb3cfa024547b847d19f16be1e1e1ab569","schema":"livegraph.boolean-propagation.v1","spec_hash":"55863054d4896d80ba34fae7965a49f24f1c9c719065be2ec7d7271fd9df6c43","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-277a16e61ae2a390fcdc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v06\n- v07 --supports--> v00\n- v06 --refutes--> v08\n- v03 --supports--> v04\n- v0b --refutes--> v0c\n- v01 --refutes--> v0a\n- v07 --refutes--> v05\n- v02 --supports--> v03\n- v0c --supports--> v01\n- v03 --refutes--> v09\n- v00 --supports--> v0b\n- v06 --refutes--> v02\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c82763178e100d1673a56fe10c175c871e653a29c793bc97cde2bec7a60bcd6b","schema":"livegraph.boolean-propagation.v1","spec_hash":"277a16e61ae2a390fcdc7cd022c90d8b2a13e30c1101c50b91d22bd3e6c3df7d","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-8cce55d9d526afaf60f7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v09\n- v00 --refutes--> v0a\n- v0c --supports--> v00\n- v01 --refutes--> v05\n- v07 --refutes--> v04\n- v0a --refutes--> v06\n- v0a --supports--> v03\n- v01 --supports--> v0b\n- v0a --refutes--> v08\n- v09 --supports--> v0c\n- v02 --refutes--> v07\n- v05 --supports--> v02\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"033ec12cac7bba9c989c1bd8a04d4b83f79360f2ab1dd0987f9c15c160606e64","schema":"livegraph.boolean-propagation.v1","spec_hash":"8cce55d9d526afaf60f7a5b0fda9125e2f97b2a99760d53e144773389aaa0b75","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-2e425e5ecb0a755a314b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v08\n- v00 --refutes--> v01\n- v05 --refutes--> v0c\n- v07 --supports--> v00\n- v01 --refutes--> v02\n- v05 --supports--> v07\n- v0b --refutes--> v04\n- v05 --refutes--> v0a\n- v06 --supports--> v05\n- v04 --supports--> v03\n- v02 --supports--> v0b\n- v05 --refutes--> v09\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"09c45dc0dbaaf72409617ddc57bb32fb0e9f70d6314dab764e9833c85142ee22","schema":"livegraph.boolean-propagation.v1","spec_hash":"2e425e5ecb0a755a314b07d9374a3b0ecfa58828ea2fb8f3a2105b463c957b4e","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-36705c1c671ab0d02c4c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v0a\n- v03 --supports--> v02\n- v01 --supports--> v09\n- v03 --refutes--> v04\n- v00 --supports--> v08\n- v07 --supports--> v03\n- v09 --refutes--> v00\n- v01 --supports--> v06\n- v0c --refutes--> v01\n- v0b --refutes--> v07\n- v04 --supports--> v05\n- v04 --supports--> v0c\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4cda6a53086e497843af54b7a1b75c25b3b9d2b50c95b44725f5579240aa33d1","schema":"livegraph.boolean-propagation.v1","spec_hash":"36705c1c671ab0d02c4cfc5d1b116508aa4d56fcf302bdfccd66ca072903cbb9","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-d6ad889929acdcf8d84e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v0a\n- v0a --supports--> v02\n- v03 --supports--> v0d\n- v01 --supports--> v06\n- v03 --refutes--> v09\n- v05 --refutes--> v01\n- v05 --refutes--> v04\n- v06 --supports--> v00\n- v08 --supports--> v07\n- v08 --refutes--> v0c\n- v07 --supports--> v05\n- v0d --supports--> v08\n- v06 --refutes--> v0b\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"92c93aa2643f07ee0eb197ec97773e885f229bf00d7531742772c2b2530f1da5","schema":"livegraph.boolean-propagation.v1","spec_hash":"d6ad889929acdcf8d84e246f3a9e46efc447f2e028ed3eaf118290553d994d06","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-3c10e5acd708f5330bb1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v07\n- v0d --supports--> v02\n- v0d --supports--> v00\n- v04 --refutes--> v08\n- v02 --refutes--> v0a\n- v0c --supports--> v0b\n- v04 --supports--> v05\n- v09 --supports--> v01\n- v0a --refutes--> v06\n- v01 --refutes--> v04\n- v03 --refutes--> v0d\n- v05 --refutes--> v03\n- v0a --supports--> v0c\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"167a0fc33cf1b80a1c3696a8a052efbb2499933cb4596f860390fd0a49e341d2","schema":"livegraph.boolean-propagation.v1","spec_hash":"3c10e5acd708f5330bb1fb4d62af293f8bc92fbb87954cdfceb237be1f5531ed","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-028e67883ea70d2c5bb0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v0b\n- v00 --refutes--> v03\n- v07 --refutes--> v0a\n- v08 --refutes--> v06\n- v0c --supports--> v00\n- v08 --supports--> v09\n- v00 --refutes--> v07\n- v04 --refutes--> v02\n- v0a --supports--> v04\n- v02 --refutes--> v08\n- v09 --supports--> v0d\n- v0d --refutes--> v05\n- v0d --refutes--> v01\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f3b8386c1fb27d3fd861074bda863cabf989964d65109679f5c0def42b12a171","schema":"livegraph.boolean-propagation.v1","spec_hash":"028e67883ea70d2c5bb018c48deff01ee5dff210906427912e4fd9ee315aa9f0","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-b11f1f8ba77b5278ed6d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v05\n- v0a --refutes--> v06\n- v04 --refutes--> v0a\n- v04 --refutes--> v02\n- v0b --supports--> v09\n- v01 --supports--> v03\n- v09 --refutes--> v08\n- v0a --refutes--> v00\n- v00 --supports--> v0b\n- v05 --refutes--> v01\n- v03 --supports--> v0c\n- v09 --refutes--> v0d\n- v0d --supports--> v07\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"35458cb161c13f29118dceae7564fcaae6003b1cbb22ad8ab06fd4b41cc0d02d","schema":"livegraph.boolean-propagation.v1","spec_hash":"b11f1f8ba77b5278ed6d997cd9b29ae1016ace75d8648315589f551aeb1ec898","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-04048805c82c34d97238","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v06\n- v09 --supports--> v0a\n- v02 --supports--> v0d\n- v04 --refutes--> v08\n- v08 --supports--> v0c\n- v02 --refutes--> v00\n- v0b --refutes--> v03\n- v0c --supports--> v0b\n- v0d --refutes--> v04\n- v01 --refutes--> v05\n- v01 --refutes--> v02\n- v0c --refutes--> v07\n- v0a --supports--> v01\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"351909203d5540b38b95984ddcda41d4a77a445f1e787e4f55f7f4de9ea5bba4","schema":"livegraph.boolean-propagation.v1","spec_hash":"04048805c82c34d972383f1e8d8318a8c4356a55162181857f1112dc20df49f7","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-a9fb6eb10bd69b770480","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v0a\n- v01 --refutes--> v02\n- v03 --supports--> v01\n- v0a --supports--> v06\n- v00 --refutes--> v03\n- v0c --supports--> v09\n- v05 --supports--> v04\n- v05 --refutes--> v0b\n- v0a --refutes--> v08\n- v05 --supports--> v0d\n- v0d --refutes--> v07\n- v0d --refutes--> v0c\n- v02 --refutes--> v05\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"52917d55cb2dac037eb0fc4f8aa0ca4cabf034195223334b5cd72573b5d02f0c","schema":"livegraph.boolean-propagation.v1","spec_hash":"a9fb6eb10bd69b770480e33f925de3c287219d35db377e7f2fac6fd659072843","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-c801066632d3d55b0a30","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v0d\n- v07 --supports--> v0b\n- v04 --refutes--> v03\n- v02 --supports--> v0c\n- v03 --supports--> v01\n- v0b --refutes--> v06\n- v09 --refutes--> v08\n- v00 --supports--> v07\n- v0a --supports--> v02\n- v03 --refutes--> v0a\n- v0b --supports--> v09\n- v02 --refutes--> v00\n- v08 --refutes--> v05\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2abb5dfd6a81e5360c831de6f01ed5a67b50c24a9f78a427e246caab99de83b8","schema":"livegraph.boolean-propagation.v1","spec_hash":"c801066632d3d55b0a309dafbf75b6320a719d52d8dd94b39f8302ca06358ae7","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":9,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-0165723f035ca72c34c5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v02\n- v01 --supports--> v04\n- v08 --refutes--> v09\n- v01 --refutes--> v00\n- v00 --refutes--> v0d\n- v02 --refutes--> v06\n- v08 --refutes--> v03\n- v0d --supports--> v05\n- v00 --supports--> v0a\n- v0c --refutes--> v08\n- v06 --supports--> v0b\n- v07 --refutes--> v0c\n- v0b --supports--> v07\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9b54c7bb0f1ab0fa300f32968e49a013f13f127e7c5dbd6b30523a71bb449350","schema":"livegraph.boolean-propagation.v1","spec_hash":"0165723f035ca72c34c58a8fbb5eacea6c6bd5b6963a604356890da72c630754","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-cbec1243835e6202c0bf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v05\n- v06 --supports--> v07\n- v09 --refutes--> v03\n- v07 --supports--> v01\n- v04 --refutes--> v06\n- v03 --refutes--> v08\n- v07 --refutes--> v0c\n- v06 --supports--> v0d\n- v0c --refutes--> v00\n- v09 --refutes--> v0a\n- v0b --supports--> v04\n- v0b --refutes--> v02\n- v00 --supports--> v09\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0e8c63c703dd4e278a794cc1b2e5bae266a0d196c05406c562384be660c6ca3f","schema":"livegraph.boolean-propagation.v1","spec_hash":"cbec1243835e6202c0bf67b166147e8671d53ae8be1c0f78d1c0eb4af9e50056","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-339e709add75b21a0852","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v08\n- v0a --supports--> v07\n- v08 --supports--> v06\n- v0d --supports--> v05\n- v0b --refutes--> v02\n- v01 --supports--> v04\n- v05 --refutes--> v01\n- v01 --supports--> v00\n- v09 --supports--> v0d\n- v08 --refutes--> v03\n- v08 --supports--> v09\n- v00 --refutes--> v0b\n- v01 --refutes--> v0c\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"da853abfad72169c41753087d4cd8c94d4609b0dad486ceb4127cf717607fda3","schema":"livegraph.boolean-propagation.v1","spec_hash":"339e709add75b21a085259880b9c07754d10dc32c25b7ab05d23973c608da88b","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-a5eecc9b67a2132afabc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v0c\n- v01 --supports--> v0b\n- v04 --refutes--> v07\n- v06 --supports--> v0a\n- v0a --supports--> v02\n- v07 --refutes--> v08\n- v00 --refutes--> v0d\n- v07 --refutes--> v06\n- v02 --supports--> v00\n- v03 --supports--> v04\n- v06 --supports--> v05\n- v00 --refutes--> v09\n- v0b --supports--> v03\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2ee8a0d784cfe38e363bbbe677d2675f66a9a9e677e39976ac85b665073d7a8a","schema":"livegraph.boolean-propagation.v1","spec_hash":"a5eecc9b67a2132afabce257d34f63e5b4986c535fc0c893ac15e0fa14a2c691","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":9,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-21d70ff5516c907a70f0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v0b\n- v01 --supports--> v09\n- v02 --supports--> v03\n- v0a --refutes--> v02\n- v06 --refutes--> v05\n- v04 --refutes--> v07\n- v09 --supports--> v0d\n- v0d --supports--> v0a\n- v00 --refutes--> v06\n- v06 --refutes--> v08\n- v03 --supports--> v0c\n- v04 --refutes--> v00\n- v03 --refutes--> v04\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f9f225858bfe8a8bffe0ffd8c8f1a54cce2618b22beab4c512af80eec44f422c","schema":"livegraph.boolean-propagation.v1","spec_hash":"21d70ff5516c907a70f0d194e8687c600ed14b98212709e7ee9101f1a839adc2","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-7002d1a6d19c0671397a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v06\n- v0c --refutes--> v08\n- v08 --refutes--> v05\n- v01 --supports--> v00\n- v00 --refutes--> v0a\n- v04 --supports--> v02\n- v0d --supports--> v09\n- v05 --refutes--> v0b\n- v0b --supports--> v04\n- v0c --supports--> v03\n- v04 --refutes--> v07\n- v06 --supports--> v0d\n- v0d --refutes--> v0c\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"133cd572483bce91b3af9c821b2b536a989215fa610f4659de624dde395476bb","schema":"livegraph.boolean-propagation.v1","spec_hash":"7002d1a6d19c0671397ab391222538e631ed8f7d2693d41fdcc23404b82b85ec","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-13b3d91ba2362e969d88","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v01\n- v05 --supports--> v08\n- v0d --supports--> v02\n- v09 --refutes--> v06\n- v06 --supports--> v07\n- v02 --supports--> v05\n- v07 --supports--> v0a\n- v03 --refutes--> v0b\n- v0b --refutes--> v0c\n- v0b --supports--> v00\n- v09 --refutes--> v04\n- v07 --refutes--> v0d\n- v0c --supports--> v09\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"764c41d005b48e45d1e50294da0672b5026c0820151085142ec49c31d7e1ac16","schema":"livegraph.boolean-propagation.v1","spec_hash":"13b3d91ba2362e969d88bc25f921765e83dbe6d37fcae4042513204f08c302a3","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-c230adaa4f8c7345da9b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v09\n- v08 --supports--> v0b\n- v09 --supports--> v03\n- v01 --refutes--> v04\n- v0b --supports--> v0a\n- v07 --refutes--> v01\n- v09 --refutes--> v07\n- v05 --refutes--> v08\n- v07 --supports--> v0d\n- v0a --refutes--> v02\n- v08 --supports--> v0c\n- v04 --refutes--> v00\n- v01 --supports--> v06\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a21b52abf0d900af6804f3190b342a9cf76768c3da772659384cc82d5ec0d5e1","schema":"livegraph.boolean-propagation.v1","spec_hash":"c230adaa4f8c7345da9baf42e0021de86217da6e62ed7692e6cc52c810f74625","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":9,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-d5d5a1b6ac9e025cb671","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v0d\n- v07 --refutes--> v00\n- v06 --refutes--> v05\n- v0d --refutes--> v0a\n- v03 --supports--> v04\n- v08 --supports--> v07\n- v03 --refutes--> v06\n- v0c --refutes--> v01\n- v03 --refutes--> v0b\n- v0d --supports--> v09\n- v0a --refutes--> v03\n- v01 --refutes--> v02\n- v00 --supports--> v0c\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"17d1bc8e53cffd81e6459850d9b84c77c8627bd765a93abb6a7e280d17913792","schema":"livegraph.boolean-propagation.v1","spec_hash":"d5d5a1b6ac9e025cb67158b1bee47d520458e4480ed7a49a1c7dbca1abb42aa2","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-67ec4dec884285d077c4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v0d\n- v03 --supports--> v05\n- v03 --refutes--> v00\n- v05 --supports--> v0b\n- v0a --refutes--> v0c\n- v0d --refutes--> v03\n- v09 --refutes--> v07\n- v05 --supports--> v06\n- v0c --supports--> v02\n- v09 --supports--> v01\n- v0c --supports--> v04\n- v07 --supports--> v0a\n- v06 --supports--> v08\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3e0b0a4015f5249b7cbe6fdaa8dbce614e2473ad5065a6a8ba61a677b609059b","schema":"livegraph.boolean-propagation.v1","spec_hash":"67ec4dec884285d077c41da1d640e69e0c4e296f2bda406afbfa395850050b63","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-3320f2c64f9658fe2e08","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v08\n- v04 --refutes--> v06\n- v0a --refutes--> v0b\n- v0d --supports--> v07\n- v0d --refutes--> v02\n- v03 --supports--> v0d\n- v05 --refutes--> v0c\n- v03 --refutes--> v09\n- v0c --supports--> v0a\n- v00 --supports--> v01\n- v00 --supports--> v03\n- v06 --refutes--> v00\n- v0b --refutes--> v04\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6b147794c1cab609c87e9f0f0342461bc3bc79c0424524edcacf78a12207e1f8","schema":"livegraph.boolean-propagation.v1","spec_hash":"3320f2c64f9658fe2e08f49fef8970be992a43b7275b5086930e8bb02f9951c3","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-ae5c3d7e031465ae314a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v02\n- v0a --supports--> v04\n- v09 --refutes--> v0b\n- v0d --supports--> v01\n- v04 --refutes--> v09\n- v06 --refutes--> v0c\n- v07 --refutes--> v0d\n- v09 --supports--> v07\n- v0d --refutes--> v08\n- v03 --refutes--> v06\n- v0d --refutes--> v00\n- v05 --supports--> v03\n- v08 --supports--> v05\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"42fc9da830f834cd9b8e6d142ec837b95d9c307e07740196032e93c3db9b447b","schema":"livegraph.boolean-propagation.v1","spec_hash":"ae5c3d7e031465ae314adfd4447f24b48d831a61ad2fc0a55224c2862e0d218b","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-b29b02f5f74f00667dfa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v0b\n- v01 --supports--> v06\n- v09 --refutes--> v0d\n- v04 --supports--> v08\n- v06 --refutes--> v09\n- v06 --supports--> v07\n- v0a --supports--> v01\n- v02 --refutes--> v0a\n- v01 --supports--> v0c\n- v03 --refutes--> v04\n- v0d --supports--> v05\n- v05 --supports--> v03\n- v03 --refutes--> v00\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5b6aa28a5110ea99d74d07038613b2d4eeefc2873c87ad7d4aa9e06bbe5620b5","schema":"livegraph.boolean-propagation.v1","spec_hash":"b29b02f5f74f00667dfac5fd4a0b9150be838824cc26c065e169eb5fd9a7c5e1","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-7c8b922758b60352693c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v00\n- v07 --supports--> v0b\n- v0b --refutes--> v02\n- v09 --refutes--> v0a\n- v0a --refutes--> v08\n- v02 --refutes--> v01\n- v0c --refutes--> v06\n- v08 --supports--> v0c\n- v00 --refutes--> v05\n- v00 --refutes--> v04\n- v08 --supports--> v0d\n- v04 --refutes--> v09\n- v0a --supports--> v03\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c6930987f5e61ac8a0aebb8f7d4cda342ce494762958b7c3e28a3fea98329ae9","schema":"livegraph.boolean-propagation.v1","spec_hash":"7c8b922758b60352693c3e65ef11660b17dce828d34c646787a2280e961e4360","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-1fee2112bd55aeac852e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v02\n- v02 --supports--> v0a\n- v09 --supports--> v03\n- v0c --refutes--> v0d\n- v04 --refutes--> v09\n- v0b --refutes--> v06\n- v02 --supports--> v0b\n- v06 --refutes--> v00\n- v0c --supports--> v05\n- v02 --refutes--> v01\n- v0c --supports--> v07\n- v08 --refutes--> v0c\n- v07 --refutes--> v04\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1e56cb592133f9913568e089be295eefbc8b0f768ebf2baf9f89a6824fd216dd","schema":"livegraph.boolean-propagation.v1","spec_hash":"1fee2112bd55aeac852ed792f452e25810137b0ae248ce7a2f91b6705edb48c1","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-84b44c40dc1a570df21a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v03\n- v0d --supports--> v06\n- v0a --refutes--> v08\n- v05 --supports--> v04\n- v07 --refutes--> v00\n- v09 --refutes--> v0a\n- v0a --refutes--> v0d\n- v0c --supports--> v01\n- v07 --refutes--> v0b\n- v0d --supports--> v05\n- v0b --refutes--> v0c\n- v02 --refutes--> v07\n- v04 --supports--> v02\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5e0232a8d227d9b4298534210c8c35d923b33fcfc3baa45cc72148140b6ae9c6","schema":"livegraph.boolean-propagation.v1","spec_hash":"84b44c40dc1a570df21af27b11ab3cb50a4b896a4c22fccf9e5f166ed89c66d1","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-c8fe3df6ff98783348f7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v0c\n- v08 --refutes--> v05\n- v0b --supports--> v08\n- v00 --refutes--> v0a\n- v03 --supports--> v0b\n- v04 --supports--> v09\n- v0b --supports--> v07\n- v0c --supports--> v04\n- v09 --refutes--> v01\n- v09 --supports--> v06\n- v01 --refutes--> v0d\n- v0d --refutes--> v00\n- v01 --refutes--> v02\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"11e8799dae1b4f6708ca7a8f168e0dde240700576b67abc708d4138ec3583741","schema":"livegraph.boolean-propagation.v1","spec_hash":"c8fe3df6ff98783348f7708a049edd5d7bf0fcfc7557d61b329c2d078bbdcf3b","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-a00b961010bfb85bd881","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v05\n- v09 --refutes--> v03\n- v07 --supports--> v04\n- v0a --refutes--> v08\n- v09 --refutes--> v07\n- v0d --supports--> v06\n- v01 --supports--> v00\n- v02 --refutes--> v0d\n- v01 --supports--> v0c\n- v04 --refutes--> v0b\n- v00 --refutes--> v02\n- v0d --supports--> v09\n- v05 --refutes--> v0a\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ba7e472f18b5b8480283de72f0b2ca1861f2193eaf069148c99c2273c42f0b54","schema":"livegraph.boolean-propagation.v1","spec_hash":"a00b961010bfb85bd881b305d4f3889378438cd3bf64cdc32934a50527207f73","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-bb85c4db8f98e30ebff4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v06\n- v06 --supports--> v0c\n- v01 --supports--> v08\n- v0a --supports--> v09\n- v01 --supports--> v02\n- v0d --supports--> v01\n- v0b --refutes--> v00\n- v00 --supports--> v04\n- v01 --refutes--> v0a\n- v0b --supports--> v07\n- v05 --supports--> v0d\n- v04 --refutes--> v05\n- v0d --refutes--> v03\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ac07da149bf8e087047e58c09c1209c17ab7bb69f90fe1b2104b70ddd33aa8d4","schema":"livegraph.boolean-propagation.v1","spec_hash":"bb85c4db8f98e30ebff45c7f2ef67126f6a4626113975dcc9e4798bd9314667a","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-ce08b78fc919174e31c7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v0d\n- v07 --supports--> v0b\n- v03 --supports--> v00\n- v04 --refutes--> v03\n- v08 --supports--> v0a\n- v05 --refutes--> v02\n- v0d --supports--> v07\n- v06 --refutes--> v04\n- v05 --supports--> v08\n- v01 --refutes--> v06\n- v08 --supports--> v0c\n- v07 --supports--> v01\n- v06 --refutes--> v09\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ccac8e5e64c0e204271eb3ada1e1dd986481b20827da9865011b98bfa34babd5","schema":"livegraph.boolean-propagation.v1","spec_hash":"ce08b78fc919174e31c72af06f5b53813264790e73511e0a6a7074e89dfe96cd","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-210eb97774098f7742eb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v04\n- v0b --supports--> v05\n- v01 --refutes--> v0a\n- v07 --supports--> v09\n- v02 --refutes--> v07\n- v04 --refutes--> v00\n- v0a --refutes--> v03\n- v02 --supports--> v0c\n- v05 --supports--> v01\n- v0d --supports--> v06\n- v00 --supports--> v0d\n- v0d --refutes--> v0b\n- v0a --refutes--> v08\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"611283f7e13d79571403193e7f5764c8e394000f30f8e65ed306ae3d8b398cb1","schema":"livegraph.boolean-propagation.v1","spec_hash":"210eb97774098f7742ebf75312b87e21c2e704bee35053b0d5a3705cdd30b1cb","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-fc838e96c47912950157","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v0c\n- v0c --refutes--> v0a\n- v07 --refutes--> v0b\n- v03 --supports--> v02\n- v06 --supports--> v0d\n- v04 --supports--> v01\n- v0b --refutes--> v04\n- v06 --refutes--> v05\n- v06 --refutes--> v08\n- v08 --supports--> v00\n- v00 --refutes--> v03\n- v03 --refutes--> v09\n- v0c --refutes--> v07\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d8ef00b9695d532b4b73566364434f3feda74194d571ab13c8419400ea58a1fa","schema":"livegraph.boolean-propagation.v1","spec_hash":"fc838e96c479129501579a85796665af5c88c483cac4cb24616a253244c386f3","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-da9943011bc90af98411","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v0d\n- v0c --supports--> v05\n- v05 --supports--> v0b\n- v03 --refutes--> v0c\n- v0b --refutes--> v0a\n- v00 --refutes--> v06\n- v07 --supports--> v00\n- v0a --supports--> v07\n- v0d --supports--> v04\n- v0c --refutes--> v01\n- v0b --supports--> v02\n- v05 --supports--> v09\n- v00 --refutes--> v08\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"17589f9d7cabf078689eaff862c08aa223248012b738d2ee241725ed9a4f5f71","schema":"livegraph.boolean-propagation.v1","spec_hash":"da9943011bc90af984114cc7721c9192aea8431cff26a2fc08446057cbd2bfbc","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-bc89ee51c58c286d5213","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v02\n- v0c --refutes--> v03\n- v06 --supports--> v00\n- v05 --refutes--> v07\n- v02 --supports--> v06\n- v0d --supports--> v05\n- v0c --supports--> v09\n- v0c --supports--> v0d\n- v01 --supports--> v04\n- v0b --refutes--> v01\n- v01 --supports--> v08\n- v06 --refutes--> v0b\n- v07 --supports--> v0a\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"205de728b09a3dd3817ce3a2e0f0f84599e0531462c767ad99cd9be62b30621e","schema":"livegraph.boolean-propagation.v1","spec_hash":"bc89ee51c58c286d521376a349ee2a2b92afe12ba80175ca77d83bb3043fc375","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-9347322bb066afb7f97d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v07\n- v09 --refutes--> v03\n- v0d --refutes--> v04\n- v0a --refutes--> v05\n- v00 --supports--> v0b\n- v01 --refutes--> v09\n- v01 --supports--> v06\n- v09 --supports--> v0c\n- v00 --supports--> v0d\n- v04 --supports--> v01\n- v03 --supports--> v0a\n- v00 --supports--> v02\n- v07 --refutes--> v08\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dc8e5a068a8c33aeba08836a63d5350b12610bb023f0e6cb8a0f35a7b89d417a","schema":"livegraph.boolean-propagation.v1","spec_hash":"9347322bb066afb7f97df3ef6d42889ca4c2073c1740c9ff8dfd60a1d510d213","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-56058d0955b0d056842c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v06\n- v01 --supports--> v04\n- v02 --supports--> v07\n- v01 --refutes--> v0b\n- v0d --refutes--> v0a\n- v0a --supports--> v05\n- v0d --supports--> v09\n- v0b --supports--> v0d\n- v06 --supports--> v00\n- v03 --refutes--> v0c\n- v06 --supports--> v08\n- v00 --refutes--> v01\n- v0a --refutes--> v02\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7c7d86d0cc345eca4d57dfcd57a3caa6c1c8d71e11c7c4ca0141d1f2cfebaf63","schema":"livegraph.boolean-propagation.v1","spec_hash":"56058d0955b0d056842c81956dc55873a9695c2f0ba4a86d8e093827b7255c9f","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-ef67dd52151665e6e4e4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v00\n- v00 --refutes--> v08\n- v07 --supports--> v0a\n- v03 --supports--> v09\n- v00 --refutes--> v0c\n- v07 --refutes--> v02\n- v02 --supports--> v04\n- v00 --supports--> v0d\n- v01 --supports--> v0b\n- v0b --supports--> v05\n- v09 --supports--> v01\n- v09 --supports--> v06\n- v04 --supports--> v03\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2b9b12eb60f4d746a72095012f9b7779b3d80f79619e3400fc71d02b22ec7896","schema":"livegraph.boolean-propagation.v1","spec_hash":"ef67dd52151665e6e4e40fddb0cd568ef24c449d85b003df267929dcef11576f","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":9,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-a40882220d1eb49de83a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v00\n- v06 --supports--> v0b\n- v0d --refutes--> v05\n- v04 --supports--> v07\n- v0b --supports--> v08\n- v05 --refutes--> v0a\n- v08 --refutes--> v0d\n- v0a --supports--> v02\n- v0a --refutes--> v09\n- v04 --supports--> v01\n- v02 --supports--> v03\n- v07 --supports--> v06\n- v02 --supports--> v0c\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2d9a967c3cfd4f76014b233583d95b4d8a1d0eabfd8897f82a0c8af3edbba0be","schema":"livegraph.boolean-propagation.v1","spec_hash":"a40882220d1eb49de83a60b462b090a979090c96b5808bfa55f3c03558e7f7d0","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-c89fd35095a5b8049504","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v05\n- v06 --supports--> v0d\n- v03 --refutes--> v0b\n- v0b --refutes--> v02\n- v04 --supports--> v09\n- v01 --refutes--> v07\n- v05 --supports--> v06\n- v04 --refutes--> v08\n- v07 --supports--> v03\n- v0b --supports--> v00\n- v0c --refutes--> v01\n- v04 --supports--> v0a\n- v0d --refutes--> v04\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f8d4b9e308a1254c7dab851faac166949ea4d127bccca5201d7f6f18fe996ef8","schema":"livegraph.boolean-propagation.v1","spec_hash":"c89fd35095a5b80495049639ca60649a60f0ffcce056c5e1b253866121b074ff","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-0a320a577d82f506bc5e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v05\n- v0d --refutes--> v08\n- v00 --refutes--> v01\n- v0d --supports--> v02\n- v09 --supports--> v04\n- v04 --refutes--> v0c\n- v03 --refutes--> v06\n- v07 --refutes--> v09\n- v0d --supports--> v03\n- v0a --supports--> v07\n- v04 --refutes--> v00\n- v01 --supports--> v0d\n- v06 --refutes--> v0b\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7b7e2ec882d7a8e5665ccc6806c92f60ebefdd4bdc8ed74bd6c52804a9b27542","schema":"livegraph.boolean-propagation.v1","spec_hash":"0a320a577d82f506bc5e1bd7c3a9b8e603cb155e5dbd0e7f9a169295eb77f14d","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-5a072af4d4da22998377","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v05\n- v00 --refutes--> v01\n- v05 --supports--> v02\n- v02 --refutes--> v00\n- v08 --supports--> v07\n- v05 --supports--> v06\n- v0b --refutes--> v0c\n- v01 --refutes--> v03\n- v0d --refutes--> v0a\n- v0d --refutes--> v08\n- v05 --refutes--> v04\n- v0c --supports--> v0d\n- v01 --refutes--> v09\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"13f2cde7accd7fafaae489705355fe65b4d63e068cda13eb6254ea487970a0fa","schema":"livegraph.boolean-propagation.v1","spec_hash":"5a072af4d4da2299837701d1db385fa0b8234dd51e26c236a0edc8f7dffed105","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-e99342b441293dd22f21","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v0a\n- v05 --supports--> v06\n- v00 --supports--> v04\n- v06 --supports--> v07\n- v03 --supports--> v01\n- v01 --supports--> v0d\n- v06 --supports--> v0b\n- v01 --supports--> v08\n- v04 --refutes--> v02\n- v02 --supports--> v09\n- v04 --supports--> v0c\n- v0b --refutes--> v03\n- v09 --refutes--> v05\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"50a041a14bc1d4d7cce4c81ac28e14c99909d7cf1d69e94660f84552cc3ca7e3","schema":"livegraph.boolean-propagation.v1","spec_hash":"e99342b441293dd22f21b359192da9d2eef1170ed3810cdc4dafdc358f056f2b","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":9,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-5d16976078035aa8e56a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v0d\n- v01 --supports--> v03\n- v0a --supports--> v09\n- v04 --supports--> v07\n- v01 --refutes--> v05\n- v07 --refutes--> v00\n- v0d --supports--> v02\n- v05 --refutes--> v06\n- v0a --refutes--> v0b\n- v0b --supports--> v01\n- v00 --supports--> v08\n- v06 --refutes--> v04\n- v07 --supports--> v0c\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1944b3f2ef4ad6ea16c5612685bf5bd8a81fb0dee13b507afbcb800a8a89f92b","schema":"livegraph.boolean-propagation.v1","spec_hash":"5d16976078035aa8e56af125cc9b24607722ae90730cd366ce40057c42ba95c0","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-3e4571df01e17a2f51a3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v04\n- v05 --supports--> v02\n- v06 --supports--> v07\n- v0d --refutes--> v0b\n- v04 --supports--> v03\n- v0a --supports--> v08\n- v01 --refutes--> v0c\n- v01 --supports--> v00\n- v0a --refutes--> v09\n- v08 --refutes--> v05\n- v0b --supports--> v06\n- v04 --refutes--> v0d\n- v01 --refutes--> v0a\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"983d01a9a38d6c1bb771c10520f8129c2c700d56e36c8afb8dc3f2af9c23d817","schema":"livegraph.boolean-propagation.v1","spec_hash":"3e4571df01e17a2f51a3aec2e42b6b50976ac192318b400ac2249b3262903265","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-e136ef6e33f8b097861a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v0a\n- v02 --refutes--> v0d\n- v0c --refutes--> v02\n- v07 --refutes--> v08\n- v07 --supports--> v01\n- v06 --supports--> v09\n- v0a --refutes--> v03\n- v04 --refutes--> v00\n- v09 --supports--> v07\n- v04 --supports--> v0b\n- v0d --supports--> v04\n- v03 --refutes--> v0c\n- v04 --refutes--> v05\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4c25ee5531627d85928b3a558f36ae308677cd6eec8cbb6299ba0dbc27dab9e1","schema":"livegraph.boolean-propagation.v1","spec_hash":"e136ef6e33f8b097861ad2bbafbc8faaddb0bfdb78f622463d17ccb1670af010","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-09d3c5aa754a8bbac3ec","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v0d\n- v08 --refutes--> v0b\n- v06 --supports--> v03\n- v04 --refutes--> v0c\n- v09 --supports--> v0a\n- v0a --supports--> v08\n- v0d --refutes--> v00\n- v05 --supports--> v01\n- v06 --supports--> v04\n- v0a --refutes--> v07\n- v0d --refutes--> v09\n- v02 --supports--> v06\n- v08 --refutes--> v05\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3ad190d8b5a1f4f17c550033f154469bf59c3060d85f74553ffd01a3055474c9","schema":"livegraph.boolean-propagation.v1","spec_hash":"09d3c5aa754a8bbac3ecf31de05413c79eaede0af22a7cfc32c5a45d16d8fefc","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-8f33a1975d3b9d97c1ea","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v04\n- v0a --supports--> v07\n- v07 --supports--> v02\n- v09 --supports--> v0b\n- v08 --refutes--> v0a\n- v0b --supports--> v05\n- v07 --refutes--> v06\n- v0b --supports--> v03\n- v06 --refutes--> v0d\n- v0b --supports--> v01\n- v04 --refutes--> v00\n- v03 --supports--> v08\n- v04 --refutes--> v09\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0d1c4daf6dde8a9ff45c433951cdce0130d90aa180b23c869a7ba06c7ca8c585","schema":"livegraph.boolean-propagation.v1","spec_hash":"8f33a1975d3b9d97c1ea939cf77f0533a47c1bdbd6fd290ef37c7fc769229642","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-8e6646c8fc602bd856ef","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v0b\n- v06 --refutes--> v03\n- v08 --supports--> v05\n- v04 --supports--> v02\n- v09 --supports--> v07\n- v06 --supports--> v04\n- v0b --refutes--> v0a\n- v02 --supports--> v09\n- v0b --supports--> v01\n- v05 --supports--> v0c\n- v0a --supports--> v0d\n- v07 --supports--> v00\n- v05 --supports--> v06\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"39621ae9f150aa700b68000c9a14aa30e2836a153a6450f0e81e400b6ed34aa3","schema":"livegraph.boolean-propagation.v1","spec_hash":"8e6646c8fc602bd856ef5c696fc0c22508b2c78cc4084901dac3fc96c4689ae1","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":9,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-b1e8487f664febc8f6d3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v0a\n- v05 --refutes--> v07\n- v00 --supports--> v05\n- v08 --refutes--> v00\n- v0d --refutes--> v04\n- v0d --refutes--> v01\n- v0c --refutes--> v02\n- v06 --supports--> v08\n- v00 --supports--> v0b\n- v0a --refutes--> v09\n- v0c --supports--> v03\n- v07 --refutes--> v0d\n- v04 --refutes--> v0c\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b6b0e0700bc1ff940ec2aaa2cbb93634ebf510b8b607b9d5481028a3c53ed2fc","schema":"livegraph.boolean-propagation.v1","spec_hash":"b1e8487f664febc8f6d321aadefcf18a94d7610b795e974a66a8e9b5012e8c56","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-dc64b49f2691b18947a2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v01\n- v04 --supports--> v06\n- v0a --supports--> v05\n- v0d --refutes--> v0b\n- v06 --refutes--> v0d\n- v0c --supports--> v02\n- v00 --supports--> v0a\n- v0a --refutes--> v04\n- v0a --supports--> v03\n- v09 --supports--> v07\n- v02 --supports--> v09\n- v0d --refutes--> v0c\n- v07 --refutes--> v08\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"733083b88c20c78a44ff41eedea98e3e4d856164088d064ec187eff60e96d5f5","schema":"livegraph.boolean-propagation.v1","spec_hash":"dc64b49f2691b18947a20ea66d5c080b8d23b284daa6e0d0826c2797f6e41c51","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-18b5c7ae603d15791c90","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v05\n- v0b --refutes--> v01\n- v01 --refutes--> v02\n- v09 --refutes--> v0c\n- v05 --supports--> v03\n- v06 --supports--> v09\n- v03 --refutes--> v0a\n- v03 --supports--> v00\n- v02 --supports--> v06\n- v06 --refutes--> v08\n- v00 --supports--> v0b\n- v0b --refutes--> v07\n- v09 --supports--> v04\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e54a70bf6b5013306a69e2155c747d89945e750929697b343bcc8eb3a0924f88","schema":"livegraph.boolean-propagation.v1","spec_hash":"18b5c7ae603d15791c900763a02b8ebd53e901b50ded6d31947c889440734ec3","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-f0b0b19bec5b311e6e02","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v0c\n- v01 --refutes--> v02\n- v04 --refutes--> v03\n- v02 --refutes--> v05\n- v09 --supports--> v00\n- v04 --refutes--> v0b\n- v05 --refutes--> v0a\n- v00 --supports--> v07\n- v02 --supports--> v06\n- v03 --refutes--> v09\n- v05 --refutes--> v08\n- v0a --refutes--> v04\n- v07 --supports--> v0d\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6609e424b181e412398807f899fbef11204abf5a496e2361ef531bbd6d4c9923","schema":"livegraph.boolean-propagation.v1","spec_hash":"f0b0b19bec5b311e6e0275385afb6f1114f33743b6c1b96a116d043a9482dc2a","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":9,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-f1b2f3605cce9d0066bf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v05\n- v05 --refutes--> v07\n- v00 --supports--> v01\n- v0d --refutes--> v06\n- v07 --supports--> v02\n- v01 --supports--> v0c\n- v07 --refutes--> v0a\n- v03 --supports--> v04\n- v08 --refutes--> v09\n- v08 --refutes--> v0b\n- v06 --supports--> v00\n- v02 --refutes--> v0d\n- v01 --supports--> v08\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"eaebf844c10307f94226148757eb627762cb2bf4b8ad1fc24fc117d63992e52d","schema":"livegraph.boolean-propagation.v1","spec_hash":"f1b2f3605cce9d0066bfe08be805cad047a254b1b758495461ab7b3de7d97fb3","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-ecf045d2dd7b4aaf60a5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v09\n- v07 --supports--> v02\n- v00 --refutes--> v01\n- v03 --refutes--> v08\n- v06 --refutes--> v07\n- v00 --refutes--> v04\n- v09 --supports--> v0d\n- v04 --refutes--> v0a\n- v03 --refutes--> v0c\n- v02 --supports--> v00\n- v0c --refutes--> v06\n- v09 --refutes--> v0b\n- v07 --supports--> v05\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c6d1299e03de9751f5cb3e8cb11b69e4179e814dd885fa446ad2998e3836f558","schema":"livegraph.boolean-propagation.v1","spec_hash":"ecf045d2dd7b4aaf60a57cfb7dad4497a62649c7c6f44b449d4abda63246632c","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-6962f5842a92ffa9b6e4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v07\n- v06 --supports--> v00\n- v0d --supports--> v04\n- v0d --supports--> v06\n- v08 --refutes--> v09\n- v00 --supports--> v0b\n- v0c --supports--> v03\n- v02 --refutes--> v01\n- v00 --refutes--> v05\n- v07 --supports--> v08\n- v0c --supports--> v0d\n- v05 --supports--> v02\n- v02 --refutes--> v0a\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0e2847314c379b18fe2763145e2a072afbf42126ecfa8d1ebbf8cdc1f231e975","schema":"livegraph.boolean-propagation.v1","spec_hash":"6962f5842a92ffa9b6e4c401f8ce3ded445537328dcc1be77293cdbd483ff370","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-39c0a8264526e3c8f4be","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v0c\n- v06 --supports--> v01\n- v0a --refutes--> v03\n- v03 --refutes--> v07\n- v01 --supports--> v08\n- v0d --refutes--> v09\n- v02 --refutes--> v0d\n- v0d --supports--> v00\n- v04 --refutes--> v06\n- v03 --refutes--> v04\n- v08 --supports--> v05\n- v09 --supports--> v0a\n- v01 --refutes--> v0b\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8b8fa7828f77b728ccd4ae2df00bb9e390873ee1b5a27253f4ab8189f6b744ab","schema":"livegraph.boolean-propagation.v1","spec_hash":"39c0a8264526e3c8f4be1b314bc2af916cc07ba4a0e5837973135441e6c9fd1e","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-e733a82ee4a6e7d568cc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v03\n- v02 --refutes--> v06\n- v08 --supports--> v0c\n- v01 --refutes--> v04\n- v09 --supports--> v0d\n- v04 --supports--> v08\n- v0a --refutes--> v0b\n- v0a --refutes--> v07\n- v02 --refutes--> v09\n- v0c --refutes--> v00\n- v0b --supports--> v02\n- v0d --refutes--> v01\n- v09 --refutes--> v05\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"59ac1875024ce96476d8f84153712f1210809bc4437e3488a850f78284822665","schema":"livegraph.boolean-propagation.v1","spec_hash":"e733a82ee4a6e7d568ccf857b6c555d47447bf48cbba5829ba7c5ecb28749abd","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-39e49509e6dd9f90c069","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v0d\n- v01 --refutes--> v09\n- v0a --refutes--> v0b\n- v0b --refutes--> v04\n- v06 --refutes--> v00\n- v01 --refutes--> v06\n- v03 --supports--> v05\n- v00 --supports--> v02\n- v05 --supports--> v0a\n- v00 --refutes--> v0c\n- v08 --supports--> v01\n- v00 --refutes--> v07\n- v02 --supports--> v03\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"70ccaf99ab4fa340e597e4ae53c6fa0794be8d0e35ee9d0902e0e4c8d42a22f9","schema":"livegraph.boolean-propagation.v1","spec_hash":"39e49509e6dd9f90c0693723aab0d34ed088ae3fb143f7833536262fbc14fa22","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":9,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-ad0fc167cd75608d08b7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v0b\n- v01 --supports--> v05\n- v0a --refutes--> v01\n- v04 --refutes--> v08\n- v06 --supports--> v0d\n- v0c --refutes--> v09\n- v0d --refutes--> v02\n- v02 --supports--> v0a\n- v08 --refutes--> v06\n- v05 --refutes--> v03\n- v0a --refutes--> v00\n- v0c --refutes--> v04\n- v0a --supports--> v07\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5f519aa673eec77966ebeb251dc4edc86286e4a71a632e73eca68ef54635deb4","schema":"livegraph.boolean-propagation.v1","spec_hash":"ad0fc167cd75608d08b75be02d7deb5b6a8ded41245eedd81a0d53a8b0b36c00","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-a0d8300d76c0eb4b2aed","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v0b\n- v01 --refutes--> v06\n- v08 --refutes--> v02\n- v06 --refutes--> v03\n- v06 --refutes--> v00\n- v09 --refutes--> v0d\n- v0c --refutes--> v05\n- v09 --supports--> v0c\n- v05 --supports--> v0a\n- v05 --refutes--> v04\n- v08 --refutes--> v07\n- v0b --supports--> v01\n- v02 --refutes--> v09\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1f34d8c9ada437276269551347d2202ce6e254a95099a0f990ea564d6465dcbc","schema":"livegraph.boolean-propagation.v1","spec_hash":"a0d8300d76c0eb4b2aed67261f9986c3eb218ca8858cd8d76bbfb360e51235eb","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-121a4897354051733e53","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v06\n- v00 --refutes--> v04\n- v01 --supports--> v0b\n- v0c --refutes--> v00\n- v07 --supports--> v02\n- v0a --supports--> v07\n- v02 --supports--> v08\n- v00 --supports--> v03\n- v00 --supports--> v01\n- v0d --supports--> v0c\n- v08 --refutes--> v0d\n- v01 --supports--> v05\n- v0b --supports--> v09\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e47faf5609bcb9e05887428e6432319567b766576c0de2d7485ceca7168562bd","schema":"livegraph.boolean-propagation.v1","spec_hash":"121a4897354051733e5361d1677c99480395bc0bb28bb6ce0c40db4fc564a1b8","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-01f0d72f35d40ce2cc37","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v0d\n- v0d --supports--> v0b\n- v0d --refutes--> v02\n- v07 --refutes--> v09\n- v08 --supports--> v04\n- v05 --refutes--> v07\n- v07 --supports--> v00\n- v08 --supports--> v01\n- v03 --supports--> v06\n- v02 --refutes--> v0a\n- v02 --supports--> v0c\n- v00 --supports--> v08\n- v0a --supports--> v05\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"84371565fdcf5f0bca40aaeb62d6b81b3ec2fb545138835f478821ee7a5295eb","schema":"livegraph.boolean-propagation.v1","spec_hash":"01f0d72f35d40ce2cc3791bde3356b8604c77adc83a0ffd76682ee0e344c7e2b","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-e42d5ea2391e9b578b7f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v02\n- v07 --refutes--> v09\n- v03 --supports--> v0a\n- v0a --supports--> v05\n- v05 --supports--> v0b\n- v03 --supports--> v08\n- v02 --supports--> v07\n- v0a --supports--> v06\n- v04 --supports--> v00\n- v0b --supports--> v04\n- v0d --refutes--> v01\n- v07 --refutes--> v0d\n- v05 --refutes--> v0c\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7e7734fb3f3af452d8d95c9a9df02a49d889209929af0bdf63ab62ef8de2012e","schema":"livegraph.boolean-propagation.v1","spec_hash":"e42d5ea2391e9b578b7f61f2fb8bf71a7fd9c4d346e07814b2ac0b709ed0be7f","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-0675c7d444af7d9d6b0e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v07\n- v0b --refutes--> v00\n- v0c --refutes--> v04\n- v09 --supports--> v06\n- v00 --supports--> v05\n- v09 --refutes--> v01\n- v02 --supports--> v0a\n- v02 --refutes--> v0b\n- v0d --refutes--> v02\n- v09 --refutes--> v03\n- v04 --refutes--> v08\n- v08 --supports--> v0d\n- v06 --refutes--> v0c\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a900401167ca62a15f01ae35e58dfe47de78db41f46b6c4a95bdac6d3649d589","schema":"livegraph.boolean-propagation.v1","spec_hash":"0675c7d444af7d9d6b0e83d8897716fde8a28f1ad57963d7c7899da2668da97c","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-92c12efc1b21b06bd2fb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v08\n- v0b --refutes--> v0a\n- v09 --refutes--> v05\n- v00 --refutes--> v02\n- v0b --supports--> v0c\n- v00 --refutes--> v0d\n- v08 --supports--> v04\n- v0c --supports--> v06\n- v03 --supports--> v01\n- v0d --supports--> v07\n- v07 --refutes--> v09\n- v06 --refutes--> v03\n- v03 --refutes--> v00\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"09753ce70f7a52d6c2d8ebdc0c378e80f790877807b2b7deddde99ca0ac32dad","schema":"livegraph.boolean-propagation.v1","spec_hash":"92c12efc1b21b06bd2fb46766290a87345f4a8492d0fc3a040b515e8a22a0d7a","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-12e361e59c70b9ceda54","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v05\n- v09 --supports--> v0a\n- v09 --supports--> v08\n- v03 --supports--> v01\n- v03 --supports--> v02\n- v0c --supports--> v07\n- v01 --refutes--> v04\n- v01 --refutes--> v0c\n- v07 --refutes--> v0b\n- v0b --supports--> v09\n- v00 --refutes--> v0d\n- v09 --refutes--> v06\n- v05 --supports--> v03\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b276343d084e2b34201b798bbd9aa6523e34d5393ca8754dbc4a22d4dbb804cf","schema":"livegraph.boolean-propagation.v1","spec_hash":"12e361e59c70b9ceda54018b24d9b8f32462f0f005a14b07aa9589785519960a","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-4000403c32564c9f1cc0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v08\n- v04 --refutes--> v0a\n- v03 --supports--> v00\n- v03 --supports--> v0c\n- v0d --refutes--> v04\n- v0a --supports--> v09\n- v01 --refutes--> v0b\n- v01 --supports--> v07\n- v05 --supports--> v03\n- v0b --supports--> v06\n- v08 --refutes--> v0d\n- v0b --supports--> v05\n- v03 --supports--> v02\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bf01c800ac447658276606bc5e7ec390e1c70a287efd86ff90003385ef066b7e","schema":"livegraph.boolean-propagation.v1","spec_hash":"4000403c32564c9f1cc0c413ce0c9cf505f574ed6e8cfc88077c4fa43df6d307","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-a39b6293521be56846f9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v08\n- v0c --refutes--> v02\n- v02 --refutes--> v03\n- v07 --refutes--> v00\n- v05 --supports--> v01\n- v00 --refutes--> v04\n- v0a --refutes--> v05\n- v01 --supports--> v07\n- v0b --supports--> v0c\n- v02 --refutes--> v09\n- v00 --refutes--> v0d\n- v0c --refutes--> v06\n- v09 --refutes--> v0a\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ac5c6f8431f1d987cf4a860fdead1a6f56d413efd653908cd47b0ccdc824924c","schema":"livegraph.boolean-propagation.v1","spec_hash":"a39b6293521be56846f9f25ba483eead1eb8581d376fa80922bdadf20b8fd332","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-949046f753a741199ef7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v02\n- v03 --supports--> v04\n- v04 --supports--> v05\n- v09 --supports--> v03\n- v06 --supports--> v0d\n- v0c --supports--> v0b\n- v06 --supports--> v07\n- v0c --refutes--> v00\n- v06 --supports--> v08\n- v03 --refutes--> v0a\n- v01 --supports--> v0c\n- v00 --supports--> v09\n- v02 --refutes--> v01\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bd46eba7f33008a163481a0db0f8a28dfef525231605b39664ef75bf6b770368","schema":"livegraph.boolean-propagation.v1","spec_hash":"949046f753a741199ef74da7fe9a5c6b08b869b99c8af8b2a95f89eb79890e44","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-758cd6af8d55e183d8e1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v08\n- v0c --refutes--> v04\n- v03 --supports--> v06\n- v04 --supports--> v0d\n- v01 --refutes--> v00\n- v03 --refutes--> v0b\n- v08 --refutes--> v02\n- v03 --refutes--> v01\n- v09 --refutes--> v0a\n- v05 --supports--> v03\n- v0a --refutes--> v0c\n- v09 --refutes--> v07\n- v0d --supports--> v05\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f42aca9abc14bd4b728178b628d40c2f8d682082f6b3fc291cb8546528d12428","schema":"livegraph.boolean-propagation.v1","spec_hash":"758cd6af8d55e183d8e112a12db035cc8dde1d41c5950d6140b336828b279b47","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-1924403c5685bde522de","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v05\n- v0b --refutes--> v08\n- v09 --supports--> v04\n- v0c --refutes--> v06\n- v05 --refutes--> v07\n- v01 --supports--> v00\n- v0b --refutes--> v0c\n- v02 --supports--> v01\n- v07 --supports--> v09\n- v07 --supports--> v03\n- v09 --supports--> v02\n- v0d --supports--> v0b\n- v0c --refutes--> v0a\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6d9144ca4bf92cbe8d1348f9f47c987a724b720d208aabad035cdeb58a1ea97f","schema":"livegraph.boolean-propagation.v1","spec_hash":"1924403c5685bde522de6a606a7ea7092eed9fb2894d453d06ccaa58e3d0fbca","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-b7eb2d47ff375671d59e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v08\n- v03 --supports--> v05\n- v07 --refutes--> v04\n- v0d --supports--> v09\n- v0a --supports--> v00\n- v03 --supports--> v06\n- v0c --supports--> v0a\n- v0b --supports--> v01\n- v00 --supports--> v07\n- v07 --supports--> v0b\n- v0d --refutes--> v0c\n- v06 --refutes--> v0d\n- v0b --supports--> v02\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c30d23ae81af0dd22a94deed3f93346a719de5731a24423918582166c6a5d136","schema":"livegraph.boolean-propagation.v1","spec_hash":"b7eb2d47ff375671d59ef957ac4aa54b0dbe60ae73052a7270b3532f0573f9c0","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-209bb1c7b97e5db2aa40","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v05\n- v01 --refutes--> v00\n- v0a --supports--> v04\n- v09 --refutes--> v02\n- v0d --supports--> v0c\n- v09 --refutes--> v07\n- v08 --refutes--> v0b\n- v0b --supports--> v09\n- v01 --supports--> v03\n- v01 --supports--> v06\n- v00 --refutes--> v0d\n- v02 --refutes--> v01\n- v05 --supports--> v08\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a3720e47507c1b22f57d30c2b25a9c1d8bc08bda5d78f86a85c3e980feae80a2","schema":"livegraph.boolean-propagation.v1","spec_hash":"209bb1c7b97e5db2aa40ee702b1f169d90521d2bfb6fa5b81235099cde82fa30","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-7688d8f7c9469ed7ad55","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v07\n- v0a --refutes--> v06\n- v09 --supports--> v05\n- v07 --supports--> v09\n- v07 --supports--> v0d\n- v0c --supports--> v08\n- v04 --supports--> v00\n- v0a --refutes--> v02\n- v05 --refutes--> v0c\n- v08 --supports--> v04\n- v0c --refutes--> v01\n- v0b --supports--> v03\n- v0a --supports--> v0b\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"76135ccad2acfe569bc1e548d6bd1052deea17665bbe95370b984645e3f6192c","schema":"livegraph.boolean-propagation.v1","spec_hash":"7688d8f7c9469ed7ad55bf3ccd83b4aa7d761bd620b6f87a028a54335fa56f3e","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-1df7dff0dcb24f314e56","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v06\n- v0b --supports--> v07\n- v0c --supports--> v03\n- v07 --supports--> v0c\n- v04 --supports--> v09\n- v0c --supports--> v05\n- v05 --supports--> v08\n- v0a --supports--> v0b\n- v04 --supports--> v01\n- v0d --refutes--> v0a\n- v01 --refutes--> v0d\n- v05 --refutes--> v02\n- v00 --refutes--> v04\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"10f57b24c569aa046cd75b1ed508302e32b96081ba031e76c144004f27bde974","schema":"livegraph.boolean-propagation.v1","spec_hash":"1df7dff0dcb24f314e56f868c58c4104f96844dadd50fe2dce2954e2649a93c0","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-032893ea83b235d9e435","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v00\n- v0c --refutes--> v0b\n- v06 --refutes--> v02\n- v02 --supports--> v09\n- v08 --supports--> v07\n- v02 --supports--> v0d\n- v01 --refutes--> v0c\n- v04 --refutes--> v0a\n- v0c --refutes--> v08\n- v09 --supports--> v04\n- v0a --refutes--> v03\n- v07 --refutes--> v06\n- v04 --refutes--> v05\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"02d7c5c69cbdbb43ab78870a093411abc0a0e15af580ec167c0ddecc2bd54614","schema":"livegraph.boolean-propagation.v1","spec_hash":"032893ea83b235d9e435409b7a1560c6331776c90d37e2dd41891ed25d4f01fb","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-9dd78ab544669c0e4ed4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v0a\n- v04 --supports--> v08\n- v01 --refutes--> v02\n- v01 --supports--> v05\n- v04 --refutes--> v0c\n- v06 --supports--> v0d\n- v0b --refutes--> v09\n- v06 --supports--> v03\n- v0a --refutes--> v06\n- v0d --refutes--> v04\n- v09 --supports--> v01\n- v08 --refutes--> v0b\n- v09 --supports--> v07\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bbd710ecf132aa6e1655dd67eaa2ccd9f7c2ecf654cbf6fccb79e89699a2ecf4","schema":"livegraph.boolean-propagation.v1","spec_hash":"9dd78ab544669c0e4ed4a70ca444469cca455e3c5a49a7c27fd88262e05cb5c3","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-5ef0081e7b9d93013f98","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v04\n- v07 --supports--> v09\n- v09 --refutes--> v0a\n- v00 --refutes--> v05\n- v01 --refutes--> v0d\n- v03 --refutes--> v00\n- v05 --refutes--> v01\n- v07 --supports--> v08\n- v02 --supports--> v06\n- v00 --refutes--> v0c\n- v06 --refutes--> v0b\n- v09 --refutes--> v03\n- v0b --supports--> v07\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"84847cc8cf6e88ba37e24ce4b689db5f61e948645a7c43f8072fa4e73499350e","schema":"livegraph.boolean-propagation.v1","spec_hash":"5ef0081e7b9d93013f98231f69ddb791a84d0a2a0146aa422a92f1f6cacb9f6c","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":9,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-aff92a3de334fabff23f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v03\n- v05 --supports--> v01\n- v0d --refutes--> v02\n- v0d --refutes--> v0c\n- v02 --supports--> v07\n- v0b --supports--> v0d\n- v08 --supports--> v09\n- v01 --refutes--> v0a\n- v01 --refutes--> v00\n- v07 --supports--> v04\n- v0d --supports--> v06\n- v03 --supports--> v0b\n- v04 --supports--> v05\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e6b35b20a090b356795fb705baab84709857c68f876385e7ecad8c5594649e7e","schema":"livegraph.boolean-propagation.v1","spec_hash":"aff92a3de334fabff23fe0c21d709b8e0b96a444914ead6d0ab58652c84f9f99","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-2444386ec0ad9260e1bc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v0d\n- v02 --refutes--> v07\n- v01 --refutes--> v06\n- v01 --refutes--> v0c\n- v09 --refutes--> v08\n- v07 --refutes--> v0b\n- v0a --refutes--> v00\n- v05 --supports--> v0a\n- v05 --supports--> v03\n- v0d --supports--> v02\n- v0c --refutes--> v04\n- v07 --supports--> v05\n- v00 --supports--> v01\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"90cf0d3b344c3362736bd57115c569e8d7a028a03e2a22e7d584fd5e95523262","schema":"livegraph.boolean-propagation.v1","spec_hash":"2444386ec0ad9260e1bc9991746538fce44d3ec193bb53f4f621efccc0081503","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-f71a6dbbff02e3e2fb22","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v01\n- v0a --supports--> v00\n- v0c --refutes--> v07\n- v0a --refutes--> v0d\n- v06 --refutes--> v03\n- v0b --supports--> v05\n- v0a --supports--> v04\n- v09 --refutes--> v08\n- v08 --supports--> v0a\n- v03 --refutes--> v0b\n- v01 --refutes--> v06\n- v03 --refutes--> v02\n- v0c --refutes--> v09\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"98092f6de5d46769969bf1581bfb5a67e50fc9eacf80a20bb35979c670e0245d","schema":"livegraph.boolean-propagation.v1","spec_hash":"f71a6dbbff02e3e2fb22c40126a38a32f635b58640811298e5e89d5cb7c3cec1","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-acc6c0bc9fc42e39071e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v05\n- v09 --supports--> v02\n- v0c --refutes--> v09\n- v01 --supports--> v0a\n- v0a --refutes--> v03\n- v00 --refutes--> v0c\n- v08 --supports--> v0d\n- v0d --refutes--> v0b\n- v08 --refutes--> v06\n- v0a --refutes--> v00\n- v05 --refutes--> v01\n- v09 --refutes--> v04\n- v05 --supports--> v07\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"25d0cafaa88bd602f42e8d02dc2743c72e53860b410959248f1a5c5513ff5c7f","schema":"livegraph.boolean-propagation.v1","spec_hash":"acc6c0bc9fc42e39071e967f518a8773c3ab58b52187005acb586b1418fe8ba0","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-aff4f46681f05688f6a2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v0c\n- v0b --supports--> v07\n- v03 --supports--> v04\n- v09 --refutes--> v00\n- v05 --refutes--> v03\n- v06 --refutes--> v0b\n- v00 --refutes--> v06\n- v01 --refutes--> v05\n- v04 --supports--> v09\n- v02 --supports--> v01\n- v02 --supports--> v08\n- v0b --supports--> v0a\n- v04 --refutes--> v0d\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9a42fb68fa53c067a69773b1dd8472d2699cc415a1dabac25da76d779916e8c7","schema":"livegraph.boolean-propagation.v1","spec_hash":"aff4f46681f05688f6a26952687382a45d1cdd7533e3e310826b37b6dd77dd73","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":9,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-3396a6fb168cca1763db","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v03\n- v0d --supports--> v02\n- v01 --refutes--> v08\n- v08 --supports--> v0d\n- v0d --refutes--> v05\n- v05 --supports--> v04\n- v0b --refutes--> v01\n- v09 --supports--> v0c\n- v05 --supports--> v06\n- v06 --supports--> v07\n- v06 --supports--> v00\n- v0c --refutes--> v0b\n- v00 --supports--> v0a\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"15d8062fe3556425abf896a5d352c4efa730631ed00a57ebad96d76bbc6c6cc6","schema":"livegraph.boolean-propagation.v1","spec_hash":"3396a6fb168cca1763dbb7049b02716223b521860272466316fc6358c1850f2d","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-3f728c6634c180e6c503","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v0d\n- v0b --refutes--> v08\n- v08 --refutes--> v09\n- v03 --refutes--> v01\n- v08 --refutes--> v0c\n- v04 --refutes--> v07\n- v00 --refutes--> v0b\n- v0a --supports--> v06\n- v06 --refutes--> v00\n- v03 --supports--> v05\n- v09 --supports--> v04\n- v03 --supports--> v0a\n- v0a --refutes--> v02\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"13e4d2d1c4bba5b406fde19101083e40e7c8877e541405a478ee52ce89d92c0a","schema":"livegraph.boolean-propagation.v1","spec_hash":"3f728c6634c180e6c503e7fc9a03f5ae9d802534a1de2a5740f37231d47226bc","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":9,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-8cb2c4eb79f145feef8b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v0a\n- v08 --refutes--> v09\n- v0a --supports--> v0c\n- v01 --refutes--> v06\n- v0c --refutes--> v04\n- v0d --supports--> v0b\n- v03 --supports--> v05\n- v04 --refutes--> v01\n- v0d --refutes--> v02\n- v03 --supports--> v08\n- v08 --refutes--> v07\n- v0b --supports--> v03\n- v01 --refutes--> v0d\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b05824c3aa757ebe229e0b24f15f008c149e3f0dcdba14b375bc025217586f28","schema":"livegraph.boolean-propagation.v1","spec_hash":"8cb2c4eb79f145feef8bc7efbe7ba84e2677c0430cef15a4eb477494bafe624d","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-d67bf93eefe91da16c8a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v05\n- v08 --refutes--> v0b\n- v07 --supports--> v0d\n- v0d --refutes--> v03\n- v0d --supports--> v01\n- v08 --supports--> v09\n- v0a --supports--> v0c\n- v00 --supports--> v07\n- v03 --supports--> v0a\n- v04 --refutes--> v02\n- v04 --supports--> v08\n- v09 --supports--> v00\n- v08 --supports--> v06\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d7178fecf6929d732ca64fd45d672944baf397c2d06cd982550cdacde073d184","schema":"livegraph.boolean-propagation.v1","spec_hash":"d67bf93eefe91da16c8a6b455fb04967343ed41bdea6fb8a4073139872eb7f91","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-f486fd5f26de3340dc76","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v04\n- v0a --refutes--> v00\n- v0d --supports--> v08\n- v05 --supports--> v06\n- v06 --supports--> v0c\n- v0c --supports--> v02\n- v09 --supports--> v0a\n- v0a --supports--> v03\n- v0c --supports--> v01\n- v0d --refutes--> v05\n- v0b --supports--> v07\n- v0b --supports--> v09\n- v04 --refutes--> v0d\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6f34784e1bec0a92e5c34bd9a311cc79e53324646a3890559ed2c3a707e5e4bd","schema":"livegraph.boolean-propagation.v1","spec_hash":"f486fd5f26de3340dc76e267e3fe0857d2486a2e3d450fb7cf5a5eff1e3da76f","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-2f6e5d822fcc93c5dab8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v0c\n- v0c --supports--> v00\n- v0b --supports--> v02\n- v02 --refutes--> v05\n- v08 --refutes--> v01\n- v01 --supports--> v0d\n- v02 --refutes--> v04\n- v04 --refutes--> v0a\n- v02 --supports--> v07\n- v0b --refutes--> v03\n- v01 --refutes--> v09\n- v00 --refutes--> v06\n- v0a --supports--> v08\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cc49dcc6df72ba6c5aa64dc68dde64f540b994f70ebff1d1b70438a2e81144e4","schema":"livegraph.boolean-propagation.v1","spec_hash":"2f6e5d822fcc93c5dab8cfcea8d82a8e83e941c2e5e008da258478a8155e819c","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-904644a226b5854dfe54","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v02\n- v03 --supports--> v04\n- v07 --refutes--> v0c\n- v0c --refutes--> v00\n- v05 --supports--> v0d\n- v09 --supports--> v08\n- v07 --refutes--> v0a\n- v00 --supports--> v01\n- v00 --refutes--> v03\n- v0b --supports--> v07\n- v08 --refutes--> v0b\n- v05 --supports--> v06\n- v0d --supports--> v09\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c7bdbd7e536ab648a2f7547d1b0204ee6e03b99bc95ec14998d1f2cdb320337d","schema":"livegraph.boolean-propagation.v1","spec_hash":"904644a226b5854dfe541a80135223e3f0bbb2006ae33e0d35082d8df1d5df85","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-281ee9148e1bfa1c39a1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v03\n- v0a --refutes--> v06\n- v08 --supports--> v07\n- v02 --refutes--> v05\n- v00 --refutes--> v01\n- v0c --refutes--> v00\n- v0a --refutes--> v09\n- v01 --supports--> v0a\n- v02 --refutes--> v0c\n- v08 --refutes--> v04\n- v0b --refutes--> v02\n- v01 --supports--> v0d\n- v06 --supports--> v08\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"95ac6a49f05761cf903331123cf107b13d617e0fad09647e04a335d582018c24","schema":"livegraph.boolean-propagation.v1","spec_hash":"281ee9148e1bfa1c39a1e96e327e3cebf11060ad476fdf8765d3305e40c39ef9","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-abddc40dc7e354b1e826","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v0d\n- v04 --supports--> v06\n- v06 --refutes--> v0a\n- v07 --supports--> v0c\n- v07 --supports--> v0b\n- v06 --supports--> v08\n- v02 --supports--> v05\n- v08 --supports--> v07\n- v0c --refutes--> v09\n- v09 --supports--> v00\n- v0d --supports--> v02\n- v04 --supports--> v01\n- v02 --refutes--> v04\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"969c75d9114fc33e38444090e258442cee315d91c93c2dd92b6e3e73761e3130","schema":"livegraph.boolean-propagation.v1","spec_hash":"abddc40dc7e354b1e826cac36548fbc5454915169fa204594ef84dc40e1926c6","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-835a155020a6351392cd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v06\n- v02 --refutes--> v09\n- v0b --supports--> v03\n- v01 --refutes--> v0b\n- v04 --refutes--> v08\n- v07 --refutes--> v05\n- v0d --refutes--> v0c\n- v03 --supports--> v0d\n- v02 --refutes--> v07\n- v02 --supports--> v00\n- v05 --refutes--> v04\n- v0a --refutes--> v01\n- v04 --supports--> v0a\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e45f9c85ad385470ad1f3837e7f76d1c342da155ce676a72046aad92f10bd0bc","schema":"livegraph.boolean-propagation.v1","spec_hash":"835a155020a6351392cda56cf71170249470e45aedd1ccc5539d2e49613c5722","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-471884c59e8ac46b3c75","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v05\n- v0b --refutes--> v08\n- v01 --supports--> v02\n- v0c --supports--> v03\n- v02 --supports--> v0c\n- v04 --refutes--> v09\n- v00 --refutes--> v06\n- v03 --refutes--> v07\n- v09 --refutes--> v01\n- v07 --refutes--> v0b\n- v09 --supports--> v0d\n- v0c --refutes--> v0a\n- v00 --supports--> v04\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"74baa85104ccbc9f642a80bcd6f1e36c8f6b1e776b6eb87b3c2d65dfe55be340","schema":"livegraph.boolean-propagation.v1","spec_hash":"471884c59e8ac46b3c75b8de9d35611e60368b3065601e4acd8ad712c41051cc","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-f67c4e4a2bb1bbfb094b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v0b\n- v0a --refutes--> v0c\n- v0d --refutes--> v03\n- v05 --supports--> v01\n- v04 --refutes--> v08\n- v0a --supports--> v00\n- v0b --supports--> v02\n- v03 --supports--> v07\n- v00 --refutes--> v04\n- v03 --refutes--> v0a\n- v0b --supports--> v06\n- v0d --refutes--> v09\n- v02 --refutes--> v0d\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9c7012a0c337098b50be127d1c6d82feb347686c6a9e50b5201858a75452cd37","schema":"livegraph.boolean-propagation.v1","spec_hash":"f67c4e4a2bb1bbfb094bff223a439bf3d8b84f476b2e49f9ea9303f855cf9d82","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-0e49f2538e07c02def40","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v0c\n- v04 --refutes--> v01\n- v03 --supports--> v04\n- v06 --supports--> v0a\n- v0a --supports--> v03\n- v06 --supports--> v07\n- v05 --supports--> v09\n- v06 --refutes--> v08\n- v09 --refutes--> v0d\n- v01 --refutes--> v05\n- v02 --supports--> v0b\n- v09 --supports--> v02\n- v0a --supports--> v00\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"79a699adbfdd160b0a5fe6ecb97495c167fc3fe131c545c6bb43a8ac7e2a3276","schema":"livegraph.boolean-propagation.v1","spec_hash":"0e49f2538e07c02def4023cdb4051da60d01eb0f1828c74249919bd4f3bebaa9","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":9,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-fcfaeb91b7840524a97d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v06\n- v0d --refutes--> v04\n- v09 --supports--> v0a\n- v0c --refutes--> v00\n- v01 --supports--> v07\n- v09 --refutes--> v0b\n- v08 --refutes--> v01\n- v07 --supports--> v0d\n- v04 --refutes--> v02\n- v0c --supports--> v08\n- v02 --refutes--> v03\n- v0a --supports--> v0c\n- v0c --supports--> v05\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"04c535448c184908b724e2a44947e5cb7331716ee33df0e9f21ec26cb562058f","schema":"livegraph.boolean-propagation.v1","spec_hash":"fcfaeb91b7840524a97da9090ae2351e5d115e5d5334f7b098f747e4b84fa66b","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-15e75607a2065e0bc8b7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v07\n- v04 --supports--> v00\n- v0b --refutes--> v0d\n- v03 --refutes--> v05\n- v06 --refutes--> v0a\n- v03 --refutes--> v09\n- v05 --supports--> v04\n- v07 --refutes--> v0c\n- v02 --refutes--> v06\n- v00 --refutes--> v01\n- v06 --supports--> v03\n- v07 --refutes--> v0b\n- v05 --supports--> v08\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"14e89ecc39197462a12c3ebe04f8350e5f19b27c09f13c3b21d4917d7e585e05","schema":"livegraph.boolean-propagation.v1","spec_hash":"15e75607a2065e0bc8b7eb67f440949a3dd03de57b5fb61192585291f42fdfa0","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":9,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-5f578386ec72a9de15c9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v03\n- v01 --refutes--> v05\n- v0d --refutes--> v02\n- v02 --supports--> v0c\n- v09 --refutes--> v07\n- v06 --supports--> v0d\n- v09 --supports--> v01\n- v0a --supports--> v08\n- v0a --supports--> v09\n- v0b --refutes--> v06\n- v04 --supports--> v0b\n- v0c --refutes--> v0a\n- v0c --refutes--> v00\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"67f4178235ffba5d6b7f34bfb39b4514537a75d0c19f756e5e5062e9e31af8f1","schema":"livegraph.boolean-propagation.v1","spec_hash":"5f578386ec72a9de15c9a310d5903627b05b6221d8e1a215db70f7a01edb115a","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-9b0c013f7b14fcb3bcf4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v05\n- v06 --supports--> v01\n- v0b --refutes--> v08\n- v05 --refutes--> v06\n- v04 --refutes--> v07\n- v03 --supports--> v00\n- v01 --supports--> v03\n- v02 --supports--> v0b\n- v00 --refutes--> v0c\n- v0c --supports--> v02\n- v03 --supports--> v09\n- v06 --supports--> v0a\n- v00 --supports--> v0d\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fb5fc2de760c0f6a59dee9e7b078f55a3661b4525062157c955d3523cbd9ae7a","schema":"livegraph.boolean-propagation.v1","spec_hash":"9b0c013f7b14fcb3bcf468304a452bf0c34b29fbbb1068be372e2d2816af909c","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-da37c7e82139c9c9d72f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v03\n- v0a --supports--> v07\n- v0a --refutes--> v0b\n- v00 --supports--> v0d\n- v06 --supports--> v08\n- v04 --supports--> v00\n- v03 --refutes--> v01\n- v00 --supports--> v05\n- v06 --refutes--> v09\n- v02 --refutes--> v0c\n- v03 --supports--> v0a\n- v07 --refutes--> v02\n- v0d --supports--> v06\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c1f328bff5dd7297e8d244d13b8b9490dafeec12f32cc4a7f224b0ac771c7e5f","schema":"livegraph.boolean-propagation.v1","spec_hash":"da37c7e82139c9c9d72f096fcd71f73f1709d1ce9d76aecb246e99bc9c6bbc8b","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-b0b96b8d6a1c36acb40d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v06\n- v0d --supports--> v02\n- v08 --refutes--> v07\n- v0b --supports--> v04\n- v0a --refutes--> v01\n- v04 --supports--> v0c\n- v02 --supports--> v08\n- v08 --refutes--> v00\n- v0c --refutes--> v0d\n- v09 --refutes--> v0a\n- v0a --supports--> v0b\n- v04 --refutes--> v03\n- v05 --refutes--> v09\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"617799d1cb89abca4896e5089bb9b70c5149cd9727941cc19be7e7a09f07fa08","schema":"livegraph.boolean-propagation.v1","spec_hash":"b0b96b8d6a1c36acb40db0db2dacec332d4ae8fcb668a2e3a733f472065d16fd","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-144c7445bdfa6e83725d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v01\n- v08 --refutes--> v07\n- v0d --refutes--> v09\n- v0c --refutes--> v0b\n- v0d --supports--> v05\n- v0b --refutes--> v04\n- v01 --supports--> v00\n- v0a --supports--> v0c\n- v07 --refutes--> v0a\n- v0b --refutes--> v06\n- v0d --supports--> v08\n- v0b --refutes--> v03\n- v00 --supports--> v02\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9156d43fc0630b36105a5557674f385f84a5c37f6109ea6ad850329835d1a465","schema":"livegraph.boolean-propagation.v1","spec_hash":"144c7445bdfa6e83725d3f6ec6afa2046b0509f84e936f1c0736e32279852531","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-cfe8bc65135620799d74","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v0c\n- v06 --supports--> v09\n- v07 --refutes--> v0b\n- v06 --supports--> v07\n- v0a --refutes--> v06\n- v00 --refutes--> v08\n- v01 --supports--> v00\n- v03 --refutes--> v04\n- v0d --refutes--> v0a\n- v05 --supports--> v0d\n- v00 --supports--> v05\n- v03 --refutes--> v02\n- v0b --refutes--> v03\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4edf8fec11acaf69ca4fbd5485cf46a55351c51d34565ca247cbacb8d9e6261f","schema":"livegraph.boolean-propagation.v1","spec_hash":"cfe8bc65135620799d7484c30809b574474e00f96117f3f6cdddb884b453f027","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-848741be5f6e0199a3d2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v00\n- v02 --refutes--> v07\n- v0c --refutes--> v03\n- v03 --refutes--> v04\n- v08 --refutes--> v06\n- v00 --supports--> v0b\n- v02 --supports--> v01\n- v08 --refutes--> v0c\n- v07 --supports--> v05\n- v05 --supports--> v09\n- v0b --supports--> v02\n- v05 --refutes--> v0a\n- v0c --refutes--> v0d\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c939d26adaaded5e2eb28f7fb728d77ccbee201c09926bd0941991a7158fd295","schema":"livegraph.boolean-propagation.v1","spec_hash":"848741be5f6e0199a3d2afc2817c4f32d9162ffe9b53290c012995826d18421f","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-75cdb4cc1fcf6b424ef1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v03\n- v07 --refutes--> v01\n- v05 --supports--> v04\n- v0a --supports--> v05\n- v09 --supports--> v08\n- v02 --refutes--> v0c\n- v09 --refutes--> v0b\n- v0d --supports--> v0a\n- v0c --supports--> v0d\n- v04 --refutes--> v07\n- v09 --refutes--> v00\n- v01 --supports--> v09\n- v0d --supports--> v06\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9cccc6553d3fdd7dad057e6152e3c480838bb12ba0d6ea2ba359467e9c3e373f","schema":"livegraph.boolean-propagation.v1","spec_hash":"75cdb4cc1fcf6b424ef1ace8b58a80cf5b03bc6d29bbf2c9034f1de5af19c22c","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-55b6a2548f42336b6182","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v01\n- v09 --refutes--> v06\n- v07 --refutes--> v08\n- v01 --supports--> v05\n- v08 --refutes--> v0a\n- v06 --refutes--> v07\n- v04 --refutes--> v00\n- v08 --supports--> v02\n- v06 --supports--> v0b\n- v09 --refutes--> v0d\n- v02 --refutes--> v04\n- v05 --refutes--> v0c\n- v05 --refutes--> v03\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"41080e6b36f6afb1d125c351fd5f3bd976b8a9e21337353485f927ebd968a52c","schema":"livegraph.boolean-propagation.v1","spec_hash":"55b6a2548f42336b618224c7b9cc7b2aced3daa7e7dc74b2a045352d99be0c5e","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":9,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-fa68f7681b2f832fc0e4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v0c\n- v0c --supports--> v0a\n- v07 --refutes--> v0d\n- v00 --supports--> v04\n- v08 --refutes--> v0b\n- v0d --supports--> v03\n- v0c --refutes--> v08\n- v09 --supports--> v06\n- v04 --supports--> v05\n- v09 --refutes--> v02\n- v08 --supports--> v07\n- v03 --supports--> v00\n- v07 --refutes--> v01\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c1ffddb255d452f45cb3cf49351c9848fc43e6d9ffaad1b88fc5052e6a21a4ff","schema":"livegraph.boolean-propagation.v1","spec_hash":"fa68f7681b2f832fc0e49eedf592f65985b69b5213e0a0afac797e49262d93b0","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-3242835a3170dca75425","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v05\n- v03 --supports--> v04\n- v00 --supports--> v0a\n- v0a --supports--> v08\n- v05 --refutes--> v02\n- v0c --supports--> v06\n- v03 --supports--> v01\n- v09 --supports--> v07\n- v03 --supports--> v0d\n- v09 --refutes--> v0c\n- v0c --refutes--> v00\n- v08 --refutes--> v03\n- v02 --refutes--> v0b\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0d0d25f43cf26e18816ee9aaa9b77df2f3b9f4ff53c2af05397776247c702312","schema":"livegraph.boolean-propagation.v1","spec_hash":"3242835a3170dca75425ea40f3465ad8cdec2543804d1c712b7dc5798d6a8214","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-374a136b49435a351864","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v06\n- v0a --supports--> v08\n- v02 --refutes--> v01\n- v00 --supports--> v03\n- v08 --supports--> v0d\n- v07 --supports--> v04\n- v07 --refutes--> v0b\n- v0a --supports--> v05\n- v0d --supports--> v07\n- v00 --supports--> v0c\n- v02 --refutes--> v09\n- v03 --refutes--> v02\n- v06 --supports--> v0a\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"505784d8ff5af1532fa91872caff7c09683cef26476124a8c531c24293894186","schema":"livegraph.boolean-propagation.v1","spec_hash":"374a136b49435a3518648a27541c64bbcd05019b1db481a4f1c88889cd64aa4e","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-439a6f5ed50e2c488838","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v0b\n- v0b --supports--> v05\n- v06 --supports--> v03\n- v03 --supports--> v0d\n- v07 --supports--> v09\n- v03 --refutes--> v02\n- v09 --refutes--> v0c\n- v05 --supports--> v01\n- v03 --supports--> v08\n- v05 --refutes--> v07\n- v0c --refutes--> v06\n- v0d --refutes--> v00\n- v04 --supports--> v0a\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3e7d6613ead0ec1e64370b7f919cda3762b7c1c98e6325d26626ecf12efa0f75","schema":"livegraph.boolean-propagation.v1","spec_hash":"439a6f5ed50e2c488838dc1aaa19a4f38b80af3ec26da783c6a7520456afb2c9","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-bcaee51aed8177cd0e5d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v0c\n- v09 --refutes--> v05\n- v08 --supports--> v0b\n- v00 --supports--> v09\n- v0c --refutes--> v07\n- v0d --refutes--> v03\n- v08 --refutes--> v01\n- v08 --supports--> v06\n- v04 --refutes--> v0d\n- v03 --supports--> v00\n- v0d --refutes--> v0a\n- v03 --refutes--> v02\n- v07 --supports--> v04\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c0626e18c9e4e5605f7ddf1ca4d311090b91e5f485523c881c981de2eb911c45","schema":"livegraph.boolean-propagation.v1","spec_hash":"bcaee51aed8177cd0e5df8446c18abf1b68d54eb1c11ef3aea09c91c0840af24","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-f84473007d60b2f99990","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v05\n- v07 --supports--> v02\n- v02 --refutes--> v0d\n- v01 --refutes--> v03\n- v0b --supports--> v09\n- v05 --supports--> v0b\n- v0a --refutes--> v06\n- v04 --supports--> v08\n- v0d --refutes--> v01\n- v04 --refutes--> v0a\n- v02 --supports--> v00\n- v0d --refutes--> v0c\n- v0a --supports--> v07\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4b17275a863719a27c559f850f5fd770fcc1e254bd809c84ca7162b14cccf089","schema":"livegraph.boolean-propagation.v1","spec_hash":"f84473007d60b2f999905d26887e415ddfc08ebf34e82571a641873d0e56ee39","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-c85777da740f309fbb3e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v07\n- v04 --supports--> v0b\n- v08 --supports--> v01\n- v03 --supports--> v0d\n- v08 --supports--> v04\n- v0b --refutes--> v06\n- v04 --supports--> v00\n- v07 --supports--> v05\n- v06 --refutes--> v0c\n- v0c --refutes--> v09\n- v07 --refutes--> v08\n- v04 --refutes--> v0a\n- v02 --refutes--> v03\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f9fb8a8979d23ee8b56aca3be9b4fa1c7fc4c9c465fa871d95fdc90fff7e4f68","schema":"livegraph.boolean-propagation.v1","spec_hash":"c85777da740f309fbb3ec2b877a04e7d0451fea21a9640887b5513e6814f1494","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-d89d0980281375c1507c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v06\n- v04 --refutes--> v05\n- v05 --supports--> v09\n- v0c --refutes--> v03\n- v04 --supports--> v08\n- v03 --supports--> v02\n- v0d --supports--> v00\n- v0a --supports--> v07\n- v03 --supports--> v0a\n- v06 --refutes--> v01\n- v0b --refutes--> v0d\n- v0a --refutes--> v0b\n- v06 --supports--> v04\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"973599cfe5bbddbe964d7e48c07132f793d304f12881543f44dcff82f0f6a915","schema":"livegraph.boolean-propagation.v1","spec_hash":"d89d0980281375c1507c054d460af9cfa8eceb765c76a87fe11605d6ade61bdd","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-f6cc59d12eecf12ff046","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v0a\n- v0b --supports--> v05\n- v06 --refutes--> v07\n- v05 --refutes--> v06\n- v0c --refutes--> v00\n- v0b --refutes--> v0d\n- v08 --refutes--> v03\n- v03 --refutes--> v02\n- v09 --supports--> v01\n- v01 --refutes--> v0c\n- v06 --supports--> v04\n- v00 --refutes--> v0b\n- v03 --supports--> v09\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fd61be1d81026a9b3de61b89a4e5eef66dbe30822436a504bc153d6eabdf3bb1","schema":"livegraph.boolean-propagation.v1","spec_hash":"f6cc59d12eecf12ff0469bbe6aae2882d2b2b12b66da116f27872b25763ae40d","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-72a78345324df32ccb69","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v05\n- v04 --supports--> v00\n- v00 --refutes--> v09\n- v06 --refutes--> v0b\n- v0b --refutes--> v07\n- v00 --refutes--> v0c\n- v04 --supports--> v01\n- v03 --refutes--> v08\n- v0a --supports--> v0d\n- v0a --supports--> v02\n- v08 --supports--> v06\n- v0a --refutes--> v04\n- v05 --refutes--> v03\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8dfb6fe9cd2c2cbfc39e09804fa3bb705d4f9a3c779bea9bd492a780472082e1","schema":"livegraph.boolean-propagation.v1","spec_hash":"72a78345324df32ccb69f7f1669a069a0994db55023dcb13ef96c6b2df29c1ba","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-98e278fa6042259e601c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v0d\n- v02 --supports--> v09\n- v00 --supports--> v06\n- v04 --supports--> v02\n- v0d --supports--> v00\n- v0b --supports--> v04\n- v08 --supports--> v0a\n- v01 --refutes--> v05\n- v01 --supports--> v0b\n- v03 --refutes--> v01\n- v00 --refutes--> v07\n- v08 --refutes--> v0c\n- v0a --supports--> v03\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0496c8154faf7b4c219335ddc371376d1419d72d883f6fc862bf48762ec0e09b","schema":"livegraph.boolean-propagation.v1","spec_hash":"98e278fa6042259e601c6416148b03d6402a7d5447a7f2d5a17e3ab91911fa82","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-9db9e2777d6279afb4b3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v01\n- v0d --refutes--> v00\n- v08 --refutes--> v02\n- v01 --refutes--> v07\n- v0a --supports--> v06\n- v07 --refutes--> v0d\n- v00 --refutes--> v03\n- v0c --supports--> v05\n- v0a --supports--> v0b\n- v0a --supports--> v08\n- v02 --refutes--> v09\n- v03 --supports--> v0a\n- v0c --supports--> v04\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9f983a4902f493154ef4390f8681ce8cfc6443d968462c7ee9a39945db5560a5","schema":"livegraph.boolean-propagation.v1","spec_hash":"9db9e2777d6279afb4b3319b530a1ebfe91c0b4c9e30b51e9c9c02111cbce413","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-e31cd79d650191bf1169","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v0d\n- v07 --refutes--> v00\n- v00 --refutes--> v0a\n- v02 --supports--> v01\n- v0b --refutes--> v09\n- v03 --supports--> v02\n- v08 --supports--> v03\n- v0d --refutes--> v08\n- v02 --refutes--> v05\n- v01 --supports--> v0b\n- v09 --refutes--> v04\n- v00 --refutes--> v06\n- v09 --refutes--> v0c\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f242cb618f815643779558a73ea3dc42c5e739b2388a8f623487da2f95a7bf73","schema":"livegraph.boolean-propagation.v1","spec_hash":"e31cd79d650191bf11691f3453e41e7d65ac212be12c142450cd520e72166ce0","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-d2070c2113cf5206c5fa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v00\n- v07 --supports--> v06\n- v00 --refutes--> v0c\n- v08 --refutes--> v0d\n- v07 --refutes--> v04\n- v08 --refutes--> v05\n- v00 --supports--> v03\n- v02 --supports--> v01\n- v05 --supports--> v0a\n- v04 --supports--> v08\n- v05 --supports--> v02\n- v0b --supports--> v07\n- v09 --refutes--> v0b\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"faaf681ce54e4869f67623ec93faabaae362fc2903dc26eb70289b0245fc7437","schema":"livegraph.boolean-propagation.v1","spec_hash":"d2070c2113cf5206c5fab07f62a175a7d9060787331b3a9fe807e21742343c5b","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":9,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-71edb900d6bc5a905d69","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v05\n- v07 --refutes--> v08\n- v02 --supports--> v0b\n- v07 --supports--> v04\n- v0b --refutes--> v03\n- v0a --refutes--> v09\n- v01 --supports--> v00\n- v07 --refutes--> v02\n- v0d --supports--> v07\n- v05 --refutes--> v0a\n- v09 --supports--> v01\n- v03 --supports--> v06\n- v0b --refutes--> v0c\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0d5b93a8e40dd334b5b6d43ff4b3b26cff0d68a6b74e8e7d46a857e1c74581ab","schema":"livegraph.boolean-propagation.v1","spec_hash":"71edb900d6bc5a905d69758e6b251c4e9df6e70f67dcb772cfb3d1a40c3982af","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-08cf9084f8070ef5a1ff","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v00\n- v0b --refutes--> v0d\n- v07 --refutes--> v06\n- v05 --refutes--> v07\n- v06 --refutes--> v0b\n- v09 --supports--> v0c\n- v01 --supports--> v09\n- v0c --supports--> v05\n- v07 --supports--> v04\n- v0c --refutes--> v03\n- v06 --refutes--> v08\n- v0d --supports--> v0a\n- v0d --supports--> v02\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d24359d3d082476fe88afe817f3c664966f55960333ba54734b7d8755253b86d","schema":"livegraph.boolean-propagation.v1","spec_hash":"08cf9084f8070ef5a1ff8c77c91fc9cd094a7e12125901e37867198549a47ec2","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-ed4a4198c9ef8f6e4ff9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v07\n- v0d --refutes--> v04\n- v06 --refutes--> v09\n- v05 --refutes--> v08\n- v03 --refutes--> v06\n- v00 --refutes--> v0b\n- v07 --refutes--> v02\n- v05 --refutes--> v0a\n- v0b --refutes--> v05\n- v07 --refutes--> v0d\n- v04 --refutes--> v0c\n- v06 --refutes--> v00\n- v01 --supports--> v03\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3066386a218e1013289fd800fabd691d602b2e70833d2f12d435b80e4fe01a89","schema":"livegraph.boolean-propagation.v1","spec_hash":"ed4a4198c9ef8f6e4ff92cbaffbf2fdd268a24af4bc8ef2ef2f6edc0f7b51d2a","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-e655feb1b33be6f9b4d2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v0c\n- v0d --refutes--> v06\n- v02 --refutes--> v0a\n- v07 --refutes--> v03\n- v00 --supports--> v09\n- v06 --supports--> v01\n- v08 --supports--> v00\n- v07 --supports--> v08\n- v0b --supports--> v0d\n- v0a --refutes--> v0b\n- v06 --supports--> v07\n- v05 --refutes--> v02\n- v00 --supports--> v04\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"992c62721161ffd2c37c367833d479999aed7e60eb708d6dca07a6532c88eaf6","schema":"livegraph.boolean-propagation.v1","spec_hash":"e655feb1b33be6f9b4d2e9698d2a1fc46199e26b204b2ec958e0fbce121a9895","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-4627088c91c46f80a717","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v07\n- v0b --refutes--> v02\n- v0a --supports--> v03\n- v07 --supports--> v0a\n- v02 --supports--> v04\n- v00 --supports--> v0b\n- v01 --supports--> v0c\n- v03 --supports--> v01\n- v02 --refutes--> v05\n- v0a --refutes--> v09\n- v00 --supports--> v0d\n- v0c --supports--> v06\n- v02 --refutes--> v08\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cd0b4d32c288c8027d62b6b76a17a5efe157c71f7a46821fb20232dcb66afdaa","schema":"livegraph.boolean-propagation.v1","spec_hash":"4627088c91c46f80a717148c91473d0d41bdc7765767cbddd371b16d55ce7461","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-fd4656d23928523ac184","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v0b\n- v0d --refutes--> v09\n- v0a --supports--> v00\n- v0c --refutes--> v07\n- v0b --refutes--> v01\n- v0d --refutes--> v0a\n- v0b --supports--> v05\n- v01 --supports--> v0c\n- v0c --refutes--> v06\n- v0c --supports--> v02\n- v06 --supports--> v03\n- v03 --refutes--> v08\n- v04 --refutes--> v0d\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e1bc786fa820faeae65101a3c7d345f6e933cc2473e78987be9eea0dc99056f2","schema":"livegraph.boolean-propagation.v1","spec_hash":"fd4656d23928523ac1844bdac10aa1913f5611081f956683914ac4af40954a4c","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-e8713cf43e1034fa7187","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v0a\n- v00 --refutes--> v06\n- v0b --supports--> v09\n- v03 --refutes--> v01\n- v0d --supports--> v07\n- v08 --supports--> v0b\n- v0d --refutes--> v02\n- v01 --refutes--> v0d\n- v0b --supports--> v0c\n- v0a --supports--> v08\n- v0b --supports--> v05\n- v09 --refutes--> v03\n- v03 --refutes--> v04\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cfefeba425ab7b1bcb1a265c5dbaf6c861c01006f778a6217e10468aef6f5d66","schema":"livegraph.boolean-propagation.v1","spec_hash":"e8713cf43e1034fa7187c9faccae15e4cbc8f7830ceb17fb96e5f5dfc085593b","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-ad938cbd9a83c0b61f6d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v05\n- v05 --supports--> v00\n- v02 --supports--> v06\n- v05 --refutes--> v03\n- v0d --supports--> v07\n- v04 --supports--> v0a\n- v0d --supports--> v04\n- v0a --supports--> v0b\n- v06 --refutes--> v08\n- v00 --refutes--> v09\n- v08 --refutes--> v01\n- v05 --refutes--> v0c\n- v0a --refutes--> v02\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"11c61043891efea3efe9cec61537aa5772ef7ef89162929121644c5dcd118043","schema":"livegraph.boolean-propagation.v1","spec_hash":"ad938cbd9a83c0b61f6d1aa47569e4ae58a9a4deed61e4eef69e785cb427e302","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-3b3a14a9335600c9a8b4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v07\n- v01 --supports--> v05\n- v0d --refutes--> v08\n- v00 --refutes--> v02\n- v0b --supports--> v01\n- v0a --supports--> v06\n- v00 --refutes--> v0d\n- v05 --refutes--> v03\n- v05 --supports--> v0c\n- v08 --supports--> v04\n- v04 --supports--> v0a\n- v05 --supports--> v09\n- v03 --refutes--> v00\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4d6f0f5ce66a38cc8d25d99e10c42ca241fee63acd34a42c815262beef6f443d","schema":"livegraph.boolean-propagation.v1","spec_hash":"3b3a14a9335600c9a8b45a297583348fdcebed528251aa2e5a81e3f908777900","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-53c1525b0ebb8306b172","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v0c\n- v01 --refutes--> v05\n- v09 --supports--> v07\n- v08 --refutes--> v01\n- v03 --refutes--> v02\n- v02 --supports--> v06\n- v0c --refutes--> v0d\n- v00 --supports--> v08\n- v03 --supports--> v0b\n- v01 --supports--> v04\n- v09 --refutes--> v0a\n- v06 --supports--> v00\n- v0c --supports--> v03\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0c85ce5172aba78ec4b2b497fd2ffb8c35bb95cc890163322528d06eb655ecfa","schema":"livegraph.boolean-propagation.v1","spec_hash":"53c1525b0ebb8306b17249ebd2f43c92e6ce627cbf77da63fea96b5426bdbd7c","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-d8a9f9c34dae32843dcd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v05\n- v0b --refutes--> v02\n- v02 --refutes--> v0d\n- v05 --supports--> v00\n- v01 --refutes--> v09\n- v0b --refutes--> v04\n- v01 --supports--> v03\n- v05 --supports--> v06\n- v08 --supports--> v0c\n- v07 --supports--> v0b\n- v08 --supports--> v01\n- v00 --refutes--> v08\n- v03 --supports--> v07\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e04bacf40755c69b50366d713514dbe9879bf287e72f309ab1009409b91bf0d6","schema":"livegraph.boolean-propagation.v1","spec_hash":"d8a9f9c34dae32843dcdde2353dd428e5ab3d1bf17ce29d50599fc8134ebf1fe","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":9,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-562976ec4bfdaf6ca310","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v09\n- v08 --refutes--> v0b\n- v0c --refutes--> v0a\n- v06 --refutes--> v00\n- v03 --refutes--> v07\n- v0b --supports--> v03\n- v03 --supports--> v02\n- v0c --supports--> v08\n- v05 --refutes--> v0c\n- v07 --refutes--> v0d\n- v04 --refutes--> v05\n- v0d --supports--> v06\n- v08 --refutes--> v01\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0391687cc1f249390ad5fce5b3d4f7d54b521c338d17f08f8eb2d85f4c3d9dd1","schema":"livegraph.boolean-propagation.v1","spec_hash":"562976ec4bfdaf6ca310486ed67d9b4215f517729b52073c8e4cdd33a18fae99","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-7326aa319d38aa2da093","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v0a\n- v01 --refutes--> v0d\n- v01 --refutes--> v00\n- v07 --refutes--> v01\n- v0c --refutes--> v08\n- v02 --refutes--> v09\n- v00 --refutes--> v05\n- v09 --supports--> v06\n- v09 --refutes--> v04\n- v0a --supports--> v0c\n- v08 --refutes--> v02\n- v08 --refutes--> v03\n- v09 --supports--> v07\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a46000a9bf4099495a3ee929c5df98e261dfacea9d46e08cb0b496ac61b01428","schema":"livegraph.boolean-propagation.v1","spec_hash":"7326aa319d38aa2da093fa54f8e940224832b9795264b10bcf2ba245ee6f1324","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-9abc1e0163493afa724f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v07\n- v0c --refutes--> v03\n- v0c --refutes--> v02\n- v08 --supports--> v0d\n- v0a --refutes--> v01\n- v08 --supports--> v00\n- v0c --refutes--> v04\n- v01 --supports--> v0c\n- v07 --supports--> v0b\n- v09 --supports--> v06\n- v05 --refutes--> v0a\n- v09 --refutes--> v08\n- v00 --refutes--> v05\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8e3fb389898fc54e372af27dedb47fe63c721b49d3dea94ffb627fcdfa273e80","schema":"livegraph.boolean-propagation.v1","spec_hash":"9abc1e0163493afa724f6fd467f319a2ebf0df84ddb20385062f96816e11dceb","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-771e2fb863228154c7f9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v05\n- v0d --supports--> v02\n- v05 --refutes--> v09\n- v08 --supports--> v06\n- v03 --refutes--> v0b\n- v0d --refutes--> v0a\n- v02 --refutes--> v01\n- v07 --refutes--> v03\n- v01 --refutes--> v00\n- v00 --refutes--> v08\n- v03 --refutes--> v0c\n- v05 --supports--> v0d\n- v08 --refutes--> v04\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"34d92974b87992a7e52a2574611d30898d27fef30f5041c40e3ece1d895c151a","schema":"livegraph.boolean-propagation.v1","spec_hash":"771e2fb863228154c7f9453c03217b3d47ecdcece931aae8e05d619504646a04","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-243145dc5e07c351514e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v07\n- v03 --refutes--> v06\n- v09 --refutes--> v01\n- v0d --supports--> v02\n- v01 --supports--> v03\n- v0a --refutes--> v05\n- v05 --supports--> v08\n- v06 --refutes--> v0c\n- v01 --supports--> v04\n- v0b --supports--> v0a\n- v0b --supports--> v00\n- v07 --refutes--> v09\n- v06 --refutes--> v0b\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ec746f97cf4e8739ab512da706e88d1fdfa52600de50230118edb7c6a973358c","schema":"livegraph.boolean-propagation.v1","spec_hash":"243145dc5e07c351514ef48dfe88e66e17d94733d57e293197f2b10beaab59f0","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-b6a28f6e40b4a29efc31","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v00\n- v00 --supports--> v06\n- v08 --refutes--> v02\n- v04 --refutes--> v08\n- v05 --refutes--> v0c\n- v03 --supports--> v09\n- v00 --supports--> v0b\n- v09 --refutes--> v04\n- v00 --supports--> v0a\n- v02 --supports--> v01\n- v0b --refutes--> v03\n- v08 --refutes--> v0d\n- v05 --supports--> v07\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"56454ae76e42adacae0d4ed9d6a30a485405c5e011dc8cf0c3eaa11dedf218f3","schema":"livegraph.boolean-propagation.v1","spec_hash":"b6a28f6e40b4a29efc31b7982b976c1ee951c60922f10be8e9acb7be9bea527f","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-51e67fb64c6357de0435","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v0b\n- v0a --supports--> v00\n- v08 --refutes--> v02\n- v06 --refutes--> v0d\n- v06 --refutes--> v0a\n- v02 --refutes--> v0c\n- v09 --refutes--> v01\n- v07 --supports--> v06\n- v00 --refutes--> v08\n- v0a --supports--> v05\n- v0c --supports--> v09\n- v0c --supports--> v03\n- v09 --refutes--> v04\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ab9de466ac5cccb4ce5ff51deb651e43f601f87ff4b21d605c85f8711a5e36ad","schema":"livegraph.boolean-propagation.v1","spec_hash":"51e67fb64c6357de043585d5e4d2f9268b06fb198a71b4b9c81a443226bc4b55","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-708a16436e63a2b00dd6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v06\n- v09 --supports--> v0a\n- v05 --refutes--> v08\n- v0d --refutes--> v07\n- v02 --refutes--> v0b\n- v08 --refutes--> v03\n- v0c --supports--> v04\n- v09 --refutes--> v02\n- v08 --refutes--> v00\n- v01 --refutes--> v0d\n- v03 --refutes--> v01\n- v0c --refutes--> v09\n- v02 --refutes--> v05\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b7531ff1fc414038ade62dbd2809fe07f73e562c47efb56bd6182f3bc7c78439","schema":"livegraph.boolean-propagation.v1","spec_hash":"708a16436e63a2b00dd65b76cd85d98ef1d6e6638c1efca389bcb723426c5967","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-7292e68c2481d3494dec","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v04\n- v00 --refutes--> v03\n- v07 --supports--> v05\n- v00 --supports--> v08\n- v05 --supports--> v0b\n- v03 --refutes--> v06\n- v0d --supports--> v00\n- v0b --refutes--> v01\n- v0a --refutes--> v09\n- v0b --supports--> v0c\n- v0b --supports--> v02\n- v02 --supports--> v0a\n- v04 --refutes--> v0d\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"13021ceb37a5b13da618bce900718c36a0e3f44504f94b11eb2a9670099e78fd","schema":"livegraph.boolean-propagation.v1","spec_hash":"7292e68c2481d3494dec839284f9ad7945b2e6621b7ff32643947419041eb3c2","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-8eb7c35818b3ef714ac6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v05\n- v08 --supports--> v07\n- v08 --supports--> v0d\n- v02 --supports--> v06\n- v05 --refutes--> v0c\n- v03 --supports--> v04\n- v0d --supports--> v09\n- v01 --refutes--> v03\n- v0a --supports--> v00\n- v03 --supports--> v0a\n- v05 --supports--> v08\n- v00 --refutes--> v02\n- v0d --supports--> v0b\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a070cef7e4853bb8ed655a669ac2ac5f15b059a8fa9ee43ea5c88ffdb4b0762f","schema":"livegraph.boolean-propagation.v1","spec_hash":"8eb7c35818b3ef714ac60438fa9bcb20ad98f764ce54488a43de01d8f144f7e6","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-9328258d10dab26863b8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v0b\n- v0d --supports--> v05\n- v01 --supports--> v00\n- v05 --refutes--> v01\n- v04 --refutes--> v06\n- v02 --refutes--> v03\n- v0a --supports--> v0d\n- v0c --refutes--> v02\n- v04 --refutes--> v08\n- v02 --refutes--> v09\n- v02 --refutes--> v04\n- v06 --refutes--> v0a\n- v07 --supports--> v0c\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b04ccac51b2579358f0aa9fcd8bd35a11060209300d78b0d4e6aef96d7d2deda","schema":"livegraph.boolean-propagation.v1","spec_hash":"9328258d10dab26863b8af2139cfbcaff1be8e313948b594cd679725c9fe27f7","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-6d9c03cb2fe31700e682","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v05\n- v00 --refutes--> v07\n- v01 --refutes--> v00\n- v04 --supports--> v0b\n- v06 --refutes--> v0d\n- v0d --refutes--> v03\n- v00 --refutes--> v02\n- v0c --refutes--> v01\n- v02 --supports--> v0a\n- v03 --refutes--> v04\n- v0a --supports--> v06\n- v00 --supports--> v08\n- v03 --refutes--> v09\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b0ca6eaec6e7fbe9621c86b18fea35d8457efc6eca837508b27cb3c2e2c44c72","schema":"livegraph.boolean-propagation.v1","spec_hash":"6d9c03cb2fe31700e682be67a6ce0ae046536c7d66f2f35100316c4f5661edb7","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-77ce6b33e0ac16642b5a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v03\n- v00 --supports--> v0b\n- v01 --refutes--> v07\n- v01 --supports--> v08\n- v0d --refutes--> v0c\n- v0b --refutes--> v09\n- v0d --supports--> v0a\n- v05 --supports--> v04\n- v05 --refutes--> v06\n- v05 --refutes--> v0d\n- v0a --refutes--> v00\n- v08 --refutes--> v05\n- v02 --supports--> v01\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"11d0880d6dd8224ca31fd5d363209a584e0d94d318dc5411da5af4743a6f3682","schema":"livegraph.boolean-propagation.v1","spec_hash":"77ce6b33e0ac16642b5a9cbfa28347bdb2ffde4491dd60497b2374f5807d43ee","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-7dae6942b7739db50a67","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v06\n- v0d --refutes--> v04\n- v04 --refutes--> v05\n- v02 --supports--> v0a\n- v09 --supports--> v01\n- v06 --refutes--> v03\n- v09 --refutes--> v0d\n- v09 --supports--> v07\n- v0c --supports--> v02\n- v00 --refutes--> v08\n- v0a --refutes--> v0b\n- v03 --refutes--> v00\n- v0a --supports--> v09\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"51260d7b25c2ec01f54f4e00b97fd870c63e7f7a8e44798dbdf70c5856ba4368","schema":"livegraph.boolean-propagation.v1","spec_hash":"7dae6942b7739db50a679a3b2d7eb8e4e9adc494dbcda97634151417220dbb40","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-4e1594264f3bedff5216","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v0d\n- v02 --refutes--> v0a\n- v02 --supports--> v0b\n- v07 --refutes--> v00\n- v04 --supports--> v01\n- v05 --refutes--> v09\n- v04 --refutes--> v06\n- v03 --refutes--> v05\n- v07 --supports--> v0c\n- v0b --refutes--> v04\n- v0d --refutes--> v02\n- v01 --supports--> v03\n- v0d --refutes--> v08\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"eecbba38bda59c26692f9e58ff47f831d833dfc20e69cdd73c4db25f213ab1e6","schema":"livegraph.boolean-propagation.v1","spec_hash":"4e1594264f3bedff52169043a17f9f8159f55503282c92fbe1e72aeaa733dc7b","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-c18c78e7d8b9596ec66e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v06\n- v0d --refutes--> v09\n- v03 --refutes--> v02\n- v00 --refutes--> v08\n- v0b --refutes--> v0d\n- v07 --supports--> v0a\n- v00 --refutes--> v05\n- v09 --refutes--> v04\n- v06 --refutes--> v0c\n- v07 --supports--> v03\n- v09 --refutes--> v01\n- v0c --refutes--> v07\n- v02 --refutes--> v0b\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ec0d9f6cda6135c577246a9d9c5e9ad8ba4ba52c10904b5b078e092c2cf56f82","schema":"livegraph.boolean-propagation.v1","spec_hash":"c18c78e7d8b9596ec66e0e3528877d8f74c1a128a1b3a02391272ec9ef79419c","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-92f7cd08c34308d2e854","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v04\n- v05 --refutes--> v08\n- v0a --refutes--> v07\n- v02 --supports--> v00\n- v04 --refutes--> v05\n- v01 --supports--> v03\n- v06 --refutes--> v0a\n- v02 --refutes--> v0b\n- v04 --refutes--> v09\n- v08 --supports--> v06\n- v0a --supports--> v02\n- v04 --refutes--> v0c\n- v0d --supports--> v01\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7f104b52ff53e8859302aa4f77b73dcbfe5a74e54a24f361aae6e177d2667094","schema":"livegraph.boolean-propagation.v1","spec_hash":"92f7cd08c34308d2e8543145e0d9d48812f9561fd5e1e093208214b67c0a81a7","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-79a3b3df4225781fb0d0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v0c\n- v0c --supports--> v0a\n- v01 --supports--> v0d\n- v05 --supports--> v08\n- v0a --refutes--> v06\n- v05 --supports--> v0b\n- v09 --refutes--> v00\n- v06 --refutes--> v02\n- v0a --supports--> v07\n- v02 --refutes--> v05\n- v00 --refutes--> v01\n- v00 --supports--> v03\n- v01 --supports--> v04\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7aa7edc560fbbe8ea3e36fd28a850675ae87350e2f665f884f4a7e408e4330eb","schema":"livegraph.boolean-propagation.v1","spec_hash":"79a3b3df4225781fb0d091e23ee631ed5f5c9289692ee1fa7831971efa420ae7","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-dfd5fac652cfbf1f89bc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v0d\n- v08 --refutes--> v05\n- v0d --supports--> v08\n- v0a --refutes--> v00\n- v0c --supports--> v04\n- v02 --supports--> v07\n- v0c --supports--> v03\n- v05 --refutes--> v0a\n- v02 --supports--> v0c\n- v00 --supports--> v01\n- v0c --refutes--> v06\n- v0b --supports--> v09\n- v04 --refutes--> v0b\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a2617c9c4fa63b0f6f2d4d4f2adfc865da460fb6bb79162c0a1747b01399b678","schema":"livegraph.boolean-propagation.v1","spec_hash":"dfd5fac652cfbf1f89bc353f94fdd10bcdf4fdfcd9db684742447de12c07cd0c","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-2b172a4ef5b85d9fbaa0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v0d\n- v00 --refutes--> v0b\n- v0c --supports--> v06\n- v07 --supports--> v0a\n- v09 --supports--> v07\n- v01 --refutes--> v02\n- v05 --refutes--> v01\n- v09 --supports--> v04\n- v08 --refutes--> v03\n- v08 --refutes--> v00\n- v0c --refutes--> v08\n- v0d --refutes--> v0c\n- v0b --supports--> v09\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"25a74ffb38abf0c1ae941031a56a110322becce328a6fd4e418e783a15d0fb21","schema":"livegraph.boolean-propagation.v1","spec_hash":"2b172a4ef5b85d9fbaa08d58756e9e512b74349c8b4b1b16ebfdfa33aa0cd7ed","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-951a3b47749a583fc7e4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v05\n- v02 --supports--> v0a\n- v06 --supports--> v0c\n- v01 --supports--> v0d\n- v05 --refutes--> v03\n- v03 --supports--> v02\n- v09 --refutes--> v06\n- v04 --supports--> v07\n- v01 --supports--> v09\n- v0a --supports--> v0b\n- v00 --supports--> v04\n- v0a --supports--> v08\n- v0a --supports--> v00\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ccce338489a5edf06d478abd2983a0f534141ed4c561fa54db00a0dfa65d65b7","schema":"livegraph.boolean-propagation.v1","spec_hash":"951a3b47749a583fc7e4c335463bfd85385df445506a5926cac00f99aa8a295d","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":9,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-4ad538a376206362eb1f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v0c\n- v07 --supports--> v08\n- v06 --refutes--> v00\n- v09 --supports--> v0a\n- v0b --refutes--> v09\n- v09 --refutes--> v01\n- v0d --supports--> v07\n- v06 --supports--> v04\n- v08 --supports--> v02\n- v03 --supports--> v0d\n- v06 --supports--> v0b\n- v0a --supports--> v05\n- v02 --supports--> v06\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"339115ac93d3710b4eb797c7e3dfc92f3c5c8bc1919c6676f42a610e1f1dae1a","schema":"livegraph.boolean-propagation.v1","spec_hash":"4ad538a376206362eb1fc182a31a244d8285193e1bf9179287fc3746d79592c8","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-87db879af9cf46531ed0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v0c\n- v07 --refutes--> v09\n- v06 --supports--> v00\n- v04 --refutes--> v02\n- v03 --refutes--> v0a\n- v0a --supports--> v07\n- v0d --refutes--> v01\n- v0b --supports--> v03\n- v03 --refutes--> v05\n- v00 --refutes--> v0b\n- v07 --supports--> v04\n- v01 --supports--> v06\n- v04 --refutes--> v08\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a074be6acbe5b78d7f6bc9288613b02c590a5e2aa950ddf9d5f8cf0c2ca852a1","schema":"livegraph.boolean-propagation.v1","spec_hash":"87db879af9cf46531ed07a4652d24f337fe4a39ba683483ca05c0a5ee0b052c4","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-d4d0010bed938e362f0d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v03\n- v06 --supports--> v08\n- v0c --refutes--> v0a\n- v08 --supports--> v00\n- v01 --supports--> v06\n- v0b --supports--> v05\n- v0d --refutes--> v04\n- v07 --supports--> v0d\n- v08 --refutes--> v07\n- v04 --refutes--> v09\n- v05 --supports--> v01\n- v04 --supports--> v0c\n- v0d --refutes--> v02\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4eeb1383b1821d57a2e1e8030d01eaeb681be7a05d3bc91bc9dd645994f31b04","schema":"livegraph.boolean-propagation.v1","spec_hash":"d4d0010bed938e362f0dadb1af09c0852f7ae64f7dc9c9b5523a33c4a89fe384","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-a033cdb79ee035396e8e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v00\n- v06 --refutes--> v09\n- v00 --supports--> v04\n- v03 --supports--> v07\n- v09 --refutes--> v05\n- v06 --refutes--> v0c\n- v04 --supports--> v01\n- v0a --refutes--> v06\n- v02 --refutes--> v0d\n- v04 --refutes--> v0b\n- v0b --supports--> v08\n- v02 --refutes--> v03\n- v08 --supports--> v02\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e612174309038d8f9f97e468b4bfaa706e796276cce6e3981d1abed1f9f3fdc4","schema":"livegraph.boolean-propagation.v1","spec_hash":"a033cdb79ee035396e8e1eb0adf3c81c6871646d7d352b214905aa486185ce6b","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":9,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-7858bef5c2f102b610dd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v00\n- v03 --refutes--> v02\n- v08 --refutes--> v03\n- v0a --refutes--> v08\n- v0b --supports--> v0d\n- v08 --supports--> v04\n- v03 --refutes--> v06\n- v03 --refutes--> v07\n- v09 --supports--> v0b\n- v01 --supports--> v0a\n- v05 --supports--> v09\n- v0a --refutes--> v0c\n- v02 --supports--> v05\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0503a39e73915197141b75bd9fde1317745147d3527bfca8496a7c4fa63c819f","schema":"livegraph.boolean-propagation.v1","spec_hash":"7858bef5c2f102b610dd90031ff54df10950be54bbea8c0a082a2b204a0f1654","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-3008f2a2b97a572f5e01","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v04\n- v03 --supports--> v06\n- v08 --refutes--> v0d\n- v07 --refutes--> v05\n- v0d --refutes--> v07\n- v02 --refutes--> v0c\n- v03 --supports--> v0a\n- v00 --refutes--> v02\n- v09 --refutes--> v01\n- v02 --supports--> v0b\n- v0a --supports--> v00\n- v09 --refutes--> v08\n- v04 --refutes--> v03\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7f75e6c6075606d7743b3cb0cd209e3643d1029488189133a7842effb6ebcf06","schema":"livegraph.boolean-propagation.v1","spec_hash":"3008f2a2b97a572f5e01824f6ff8623c04e0f3a6be85438d173cbd5808506427","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-f19442b6cd431ec7acb5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v04\n- v0c --supports--> v09\n- v02 --supports--> v01\n- v0d --supports--> v0c\n- v09 --refutes--> v0a\n- v0c --supports--> v00\n- v03 --supports--> v0d\n- v0a --refutes--> v0b\n- v0b --supports--> v05\n- v05 --supports--> v06\n- v03 --supports--> v07\n- v01 --supports--> v03\n- v02 --supports--> v08\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4a51bde93202c55980c1d19c2f3aa4d9132af509467abd498c7be0b12ea39903","schema":"livegraph.boolean-propagation.v1","spec_hash":"f19442b6cd431ec7acb5520027278362750b7edea72a4d6fa72b6fd68ee8a14c","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-34e1fe10643c612729df","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v0d\n- v07 --supports--> v00\n- v05 --refutes--> v0c\n- v00 --refutes--> v09\n- v09 --refutes--> v0b\n- v03 --supports--> v08\n- v0a --refutes--> v03\n- v05 --refutes--> v04\n- v05 --refutes--> v02\n- v01 --refutes--> v05\n- v04 --supports--> v07\n- v07 --refutes--> v06\n- v0b --refutes--> v0a\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fe3bdcd9416bee01d939ba9468fb9677bf6dc5ccede8d468d6040d7ac37adb52","schema":"livegraph.boolean-propagation.v1","spec_hash":"34e1fe10643c612729dfe3e5afd13c518e1d8cee6b233de317723a27c5bdbfa4","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":9,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-1413074e3ba3a75f18b3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v06\n- v04 --supports--> v0a\n- v07 --refutes--> v0d\n- v0c --supports--> v00\n- v08 --refutes--> v0b\n- v0c --supports--> v02\n- v03 --refutes--> v04\n- v01 --refutes--> v03\n- v06 --refutes--> v09\n- v06 --supports--> v08\n- v04 --supports--> v0c\n- v07 --supports--> v05\n- v00 --supports--> v07\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2c04e8198aae56e7216402beac72027a7950681421e6e6edada81b01056fefb0","schema":"livegraph.boolean-propagation.v1","spec_hash":"1413074e3ba3a75f18b3596fbd8af781df6b12ad31e6e220a326292278ea1d3f","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-d8ddc325ccf875d713db","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v0d\n- v08 --refutes--> v0c\n- v00 --supports--> v06\n- v00 --supports--> v03\n- v07 --supports--> v04\n- v0d --supports--> v02\n- v0d --supports--> v01\n- v01 --refutes--> v0a\n- v07 --supports--> v0b\n- v06 --supports--> v09\n- v0c --refutes--> v07\n- v00 --supports--> v05\n- v09 --supports--> v08\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"315e2990121d4f770c5a83fa09bffb12bc84152bf84c6d21d588334ab048181e","schema":"livegraph.boolean-propagation.v1","spec_hash":"d8ddc325ccf875d713db36db4fe9575ce9c3614341d5af400f98e94328a3d4b3","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-bb6ccb266af1d4d56091","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v0c\n- v0c --refutes--> v09\n- v00 --refutes--> v08\n- v04 --refutes--> v05\n- v0b --refutes--> v01\n- v0a --supports--> v0d\n- v03 --supports--> v0b\n- v0d --refutes--> v07\n- v08 --refutes--> v03\n- v00 --refutes--> v02\n- v0c --refutes--> v06\n- v05 --supports--> v0a\n- v0b --refutes--> v04\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cd9aeb3385157fc48d6f887c6cfcb213996f8c11cd1f0217b67e9a4b47fdb0b9","schema":"livegraph.boolean-propagation.v1","spec_hash":"bb6ccb266af1d4d560911fe6f474df7e8add83ff27bdc0d95719c3483b0dc952","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-c7efeaf4cdfcda7ef97f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v01\n- v0d --refutes--> v0a\n- v02 --supports--> v09\n- v01 --supports--> v05\n- v00 --refutes--> v08\n- v0c --supports--> v03\n- v04 --refutes--> v0c\n- v0b --supports--> v07\n- v01 --refutes--> v02\n- v08 --supports--> v04\n- v09 --supports--> v0b\n- v02 --supports--> v06\n- v0d --refutes--> v00\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3cf6c34b7bb4d5297b8a9be4680765061f12ade9554571d02bed593549c58869","schema":"livegraph.boolean-propagation.v1","spec_hash":"c7efeaf4cdfcda7ef97fbf2ae4eacca07c671b8c6b1622323330d8e5f2e81b08","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-8511085f81543a9b291b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v07\n- v03 --supports--> v0b\n- v09 --supports--> v05\n- v03 --supports--> v00\n- v07 --refutes--> v09\n- v04 --refutes--> v01\n- v03 --refutes--> v0c\n- v05 --refutes--> v02\n- v0c --refutes--> v0d\n- v08 --refutes--> v04\n- v0a --supports--> v03\n- v05 --refutes--> v08\n- v07 --supports--> v06\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bd4db0acfe65c336a103e46f0c997be94248e58320fc18bb04724ccf228357dc","schema":"livegraph.boolean-propagation.v1","spec_hash":"8511085f81543a9b291b94a24a01dce2ffd18b229579b1bcbcb5e0057b9632cc","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-cdd89f610417189e72a5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v03\n- v00 --supports--> v02\n- v0c --refutes--> v0d\n- v0a --refutes--> v00\n- v05 --supports--> v06\n- v08 --supports--> v0b\n- v01 --supports--> v05\n- v06 --supports--> v07\n- v02 --refutes--> v08\n- v0c --supports--> v01\n- v09 --refutes--> v0a\n- v08 --refutes--> v04\n- v08 --supports--> v0c\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"77fe8652554ebc4ff55e8581bf4101c3cb700f994758010a6e68072112f9ecab","schema":"livegraph.boolean-propagation.v1","spec_hash":"cdd89f610417189e72a53b72f775bc262bd89e3139abcb8d5f46960cd759c9b8","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-04dfa84f272902e6d070","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v04\n- v03 --refutes--> v08\n- v0d --refutes--> v02\n- v09 --refutes--> v07\n- v05 --refutes--> v03\n- v06 --supports--> v0b\n- v00 --refutes--> v05\n- v02 --supports--> v0c\n- v08 --refutes--> v0d\n- v02 --refutes--> v09\n- v08 --refutes--> v0a\n- v04 --supports--> v06\n- v06 --refutes--> v01\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ae72a23cb008edc96158d430d171913acf1b6cc98304d1dd34e0afb729653164","schema":"livegraph.boolean-propagation.v1","spec_hash":"04dfa84f272902e6d070c493f85f74a7e3f9b72d069753b18055ee4da8923449","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-c6b8a86a4d8e51c58ea3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v07\n- v07 --refutes--> v0a\n- v05 --supports--> v04\n- v01 --supports--> v03\n- v07 --refutes--> v06\n- v05 --supports--> v02\n- v01 --refutes--> v0d\n- v00 --refutes--> v05\n- v09 --supports--> v00\n- v04 --refutes--> v01\n- v03 --refutes--> v0c\n- v03 --refutes--> v08\n- v0c --supports--> v0b\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"226bd21cbb74c05af6010d68d43170e1b96cdb76d665d0dd8031c7bff289f0e5","schema":"livegraph.boolean-propagation.v1","spec_hash":"c6b8a86a4d8e51c58ea3240ea71ccd61d45fcb676d21857b7228ec7d92329bc9","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-afb7a584cf228e120220","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v02\n- v04 --refutes--> v08\n- v07 --refutes--> v05\n- v00 --supports--> v04\n- v02 --supports--> v07\n- v02 --refutes--> v01\n- v0a --refutes--> v00\n- v05 --refutes--> v0c\n- v05 --refutes--> v09\n- v0b --refutes--> v0a\n- v08 --supports--> v0d\n- v03 --refutes--> v06\n- v05 --supports--> v03\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fb4ae371f7f339edc7eb85c499a36c9df86f6126f4272c283933e8c33ac05a8a","schema":"livegraph.boolean-propagation.v1","spec_hash":"afb7a584cf228e120220deeb8f91cd7bc635b0574ed7f495d09a97532b62b3a8","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":9,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-7a3948228babb9779acb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v06\n- v0a --supports--> v0d\n- v05 --refutes--> v01\n- v09 --supports--> v05\n- v0d --refutes--> v09\n- v04 --refutes--> v00\n- v03 --supports--> v07\n- v0d --refutes--> v08\n- v0b --refutes--> v03\n- v04 --supports--> v02\n- v00 --supports--> v0a\n- v07 --supports--> v04\n- v03 --refutes--> v0c\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c5dd44de8b4b7e334f247d7e9c4d4d41ec9057e717e892862d4c8f992a57a9ee","schema":"livegraph.boolean-propagation.v1","spec_hash":"7a3948228babb9779acb5c10da94cc8fec07c5c23cf9f53fc8650fbdf12cdc98","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-742d587e16ee68a2ec38","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v07\n- v03 --supports--> v05\n- v08 --refutes--> v01\n- v03 --supports--> v0c\n- v00 --refutes--> v08\n- v03 --refutes--> v0d\n- v04 --refutes--> v03\n- v0c --refutes--> v0b\n- v07 --refutes--> v04\n- v0b --supports--> v0a\n- v02 --refutes--> v00\n- v02 --supports--> v06\n- v03 --supports--> v09\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"381ebfefef9868d8dc72547aa92d430a47bd98ecfcd7bf5f5db7a05b8ec38ff0","schema":"livegraph.boolean-propagation.v1","spec_hash":"742d587e16ee68a2ec38fdba51123b597d90f7cb9452e0c9046086e19b92417c","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-7b0cd0b4e9196dce0bbb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v04\n- v06 --refutes--> v00\n- v03 --supports--> v0b\n- v08 --supports--> v0a\n- v02 --supports--> v07\n- v03 --refutes--> v0c\n- v07 --refutes--> v08\n- v00 --supports--> v02\n- v01 --refutes--> v06\n- v0a --supports--> v03\n- v09 --supports--> v01\n- v07 --supports--> v05\n- v06 --refutes--> v0d\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4624825b807392d54b6f6f23b577f4350a4f8c863737db99896cc9a7b2a2685f","schema":"livegraph.boolean-propagation.v1","spec_hash":"7b0cd0b4e9196dce0bbbe4826e3db749350c6d080f6b435c4e4bc10d63d1e0da","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":9,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-10b61c848bcd40607cd8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v03\n- v0c --refutes--> v08\n- v0b --supports--> v05\n- v06 --refutes--> v0b\n- v03 --supports--> v04\n- v0c --supports--> v01\n- v05 --refutes--> v0d\n- v02 --refutes--> v00\n- v0d --refutes--> v0c\n- v07 --refutes--> v09\n- v03 --supports--> v02\n- v09 --supports--> v06\n- v05 --refutes--> v0a\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"280d2c4e20670b5731dd807b099c2f40ff4086820a2e491623755673fbd68289","schema":"livegraph.boolean-propagation.v1","spec_hash":"10b61c848bcd40607cd82612e7cb449f227d9acda2c5dd577755b8ae16840a78","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-5e990689868f2a484d64","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v00\n- v04 --supports--> v0b\n- v0b --supports--> v06\n- v04 --refutes--> v01\n- v0c --refutes--> v08\n- v03 --supports--> v04\n- v08 --supports--> v02\n- v05 --supports--> v0a\n- v06 --refutes--> v05\n- v05 --supports--> v07\n- v0b --supports--> v09\n- v07 --supports--> v0c\n- v02 --supports--> v0d\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"39b9ef6f60e05a5dc784b6244a78c868b1f11ae256fb4b34c25d774f14efec7d","schema":"livegraph.boolean-propagation.v1","spec_hash":"5e990689868f2a484d649e64072ebef82ff4305ea6e22d1553ed53ef873bc33f","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-b0f21f98d5f5058de906","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v08\n- v04 --refutes--> v09\n- v0d --supports--> v0a\n- v05 --supports--> v03\n- v0c --refutes--> v06\n- v0c --supports--> v0b\n- v07 --supports--> v0d\n- v05 --supports--> v00\n- v03 --refutes--> v0c\n- v02 --supports--> v07\n- v06 --supports--> v01\n- v04 --supports--> v02\n- v01 --refutes--> v04\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e0769a14768a03e0e11deb159c7f4bd27db7439b835dd2af5563bd5e0be8c1e4","schema":"livegraph.boolean-propagation.v1","spec_hash":"b0f21f98d5f5058de9065ed3d48f9a8e3a70c819929fed6f4a765c6efc54339c","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-72e31185d7f088d8e11b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v03\n- v00 --supports--> v0d\n- v06 --refutes--> v05\n- v03 --refutes--> v00\n- v02 --refutes--> v0b\n- v02 --refutes--> v0a\n- v09 --refutes--> v06\n- v08 --refutes--> v09\n- v05 --refutes--> v04\n- v08 --supports--> v01\n- v00 --refutes--> v08\n- v05 --supports--> v07\n- v07 --supports--> v02\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"15fb4996f859ff742e9d7ecded7706be50461bb6af7fe25d02b752f167c5fa6f","schema":"livegraph.boolean-propagation.v1","spec_hash":"72e31185d7f088d8e11b8c408d1e390693fd4dbcf9e4fbb3c8c96b1242d3da61","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-f42181af73f76ed58e1a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v01\n- v02 --supports--> v06\n- v08 --refutes--> v03\n- v03 --refutes--> v09\n- v00 --supports--> v08\n- v0b --supports--> v0c\n- v0b --supports--> v00\n- v0d --refutes--> v07\n- v02 --refutes--> v0b\n- v04 --supports--> v0a\n- v09 --refutes--> v0d\n- v0d --supports--> v04\n- v05 --refutes--> v02\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"69b448b6c9d4c518f41964dd018c7a7f96945b92a955a41558c273c69e3e9997","schema":"livegraph.boolean-propagation.v1","spec_hash":"f42181af73f76ed58e1af82ff747ad03520648510e340cc515e76aaaa083b2d3","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-ba91798fe9dcaf78ea5f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v07\n- v05 --refutes--> v02\n- v0c --refutes--> v05\n- v03 --supports--> v09\n- v01 --refutes--> v06\n- v0d --refutes--> v04\n- v0c --refutes--> v0b\n- v02 --supports--> v03\n- v03 --supports--> v00\n- v00 --refutes--> v0d\n- v0c --supports--> v08\n- v04 --refutes--> v01\n- v06 --refutes--> v0a\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"09aa7df4f95edad4b67dfa3f94205e2fb80cd39883151628acd7b6710b7ac928","schema":"livegraph.boolean-propagation.v1","spec_hash":"ba91798fe9dcaf78ea5f44ec66d3c45380afbe79de94eedae5234a2154d9c5f8","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-1080ec31269a2a9acdfe","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v03\n- v03 --supports--> v01\n- v06 --refutes--> v0b\n- v05 --refutes--> v09\n- v02 --refutes--> v0c\n- v0b --refutes--> v08\n- v0b --supports--> v05\n- v01 --refutes--> v0a\n- v00 --refutes--> v02\n- v03 --refutes--> v07\n- v0d --supports--> v06\n- v06 --supports--> v04\n- v02 --supports--> v0d\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5a596aa4137ac0f5bc1259ab8decb2e7cfaa023fb38c2bb75fa4306aa5f4f10d","schema":"livegraph.boolean-propagation.v1","spec_hash":"1080ec31269a2a9acdfed60e8a0dd1ce69dbd527acc0e9a7d76e68ce772f08cd","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-2c6c23c3b40a2da436cd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v02\n- v08 --supports--> v01\n- v00 --supports--> v0c\n- v04 --refutes--> v0a\n- v03 --refutes--> v08\n- v05 --supports--> v04\n- v0b --refutes--> v09\n- v04 --supports--> v07\n- v06 --refutes--> v0d\n- v08 --supports--> v05\n- v07 --supports--> v00\n- v02 --supports--> v0b\n- v0b --refutes--> v06\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a1bc6897b09eb83245b72845fea569a41a4559f822de29c0c250a666a302de45","schema":"livegraph.boolean-propagation.v1","spec_hash":"2c6c23c3b40a2da436cd36af6ea2c9d2d9ce0fa43709a2d5a2e9bab1bd6ac1ba","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":9,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-15510255736f27cdb45a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v06\n- v02 --supports--> v04\n- v01 --refutes--> v05\n- v09 --supports--> v02\n- v03 --supports--> v09\n- v06 --refutes--> v08\n- v07 --supports--> v0c\n- v07 --refutes--> v00\n- v04 --refutes--> v07\n- v06 --supports--> v03\n- v04 --supports--> v0a\n- v05 --refutes--> v0d\n- v05 --refutes--> v0b\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"97eacde33ebb58099d3be998c3d02efb755280b2a31ec18407bf00170ed4f194","schema":"livegraph.boolean-propagation.v1","spec_hash":"15510255736f27cdb45a764fb2132c1dc80b26e93bf533a74483934f5b164404","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-71d05de93503befb0edd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v0c\n- v01 --supports--> v02\n- v00 --refutes--> v09\n- v01 --supports--> v07\n- v09 --supports--> v05\n- v00 --refutes--> v04\n- v0c --supports--> v0d\n- v03 --refutes--> v01\n- v09 --supports--> v0b\n- v06 --supports--> v0a\n- v0b --supports--> v08\n- v0a --refutes--> v03\n- v0c --refutes--> v00\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5390bfcf423d626f7dfae934d0b5a18911b036efa52df39083b9ae62359e0ac1","schema":"livegraph.boolean-propagation.v1","spec_hash":"71d05de93503befb0eddf50970aec8bdd9455ceca42309c0a65f28eb507ff83f","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-98c52a513587190efe83","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v00\n- v00 --refutes--> v0a\n- v05 --supports--> v08\n- v06 --refutes--> v02\n- v06 --supports--> v01\n- v09 --refutes--> v07\n- v06 --supports--> v0b\n- v03 --refutes--> v09\n- v0c --supports--> v04\n- v07 --refutes--> v0c\n- v00 --supports--> v05\n- v04 --supports--> v0d\n- v0d --supports--> v06\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"92195fa44d12077668540017d324af5cd24c1dc8358c2fdf9d7594a1a9e8807d","schema":"livegraph.boolean-propagation.v1","spec_hash":"98c52a513587190efe83d1fe6c9b333dd3d85443d0760234c0a71805bdd5dec2","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-8d58f9381bef1b8b8939","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v0c\n- v0b --refutes--> v07\n- v04 --supports--> v0b\n- v0c --refutes--> v01\n- v0d --refutes--> v06\n- v00 --supports--> v09\n- v01 --supports--> v04\n- v08 --refutes--> v03\n- v0c --supports--> v02\n- v06 --refutes--> v00\n- v00 --supports--> v08\n- v00 --refutes--> v05\n- v08 --refutes--> v0a\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9f996aab216e734785cf99a787ae424d2b5fc5a0601e45e0526ef40fd01d6fe0","schema":"livegraph.boolean-propagation.v1","spec_hash":"8d58f9381bef1b8b8939a7dc96070b8a920117460fd2278b58ea38acb00f4116","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-ddcc31b7e9e43ab59a50","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v00\n- v0c --refutes--> v0b\n- v07 --refutes--> v01\n- v01 --refutes--> v0a\n- v02 --supports--> v07\n- v0c --supports--> v03\n- v0d --supports--> v05\n- v06 --supports--> v08\n- v04 --refutes--> v06\n- v08 --refutes--> v02\n- v03 --supports--> v04\n- v0d --supports--> v09\n- v01 --refutes--> v0d\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"48f3d82050a4cc2b12afa1ed099f21a79d4df127ad75e99624637767851dc6f4","schema":"livegraph.boolean-propagation.v1","spec_hash":"ddcc31b7e9e43ab59a50c31fbf9bea0df6aa91b578dcc3d9d66731fc50eb8be4","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-20e694a2fd1a45a00804","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v04\n- v08 --refutes--> v0b\n- v0b --refutes--> v06\n- v07 --supports--> v0d\n- v0d --refutes--> v05\n- v0b --refutes--> v01\n- v06 --supports--> v0a\n- v00 --supports--> v08\n- v08 --supports--> v02\n- v05 --supports--> v09\n- v0c --refutes--> v03\n- v09 --supports--> v00\n- v06 --refutes--> v0c\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8d37a6bdf57ca1bc27cd718670fecf2a45e5f32c245c3c9cc342b811f6754273","schema":"livegraph.boolean-propagation.v1","spec_hash":"20e694a2fd1a45a0080443a976fcf804ced99649140abdb9731aed39c6ce6b10","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-7b1153f174e7f7286a99","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v0b\n- v04 --refutes--> v03\n- v07 --supports--> v0d\n- v05 --refutes--> v07\n- v0a --refutes--> v06\n- v04 --refutes--> v09\n- v05 --supports--> v0c\n- v0d --refutes--> v01\n- v09 --refutes--> v05\n- v07 --supports--> v08\n- v02 --supports--> v04\n- v06 --refutes--> v00\n- v00 --supports--> v02\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5c89e4cede41286b74ac4d378942ff1607812af97b8d5c519496bdb7928e3558","schema":"livegraph.boolean-propagation.v1","spec_hash":"7b1153f174e7f7286a99b4d4b38bee7967dcfb875203e5c7122925480d8b9d01","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-2e8995800d7d608790b8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v03\n- v0c --refutes--> v02\n- v0a --refutes--> v04\n- v07 --refutes--> v01\n- v07 --supports--> v0d\n- v08 --refutes--> v0c\n- v0c --supports--> v05\n- v04 --supports--> v06\n- v05 --supports--> v07\n- v07 --refutes--> v00\n- v0d --supports--> v0b\n- v04 --refutes--> v09\n- v09 --refutes--> v08\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"648d75f74f3edc6663bc98e17ab0724cd1b1830487d24647357011e6d8f5e227","schema":"livegraph.boolean-propagation.v1","spec_hash":"2e8995800d7d608790b8671f23fb393c36140aa3643421b3d53be5ca301b166c","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-ed56be7934ab20e91958","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v04\n- v0b --supports--> v00\n- v03 --refutes--> v01\n- v0d --refutes--> v0a\n- v0a --refutes--> v02\n- v01 --refutes--> v0d\n- v0b --refutes--> v0c\n- v0a --refutes--> v07\n- v09 --supports--> v06\n- v0c --refutes--> v09\n- v07 --refutes--> v05\n- v03 --refutes--> v08\n- v05 --supports--> v0b\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"95dab4c75a5b42d041a170d9034777247e717ca8ba84f8dcf96451fecb09c7f1","schema":"livegraph.boolean-propagation.v1","spec_hash":"ed56be7934ab20e919580b040b337ab7a362367bbaae3431a6b60dedd7b86bdb","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-1ff56f226561bfa84681","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v03\n- v03 --supports--> v07\n- v0b --refutes--> v06\n- v02 --refutes--> v0b\n- v06 --supports--> v0a\n- v0c --refutes--> v09\n- v04 --supports--> v02\n- v09 --refutes--> v05\n- v06 --supports--> v00\n- v05 --refutes--> v08\n- v03 --refutes--> v04\n- v00 --refutes--> v01\n- v09 --refutes--> v0d\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"815c9f6188bfb616dbe55d1d02066de684b9a91c8779ee644b776aec1a0d3252","schema":"livegraph.boolean-propagation.v1","spec_hash":"1ff56f226561bfa84681a9bbe9c206fca8e1d4be0720b6572eef5ee1de1b0766","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-dffc982f53e4a757114d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v03\n- v08 --refutes--> v01\n- v04 --refutes--> v0a\n- v0c --supports--> v0d\n- v05 --supports--> v04\n- v09 --refutes--> v06\n- v01 --supports--> v0c\n- v02 --refutes--> v09\n- v0d --supports--> v02\n- v09 --supports--> v07\n- v06 --refutes--> v05\n- v09 --supports--> v00\n- v06 --supports--> v0b\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"70f7be0ba0ed9cc251c8d197cece124bf5975026525871c702db1d2305616292","schema":"livegraph.boolean-propagation.v1","spec_hash":"dffc982f53e4a757114d24600864b51bcc4287438a73bc4dcb5fe0bd0f2149f4","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-e974a23c1391f7e4a9ce","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v09\n- v00 --refutes--> v0b\n- v02 --refutes--> v05\n- v09 --supports--> v04\n- v0b --supports--> v02\n- v0a --supports--> v00\n- v02 --supports--> v08\n- v03 --refutes--> v0a\n- v00 --refutes--> v0d\n- v04 --supports--> v06\n- v05 --refutes--> v07\n- v02 --supports--> v01\n- v04 --supports--> v03\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"67f68f89966ce03532136885c9a66323c5ed2492ab9502b720e18c79e718040e","schema":"livegraph.boolean-propagation.v1","spec_hash":"e974a23c1391f7e4a9ceb161900060cdbfb35b3943aff16d61e2249b59df49d6","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-13cbd82f4cd15faaf4d6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v0b\n- v05 --supports--> v0d\n- v00 --refutes--> v0c\n- v06 --supports--> v08\n- v02 --supports--> v01\n- v09 --refutes--> v00\n- v05 --refutes--> v03\n- v0c --refutes--> v06\n- v08 --supports--> v0a\n- v02 --supports--> v09\n- v05 --refutes--> v07\n- v03 --refutes--> v02\n- v04 --supports--> v05\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9c928a635a9d66275410ac7ddc70aacfea15632f297ff86dd24f51f3046fff60","schema":"livegraph.boolean-propagation.v1","spec_hash":"13cbd82f4cd15faaf4d698ddee8519c86bd49fcba5c78f07ad43e1238850b310","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-ab0ac809a3b2c9fd1f46","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v03\n- v02 --supports--> v04\n- v05 --supports--> v06\n- v0a --refutes--> v00\n- v01 --supports--> v0a\n- v03 --refutes--> v02\n- v0c --refutes--> v05\n- v04 --refutes--> v0c\n- v01 --supports--> v08\n- v05 --refutes--> v07\n- v0c --refutes--> v0d\n- v00 --supports--> v09\n- v05 --refutes--> v01\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"db12e2cc18f1881d6696690bc2abb79daa02f6dca9a8c78e09a293dd38cb5b68","schema":"livegraph.boolean-propagation.v1","spec_hash":"ab0ac809a3b2c9fd1f46efd9879aad9c842d3f00c82d89b9d7d7f37a5dcec937","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-6cef42990f80b48bea92","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v05\n- v04 --supports--> v03\n- v00 --refutes--> v0a\n- v0d --refutes--> v04\n- v0b --supports--> v02\n- v0b --refutes--> v0c\n- v01 --supports--> v0b\n- v07 --refutes--> v00\n- v09 --refutes--> v01\n- v05 --supports--> v06\n- v00 --supports--> v08\n- v03 --supports--> v07\n- v05 --refutes--> v09\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6589a9e9c9565cdcd284c274190d61e0926cb8235602d92b30a6c5fe33beab97","schema":"livegraph.boolean-propagation.v1","spec_hash":"6cef42990f80b48bea926b518aef5218c591cc1b6484e4ba1ce3c0c4ea6249b7","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-c4f2a1d89b69486ad343","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v06\n- v08 --refutes--> v09\n- v0d --refutes--> v04\n- v01 --refutes--> v05\n- v01 --refutes--> v08\n- v00 --refutes--> v0c\n- v02 --supports--> v03\n- v08 --supports--> v0b\n- v0a --supports--> v0d\n- v02 --supports--> v00\n- v04 --refutes--> v07\n- v0b --refutes--> v0a\n- v0c --supports--> v01\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6d2c09a1d5c83be70be7e2819000f29e0d8ec779f85ba9ab8146a9007f91c947","schema":"livegraph.boolean-propagation.v1","spec_hash":"c4f2a1d89b69486ad343022593eff0ec964d10489053097ecf6b3ca317bdf2f5","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-d370e4b1ef5f28928c19","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v03\n- v0a --refutes--> v0b\n- v0b --supports--> v05\n- v04 --supports--> v02\n- v07 --supports--> v06\n- v09 --supports--> v01\n- v04 --refutes--> v00\n- v05 --refutes--> v07\n- v06 --refutes--> v04\n- v05 --supports--> v08\n- v04 --supports--> v09\n- v0c --supports--> v0a\n- v09 --supports--> v0d\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"766d5f20300a34a41bddb23257fe68e97ed3f47f088710b34f3e618e930a5fd7","schema":"livegraph.boolean-propagation.v1","spec_hash":"d370e4b1ef5f28928c190c881645e4ac8dea8c6e7d3a8b1414e0d9f19edde1b7","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-38e156c01725f4fbd914","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v03\n- v0b --supports--> v0d\n- v08 --refutes--> v05\n- v08 --refutes--> v0a\n- v06 --supports--> v00\n- v0b --supports--> v08\n- v03 --refutes--> v06\n- v0c --supports--> v07\n- v00 --supports--> v02\n- v0c --supports--> v01\n- v0a --refutes--> v09\n- v00 --refutes--> v0c\n- v01 --refutes--> v0b\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"91d4920c30f89253f28a0092c93cc4ea70775b274522202ba1ab58d0bad43fc7","schema":"livegraph.boolean-propagation.v1","spec_hash":"38e156c01725f4fbd9149c06c966d8ab78bf3fcd46b4cadb6fb0c652fc25c2b2","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-8b31be34edeadba98543","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v09\n- v00 --refutes--> v05\n- v07 --supports--> v0d\n- v06 --supports--> v00\n- v05 --refutes--> v0c\n- v0d --refutes--> v02\n- v0a --supports--> v03\n- v0a --refutes--> v0b\n- v02 --supports--> v08\n- v05 --refutes--> v07\n- v06 --refutes--> v01\n- v03 --refutes--> v04\n- v08 --supports--> v0a\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0d5eeb6a8f616f8a80b2ce7dee5338c77f7994c4d1674ab5c86dd3b14c304fb2","schema":"livegraph.boolean-propagation.v1","spec_hash":"8b31be34edeadba985435577a2a0eec050b67a467a203ed53231d3abb59f67bb","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-a839bd40a5b9eecad396","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v0d\n- v0d --refutes--> v07\n- v05 --refutes--> v06\n- v0c --supports--> v00\n- v01 --supports--> v05\n- v04 --refutes--> v09\n- v09 --supports--> v01\n- v07 --supports--> v0a\n- v00 --refutes--> v04\n- v05 --supports--> v02\n- v0d --supports--> v03\n- v05 --refutes--> v0b\n- v0c --supports--> v08\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e8d775eaff33289e27eac34237c678d86fe9664bea087a881165d52b55e840a4","schema":"livegraph.boolean-propagation.v1","spec_hash":"a839bd40a5b9eecad396b8d87b06a5e55dba756f566f15cbb6bf4b73883fbf49","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-372f121a604b20475200","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v0b\n- v0c --supports--> v04\n- v00 --supports--> v07\n- v0c --refutes--> v09\n- v01 --refutes--> v06\n- v09 --supports--> v05\n- v06 --refutes--> v00\n- v07 --refutes--> v0a\n- v05 --refutes--> v03\n- v07 --refutes--> v02\n- v0d --supports--> v0c\n- v01 --refutes--> v08\n- v03 --refutes--> v01\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"76298bee198781b84654b6b3d7b5cbf3a8f6782b80cee03d4be8838659a86cb7","schema":"livegraph.boolean-propagation.v1","spec_hash":"372f121a604b2047520025d061547e57e0643401ccbbc69d6cc7944f19f0aa08","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-67d9af6dedcbb6457843","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v0d\n- v0c --refutes--> v05\n- v0d --refutes--> v06\n- v01 --refutes--> v03\n- v03 --supports--> v04\n- v09 --refutes--> v0a\n- v04 --supports--> v09\n- v06 --refutes--> v07\n- v09 --supports--> v00\n- v07 --supports--> v01\n- v07 --refutes--> v08\n- v06 --refutes--> v02\n- v03 --refutes--> v0b\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"673f4422bc2586aba303584cb46a76716eaea73aab738e8c0d42cbc6daaf13b2","schema":"livegraph.boolean-propagation.v1","spec_hash":"67d9af6dedcbb64578430d4efe6fcf63cb77c980bc6fef724eb0cb06740fd789","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-e8e72009e30c265582a3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v03\n- v01 --refutes--> v0b\n- v0d --supports--> v00\n- v05 --supports--> v06\n- v0d --refutes--> v04\n- v06 --refutes--> v07\n- v01 --refutes--> v08\n- v00 --supports--> v01\n- v0b --supports--> v09\n- v0a --refutes--> v0c\n- v0a --refutes--> v0d\n- v09 --supports--> v05\n- v07 --refutes--> v02\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"df096e20305be37c1f930404da41e9398c0e9bf70155dd96d890b6af2a6fc93c","schema":"livegraph.boolean-propagation.v1","spec_hash":"e8e72009e30c265582a3d73eb51b7b2da659bc6fd8467b60db3a2d7e919b5d09","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-20bb3d3657682aed7101","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v0d\n- v0c --supports--> v06\n- v09 --refutes--> v0c\n- v07 --supports--> v00\n- v0a --supports--> v07\n- v0b --refutes--> v0a\n- v06 --supports--> v04\n- v04 --refutes--> v03\n- v0a --refutes--> v08\n- v00 --supports--> v09\n- v0c --refutes--> v05\n- v03 --supports--> v02\n- v00 --refutes--> v01\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"beed78128617a85ecc38e887bd122409d13e269208d7b6e6b8171138ab3208b3","schema":"livegraph.boolean-propagation.v1","spec_hash":"20bb3d3657682aed71019885aa7e0cbf81eb9eef94d85e84695dd25428b374fb","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-57a1e6f91bac2f7fab73","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v08\n- v0c --refutes--> v02\n- v04 --refutes--> v0b\n- v01 --refutes--> v04\n- v06 --refutes--> v09\n- v0b --supports--> v03\n- v02 --refutes--> v06\n- v06 --supports--> v07\n- v09 --refutes--> v01\n- v0d --supports--> v0c\n- v09 --supports--> v05\n- v06 --supports--> v0a\n- v00 --supports--> v0d\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"49dceb9fbae7070b5f05a6eb7d1ed9cbd83b6c887143559640e0e53b5b45b89c","schema":"livegraph.boolean-propagation.v1","spec_hash":"57a1e6f91bac2f7fab7315c2608ceaa72f5d88a421fc85984ad90b75346169cf","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-4ef2aa3ff36dd102d4bd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v03\n- v05 --refutes--> v09\n- v0d --supports--> v0c\n- v0a --refutes--> v07\n- v05 --refutes--> v01\n- v04 --refutes--> v02\n- v0a --supports--> v08\n- v0c --refutes--> v0a\n- v03 --refutes--> v0b\n- v08 --supports--> v05\n- v08 --supports--> v06\n- v08 --refutes--> v00\n- v0b --refutes--> v0d\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3d83aeb041fdc609e362e65d2213355fdcc6a5eae93a8a627f798c3245a80469","schema":"livegraph.boolean-propagation.v1","spec_hash":"4ef2aa3ff36dd102d4bd8f99f81271321fcd0d68516f662c36e2ad854e0ee994","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-50a0c402f629a80cc5fc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v07\n- v09 --refutes--> v02\n- v0b --refutes--> v0c\n- v07 --refutes--> v03\n- v08 --refutes--> v09\n- v0a --refutes--> v06\n- v07 --refutes--> v0d\n- v0b --supports--> v04\n- v00 --supports--> v01\n- v0a --supports--> v08\n- v05 --supports--> v0a\n- v0b --refutes--> v00\n- v03 --refutes--> v05\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"78d2c25d4eb35572179786666673b4357b04ac6a7f88ec8e5096911f9d8cc932","schema":"livegraph.boolean-propagation.v1","spec_hash":"50a0c402f629a80cc5fcc524c47179eb3a2c2be22877ef07cd116c6daeeaee17","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":9,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-c503ad1f44a85025ba11","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v0b\n- v0d --refutes--> v04\n- v05 --supports--> v07\n- v05 --refutes--> v0a\n- v03 --refutes--> v0c\n- v06 --refutes--> v05\n- v0d --refutes--> v01\n- v07 --supports--> v09\n- v01 --refutes--> v00\n- v07 --refutes--> v03\n- v0b --refutes--> v0d\n- v01 --refutes--> v08\n- v02 --refutes--> v06\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"015e524c755e0f755d16754337eba802d96c335791d60c0572e6ad32b94ab50b","schema":"livegraph.boolean-propagation.v1","spec_hash":"c503ad1f44a85025ba11b2094d4628004adfb4eb87831ba70a16b1679322ee8a","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-d03600c72b3f7b1bdd22","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v04\n- v08 --supports--> v09\n- v00 --supports--> v0b\n- v02 --refutes--> v05\n- v0c --refutes--> v0d\n- v0d --refutes--> v03\n- v0c --refutes--> v06\n- v05 --supports--> v0a\n- v0d --supports--> v00\n- v09 --refutes--> v01\n- v0a --supports--> v07\n- v0a --refutes--> v08\n- v0b --supports--> v02\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6558fc79fc77b8d8352a6bb0f5fd2ad42ca1c7085fe13b551b239c91b9d50eb8","schema":"livegraph.boolean-propagation.v1","spec_hash":"d03600c72b3f7b1bdd22a028c7d15bd29432c21e9f5b4839b5c0fe7e5465315d","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-3a1201ffffaf288de992","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v0b\n- v06 --refutes--> v03\n- v01 --refutes--> v07\n- v06 --supports--> v0d\n- v0c --supports--> v04\n- v03 --supports--> v0a\n- v08 --refutes--> v01\n- v0b --refutes--> v08\n- v07 --refutes--> v0c\n- v0a --supports--> v09\n- v0c --supports--> v00\n- v07 --refutes--> v02\n- v01 --supports--> v05\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b4e3a294548dcc1f775d49048e9d788be1f80c1851970033c522975c1ae9bfd2","schema":"livegraph.boolean-propagation.v1","spec_hash":"3a1201ffffaf288de9926dce1ad7330cdb7d324a6c1b788de2bbce2d4e373ce3","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-ffde0d345482ce61fcad","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v00\n- v0b --refutes--> v0a\n- v0b --supports--> v09\n- v08 --supports--> v02\n- v06 --supports--> v05\n- v01 --refutes--> v08\n- v0d --refutes--> v06\n- v07 --refutes--> v04\n- v0b --supports--> v0c\n- v02 --refutes--> v07\n- v03 --supports--> v0d\n- v06 --refutes--> v0b\n- v0c --refutes--> v01\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3d9023b7fa50c6409cb7eb7decfb59ea4e8e6c9a6d3f7b6153c00837ec0f9d69","schema":"livegraph.boolean-propagation.v1","spec_hash":"ffde0d345482ce61fcadc4756922ee9ff3c3ba1f9ae718fddb8b991ecdc8bc74","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-e8d5e986d90caea259e6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v01\n- v0a --refutes--> v00\n- v07 --supports--> v0d\n- v09 --supports--> v03\n- v09 --refutes--> v08\n- v0c --supports--> v0a\n- v00 --supports--> v04\n- v08 --supports--> v0b\n- v08 --supports--> v06\n- v06 --refutes--> v07\n- v0d --refutes--> v05\n- v0c --supports--> v02\n- v04 --supports--> v09\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"800bcefe7ff5404928100e9a3588cb52fd53c780109087e9ce4d5b2cbff0a6b8","schema":"livegraph.boolean-propagation.v1","spec_hash":"e8d5e986d90caea259e63dba1fb1fa677a14f24cfa46e10ac51b4537e58ebaad","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-36899564f0063f329314","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v08\n- v04 --supports--> v01\n- v0c --refutes--> v0a\n- v09 --refutes--> v00\n- v0a --supports--> v03\n- v07 --refutes--> v0d\n- v03 --supports--> v09\n- v00 --refutes--> v04\n- v02 --supports--> v0b\n- v0d --supports--> v0c\n- v04 --supports--> v02\n- v04 --supports--> v05\n- v02 --supports--> v06\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2dad46a66508882d5ec3d5c133396720c20f3b4c6efdc4eca758e7d4fe256190","schema":"livegraph.boolean-propagation.v1","spec_hash":"36899564f0063f329314d209529cb2324872708639e87ef84c56e036a0fbef2e","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-f6b5d365c7a05a99871a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v00\n- v0d --refutes--> v07\n- v00 --supports--> v08\n- v07 --refutes--> v0c\n- v00 --supports--> v05\n- v0a --refutes--> v0d\n- v06 --refutes--> v03\n- v0b --refutes--> v0a\n- v07 --refutes--> v09\n- v0d --supports--> v01\n- v08 --supports--> v02\n- v03 --supports--> v0b\n- v08 --supports--> v04\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e593c438f31bec205d2d336b32c7c46af4b8f541ea6811737383f8939e28ecf3","schema":"livegraph.boolean-propagation.v1","spec_hash":"f6b5d365c7a05a99871ad71fc3a192465a1ab3f0d13983708859da37b55c58b8","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-71916edba377e9f420bb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v01\n- v0b --supports--> v08\n- v0d --refutes--> v06\n- v09 --refutes--> v05\n- v00 --refutes--> v0c\n- v0d --supports--> v02\n- v05 --refutes--> v0a\n- v00 --refutes--> v07\n- v04 --refutes--> v0d\n- v0a --refutes--> v00\n- v08 --refutes--> v04\n- v02 --refutes--> v09\n- v05 --supports--> v03\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c566b632e64b8611c14e58e1bd1b942a80f87642b5aef1ceb5e1f300ca78b566","schema":"livegraph.boolean-propagation.v1","spec_hash":"71916edba377e9f420bbf37cfdd2518a9eaea4672a889592bab6a8a10befe90c","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-010a8b9220225dc91e57","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v00\n- v07 --refutes--> v04\n- v09 --supports--> v02\n- v02 --refutes--> v0c\n- v06 --supports--> v08\n- v0c --supports--> v03\n- v0c --refutes--> v06\n- v08 --supports--> v0b\n- v01 --refutes--> v0a\n- v0b --refutes--> v0d\n- v0a --refutes--> v09\n- v0b --refutes--> v07\n- v0a --supports--> v05\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"21228bc1b5702cc698203b2c78fc30f9890488fc855f7924852290761f0bc634","schema":"livegraph.boolean-propagation.v1","spec_hash":"010a8b9220225dc91e57f2d15bc7e2d6a37bde315590cf2c3fb0d07219701bff","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-1639415f5fadb183a1d1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v03\n- v07 --refutes--> v02\n- v01 --refutes--> v05\n- v04 --refutes--> v0a\n- v0a --supports--> v0c\n- v01 --supports--> v0d\n- v04 --supports--> v08\n- v05 --refutes--> v07\n- v02 --refutes--> v00\n- v0c --supports--> v09\n- v09 --refutes--> v0b\n- v07 --refutes--> v06\n- v00 --supports--> v04\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9dc3a1ef11c969d949a7a6e32b238f65e72c2c472624381a643df2b637668727","schema":"livegraph.boolean-propagation.v1","spec_hash":"1639415f5fadb183a1d1a00ed15edd70d96bc794f46ef5252f5234b89112802d","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-2630a3542441bf735e1a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v0b\n- v0a --refutes--> v06\n- v02 --supports--> v0a\n- v04 --supports--> v08\n- v07 --refutes--> v09\n- v09 --refutes--> v0d\n- v06 --supports--> v05\n- v04 --refutes--> v07\n- v02 --supports--> v03\n- v05 --supports--> v04\n- v0c --refutes--> v00\n- v01 --supports--> v0c\n- v09 --refutes--> v01\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"534580fabd508226b0dca622bcb0ed3598e1763321210e84455b817e89b6726a","schema":"livegraph.boolean-propagation.v1","spec_hash":"2630a3542441bf735e1a8ca6ba4d79748a482b3853066ae81aa67d4198c7bc7f","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-662fadbf0d3fde82c4f5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v0a\n- v07 --refutes--> v03\n- v06 --supports--> v01\n- v04 --supports--> v06\n- v0b --refutes--> v09\n- v09 --refutes--> v0c\n- v0b --supports--> v02\n- v0d --refutes--> v0b\n- v03 --refutes--> v05\n- v0d --refutes--> v00\n- v09 --supports--> v08\n- v0a --supports--> v04\n- v01 --supports--> v0d\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"74048054c2121d4741892b47a467360bc241a250e0389a4f2d51f4e06690dcfc","schema":"livegraph.boolean-propagation.v1","spec_hash":"662fadbf0d3fde82c4f5509e5d681c256492457d514cd5ae02a9d1c7c4142604","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-7c31ca7ca9c5060c0fc0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v03\n- v06 --refutes--> v0b\n- v02 --supports--> v0d\n- v01 --supports--> v02\n- v0c --refutes--> v09\n- v0a --supports--> v06\n- v08 --supports--> v00\n- v09 --supports--> v0a\n- v00 --supports--> v0c\n- v08 --supports--> v05\n- v07 --supports--> v08\n- v02 --supports--> v07\n- v06 --supports--> v04\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1ea33fde0990c80b001fd93abda4db48f72ee32fdd752779c37e4de10e1fe07b","schema":"livegraph.boolean-propagation.v1","spec_hash":"7c31ca7ca9c5060c0fc006024e7ef4689af9304358d690b3d3c2cf68fdb4eca6","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-5324fa1d6629769373ab","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v02\n- v04 --refutes--> v0a\n- v0d --supports--> v01\n- v0c --supports--> v09\n- v00 --refutes--> v04\n- v09 --supports--> v08\n- v06 --supports--> v0b\n- v04 --refutes--> v06\n- v03 --refutes--> v07\n- v04 --supports--> v05\n- v03 --refutes--> v0d\n- v0b --refutes--> v03\n- v01 --supports--> v0c\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1d8192951af486fefedc696409cced2034957075d008de05b6b11f747fe461fd","schema":"livegraph.boolean-propagation.v1","spec_hash":"5324fa1d6629769373ab12c06c2c66b267e97f742462e187b85b93ba3b778f3f","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-1ef8c8b61e9bd0b50753","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v02\n- v06 --refutes--> v00\n- v08 --refutes--> v0d\n- v0b --refutes--> v04\n- v07 --refutes--> v0b\n- v01 --supports--> v09\n- v04 --supports--> v05\n- v0d --supports--> v03\n- v03 --refutes--> v06\n- v01 --supports--> v0c\n- v06 --supports--> v0a\n- v06 --supports--> v01\n- v09 --supports--> v07\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c2079603eeb3bfd80d740cacba6d0ce8dcdcc48b7854ee869c1ff9fd32cd6b32","schema":"livegraph.boolean-propagation.v1","spec_hash":"1ef8c8b61e9bd0b50753b0247a802dfea9046f046d8a4670c490beec147f7b9a","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-43644b64eac19df59a30","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v01\n- v05 --supports--> v08\n- v07 --refutes--> v04\n- v02 --supports--> v00\n- v07 --refutes--> v0b\n- v02 --supports--> v0d\n- v00 --refutes--> v03\n- v0c --refutes--> v05\n- v06 --refutes--> v02\n- v0a --refutes--> v09\n- v0b --supports--> v0c\n- v09 --supports--> v07\n- v00 --refutes--> v0a\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"644c09612e7c216501762db75bc8e4dc62c7f7c5cc9035d516b90c9367f0b677","schema":"livegraph.boolean-propagation.v1","spec_hash":"43644b64eac19df59a30df1cefcc22a776e62fb534bee200a56d6b0702dc4408","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-92ed2eea49fc8ab98b63","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v0b\n- v09 --supports--> v0a\n- v03 --supports--> v01\n- v01 --refutes--> v07\n- v08 --supports--> v03\n- v02 --refutes--> v08\n- v05 --supports--> v04\n- v04 --refutes--> v00\n- v00 --supports--> v02\n- v09 --supports--> v05\n- v07 --refutes--> v0c\n- v02 --supports--> v0d\n- v03 --refutes--> v06\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1b9db070d4877f1eb716502abbcb1248932829cd07bfd18a4c0d5fcacd606fcd","schema":"livegraph.boolean-propagation.v1","spec_hash":"92ed2eea49fc8ab98b6341581ed7c27533101a70c3a16df843973876511b984c","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":9,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-1290c720b4fa3140f886","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v0a\n- v02 --refutes--> v04\n- v05 --supports--> v0b\n- v07 --refutes--> v0c\n- v02 --refutes--> v07\n- v00 --supports--> v08\n- v06 --supports--> v05\n- v0c --refutes--> v00\n- v09 --supports--> v03\n- v0a --supports--> v01\n- v05 --refutes--> v0d\n- v09 --refutes--> v06\n- v0a --supports--> v02\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0da99635327c302afc2bfce2552cb8eb7b07ac4a16299817841070baa3fac49e","schema":"livegraph.boolean-propagation.v1","spec_hash":"1290c720b4fa3140f88690d86ff47348e8f278404ad3a0c7589b9f500825a0f1","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-e19e1c47897c4d543d16","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v05\n- v0a --refutes--> v03\n- v00 --refutes--> v0d\n- v0d --refutes--> v06\n- v02 --supports--> v0b\n- v0a --supports--> v01\n- v02 --refutes--> v04\n- v0d --supports--> v07\n- v07 --supports--> v0c\n- v01 --supports--> v00\n- v01 --refutes--> v08\n- v09 --refutes--> v02\n- v05 --supports--> v0a\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"75c5e437b3f91104f3f417d65e085536a3c4c85fa565eb215476e8e89db1e389","schema":"livegraph.boolean-propagation.v1","spec_hash":"e19e1c47897c4d543d16e74c5e706350cc35c611a147ee1c9d51153fbb9c08b5","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-c43be2733fa892346ee1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v01\n- v03 --refutes--> v0a\n- v03 --supports--> v08\n- v01 --supports--> v04\n- v08 --supports--> v00\n- v01 --refutes--> v0b\n- v0d --supports--> v05\n- v0b --refutes--> v0c\n- v06 --refutes--> v07\n- v00 --refutes--> v06\n- v0d --refutes--> v09\n- v0c --refutes--> v0d\n- v07 --supports--> v02\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c6a2c2f90fb701139b2a435c22867b7e01740feb6b6d1c467c9d4b65b91171eb","schema":"livegraph.boolean-propagation.v1","spec_hash":"c43be2733fa892346ee1fd88af33fd543c0585e74058584f858057bae2920308","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-622ad6ec03d7ec419f5b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v0d\n- v02 --refutes--> v04\n- v0c --supports--> v0b\n- v08 --supports--> v0c\n- v03 --refutes--> v05\n- v0a --refutes--> v03\n- v0b --supports--> v02\n- v05 --supports--> v01\n- v03 --refutes--> v06\n- v03 --refutes--> v07\n- v0d --refutes--> v08\n- v0d --refutes--> v09\n- v08 --refutes--> v00\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a3b4712024cb9d72dc28e347aab60d65e2acbd378f68139f91ad15282ddf985f","schema":"livegraph.boolean-propagation.v1","spec_hash":"622ad6ec03d7ec419f5b889526a2af977d8c25e5c4b1f7f9b08b445acbd151f3","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-c13e62becec52f93ea3d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v07\n- v01 --supports--> v0d\n- v04 --supports--> v01\n- v06 --supports--> v04\n- v03 --supports--> v0b\n- v05 --refutes--> v0c\n- v07 --refutes--> v0a\n- v01 --refutes--> v09\n- v0b --refutes--> v08\n- v01 --supports--> v00\n- v01 --supports--> v03\n- v05 --refutes--> v02\n- v0a --refutes--> v05\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"563275b45e0e101f966325d52a7db3bb5f4155819007066936359520bdc6a813","schema":"livegraph.boolean-propagation.v1","spec_hash":"c13e62becec52f93ea3d5e7fe6c438a105bf2b1effe6a3498aef33abe4d2f889","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":9,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-7a15388eda0f952ca86f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v00\n- v0b --refutes--> v02\n- v0d --refutes--> v0a\n- v06 --supports--> v0d\n- v0c --refutes--> v01\n- v01 --refutes--> v09\n- v02 --refutes--> v03\n- v04 --refutes--> v08\n- v07 --supports--> v0c\n- v09 --refutes--> v04\n- v0d --refutes--> v05\n- v0d --supports--> v0b\n- v02 --supports--> v07\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3aa5ec58b51b857eb380a36b27a25ecb10a6979c15bd363d2cf194bc37477960","schema":"livegraph.boolean-propagation.v1","spec_hash":"7a15388eda0f952ca86fce91ac0f9ad7334e8aeea5be8a6314583b1ce153724f","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-120eaf186d6676ae8148","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v0a\n- v08 --supports--> v0d\n- v03 --supports--> v02\n- v0d --supports--> v01\n- v04 --supports--> v05\n- v01 --refutes--> v07\n- v00 --refutes--> v09\n- v02 --supports--> v00\n- v08 --supports--> v06\n- v08 --supports--> v0b\n- v05 --refutes--> v03\n- v07 --supports--> v04\n- v08 --supports--> v0c\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2d345cf2bb1e48434fbfe8c349faa6bb464c12bd47946ee63731f35b16e31e4c","schema":"livegraph.boolean-propagation.v1","spec_hash":"120eaf186d6676ae814831465d13ecfd888913de4e6e14cd60eaa1af65263603","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-e8311fc2f02bf8350d39","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v0a\n- v09 --refutes--> v0d\n- v0c --supports--> v04\n- v0a --refutes--> v05\n- v05 --refutes--> v00\n- v05 --refutes--> v02\n- v02 --supports--> v0c\n- v06 --refutes--> v03\n- v03 --supports--> v09\n- v03 --supports--> v07\n- v09 --supports--> v01\n- v04 --supports--> v06\n- v06 --supports--> v08\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d2d34b7b17d10134b0dc0d5bbc2cdb04ed6e8e30265f3a2722f9197bd2a1f74f","schema":"livegraph.boolean-propagation.v1","spec_hash":"e8311fc2f02bf8350d398e6ae569e396c53d6edfac44461b364c1759846bbe5f","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-c9b6266ac635949ebb37","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v01\n- v0a --refutes--> v09\n- v05 --refutes--> v03\n- v0c --supports--> v00\n- v06 --refutes--> v04\n- v00 --refutes--> v0b\n- v0a --refutes--> v0d\n- v0a --refutes--> v08\n- v03 --supports--> v07\n- v02 --refutes--> v0a\n- v0a --supports--> v05\n- v01 --refutes--> v02\n- v07 --refutes--> v0c\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"259cb62f0a3e9d90074d52e442c7923e4ece64923ab4a975ca5880de2c1a6fe2","schema":"livegraph.boolean-propagation.v1","spec_hash":"c9b6266ac635949ebb379f9cb33c23fd2a6cfcdb0f9f83f619251d1614c4b8f0","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-bdfe5ac830d61e1c0479","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v0a\n- v09 --supports--> v07\n- v00 --supports--> v04\n- v03 --supports--> v00\n- v02 --refutes--> v0d\n- v01 --refutes--> v03\n- v0a --refutes--> v0b\n- v04 --supports--> v08\n- v04 --refutes--> v09\n- v05 --supports--> v0c\n- v06 --refutes--> v05\n- v05 --supports--> v01\n- v02 --supports--> v06\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6c4aa349f3f338ad4c0f9f639bf255f3cd082c86cecf83de8b0b98c1ededd948","schema":"livegraph.boolean-propagation.v1","spec_hash":"bdfe5ac830d61e1c047953b9b47cbbb7c5c9db50246cb70dc5e2471df899b85a","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":9,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-f920eab9a9424c6bd3fd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v01\n- v0a --supports--> v00\n- v04 --refutes--> v05\n- v05 --supports--> v09\n- v00 --supports--> v0c\n- v00 --supports--> v04\n- v06 --refutes--> v08\n- v0d --supports--> v03\n- v01 --refutes--> v07\n- v01 --supports--> v02\n- v01 --supports--> v0d\n- v05 --supports--> v0b\n- v03 --supports--> v0a\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7ecf34a7ade5fe645ee056f75f50a3b7d15a8b71d0b7376750b949bb4fc4135e","schema":"livegraph.boolean-propagation.v1","spec_hash":"f920eab9a9424c6bd3fd9fab995c6579718a47541168013fac1f74a3d2fabb9a","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-03254183b33ba0a6c95e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v07\n- v00 --refutes--> v0b\n- v02 --supports--> v00\n- v00 --refutes--> v0d\n- v02 --refutes--> v03\n- v09 --refutes--> v02\n- v0c --supports--> v0a\n- v04 --supports--> v05\n- v05 --refutes--> v08\n- v01 --refutes--> v09\n- v09 --supports--> v06\n- v07 --refutes--> v04\n- v0c --refutes--> v01\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"11b785d97420c61a3ace47758af0758bf0df41d7c10792eb0a8b74dc74da5bff","schema":"livegraph.boolean-propagation.v1","spec_hash":"03254183b33ba0a6c95ef5f1038789b35f7f3c1047b31469e0c45d622a5e98ae","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-909ce89601c0fe57ccea","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v09\n- v02 --refutes--> v08\n- v05 --supports--> v07\n- v0a --refutes--> v0c\n- v07 --refutes--> v0b\n- v0d --supports--> v01\n- v05 --refutes--> v00\n- v04 --supports--> v0a\n- v02 --supports--> v03\n- v0b --refutes--> v06\n- v0a --supports--> v0d\n- v09 --supports--> v04\n- v0d --supports--> v02\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e9b09890359fb307be5c06de59885658db2f3599fab06d9936bf627a6e9207af","schema":"livegraph.boolean-propagation.v1","spec_hash":"909ce89601c0fe57cceafdf01ce591d1a782f39ad1cb22124f0441e439077057","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-18e9d783cef48cb49c8e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v0a\n- v03 --refutes--> v09\n- v04 --supports--> v02\n- v0c --refutes--> v0d\n- v0a --refutes--> v08\n- v0d --supports--> v04\n- v01 --refutes--> v0b\n- v04 --supports--> v07\n- v03 --refutes--> v0c\n- v05 --supports--> v01\n- v07 --supports--> v05\n- v07 --refutes--> v00\n- v07 --supports--> v06\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4afabeb7238b9107f056e034191c8ba3aaeb54f9a295592518c85fcc1a867b4d","schema":"livegraph.boolean-propagation.v1","spec_hash":"18e9d783cef48cb49c8efaadc3ff5e7c61210c6e19d2114f8fb76f100a3e8ba2","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-ef8eeace33de14dbba94","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v0b\n- v01 --refutes--> v07\n- v02 --refutes--> v00\n- v05 --supports--> v06\n- v0a --refutes--> v09\n- v05 --supports--> v0c\n- v0c --refutes--> v0a\n- v04 --supports--> v02\n- v0a --refutes--> v03\n- v01 --refutes--> v08\n- v00 --refutes--> v0d\n- v08 --supports--> v05\n- v0b --refutes--> v04\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"780a953f5b29d3b95588be8222e3e39e3bf0669b046389f2e58beeed22e73f3f","schema":"livegraph.boolean-propagation.v1","spec_hash":"ef8eeace33de14dbba94736f632e5ebcb6652bbf0ff48b62e75cb414010fd4a5","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-38799ab07233abd4d551","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v0b\n- v0b --supports--> v0a\n- v05 --supports--> v0d\n- v0b --supports--> v09\n- v0a --supports--> v02\n- v02 --supports--> v03\n- v08 --refutes--> v05\n- v05 --supports--> v01\n- v04 --refutes--> v07\n- v02 --supports--> v00\n- v06 --refutes--> v04\n- v02 --supports--> v08\n- v0d --refutes--> v0c\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0c95ebc5cc9a6ca56b33a141af93aa535b14dd2a6cb9bf69b1068d1b35ba2ad0","schema":"livegraph.boolean-propagation.v1","spec_hash":"38799ab07233abd4d5513e3b78045808e55b77915657f2d65a5eba6ccd1abd33","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-3f4b3fd6742aad7135e3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v02\n- v07 --refutes--> v01\n- v01 --supports--> v08\n- v05 --supports--> v09\n- v03 --refutes--> v07\n- v01 --refutes--> v05\n- v0a --supports--> v0b\n- v0b --refutes--> v06\n- v04 --refutes--> v0a\n- v02 --refutes--> v03\n- v03 --supports--> v0d\n- v04 --refutes--> v00\n- v07 --refutes--> v0c\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"24614209bb76e436a6ccc0451faeb80ab85e00afc8c34e5368febe69955d2f55","schema":"livegraph.boolean-propagation.v1","spec_hash":"3f4b3fd6742aad7135e3a995d83ba25130f2b393b19591c7d5e546ba5eab532c","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":9,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-531e277ea8a340c28b6f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v00\n- v02 --refutes--> v09\n- v0d --supports--> v01\n- v04 --refutes--> v0c\n- v0d --supports--> v06\n- v04 --refutes--> v08\n- v06 --refutes--> v03\n- v08 --supports--> v07\n- v00 --supports--> v05\n- v06 --refutes--> v0a\n- v02 --refutes--> v0b\n- v05 --refutes--> v0d\n- v0a --refutes--> v04\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a437daa84141bd0783ae0cdc9cd32e68dbebdbb81146b8c2850f8f42e3e5c8ee","schema":"livegraph.boolean-propagation.v1","spec_hash":"531e277ea8a340c28b6ffe75e3527c5c7f0363b61397da3f83f3c159dfd7c1bd","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-23c9567f81136aa2ec33","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v07\n- v03 --refutes--> v09\n- v0d --refutes--> v00\n- v07 --supports--> v01\n- v02 --supports--> v04\n- v0d --supports--> v06\n- v02 --refutes--> v05\n- v02 --refutes--> v03\n- v0b --refutes--> v08\n- v06 --supports--> v0b\n- v03 --supports--> v0d\n- v0a --refutes--> v02\n- v0c --refutes--> v0a\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"89d06dfa4e6d9d2f080a0ac6003d041a39afa566219a7059f34a3f9bfedde266","schema":"livegraph.boolean-propagation.v1","spec_hash":"23c9567f81136aa2ec33035e2ba430c5cf9a716f3886de7e5cfb885327283299","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-fb809703e8f24e7c4c26","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v04\n- v04 --supports--> v0c\n- v08 --supports--> v0b\n- v03 --refutes--> v05\n- v02 --supports--> v06\n- v07 --supports--> v02\n- v03 --supports--> v09\n- v09 --supports--> v08\n- v0a --refutes--> v01\n- v0c --supports--> v07\n- v02 --supports--> v03\n- v08 --refutes--> v0d\n- v0b --refutes--> v00\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e82c02456899030abe8873d1a751057d275de2937d9c3dca93bca67040d9d29b","schema":"livegraph.boolean-propagation.v1","spec_hash":"fb809703e8f24e7c4c269a74cd8b6c69a67cc37518b024a5a4d8bb4b58a3fc7f","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-85289284ee06721bf463","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v0a\n- v0b --refutes--> v02\n- v00 --refutes--> v08\n- v03 --refutes--> v01\n- v00 --supports--> v07\n- v0d --refutes--> v00\n- v05 --refutes--> v03\n- v0b --supports--> v0c\n- v07 --supports--> v05\n- v0d --supports--> v04\n- v0d --supports--> v06\n- v01 --refutes--> v09\n- v02 --supports--> v0d\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"51f986b161e369ea7d4cc85db8c40277c102fcfd28a23311e2bec907b3448e47","schema":"livegraph.boolean-propagation.v1","spec_hash":"85289284ee06721bf4630fbc0e78d9b116e65b578211ea670b82541143b6c236","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-90ee6b2c897f30792322","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v08\n- v05 --supports--> v01\n- v03 --refutes--> v09\n- v0b --refutes--> v0a\n- v0d --refutes--> v02\n- v05 --refutes--> v07\n- v04 --refutes--> v05\n- v03 --supports--> v0c\n- v02 --supports--> v06\n- v09 --refutes--> v0b\n- v01 --supports--> v03\n- v06 --refutes--> v00\n- v06 --supports--> v04\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3f179d529c0dc118041474afba23d275753584ee7a27aa325a4f5d2d8b1df2a8","schema":"livegraph.boolean-propagation.v1","spec_hash":"90ee6b2c897f30792322f8b31cae4195da2b289fb1635bdeb1716f8f5506335e","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-4d43786247e24f9452ef","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v05\n- v09 --refutes--> v06\n- v04 --supports--> v07\n- v05 --supports--> v0c\n- v04 --refutes--> v08\n- v00 --supports--> v03\n- v0d --supports--> v04\n- v08 --refutes--> v00\n- v0d --refutes--> v01\n- v06 --refutes--> v0d\n- v00 --refutes--> v0a\n- v08 --supports--> v0b\n- v0c --refutes--> v09\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"37ea1bcac1d9d580bfb2324350823487e1843a75a244f8c7651ace7f8761101d","schema":"livegraph.boolean-propagation.v1","spec_hash":"4d43786247e24f9452ef983f9f7c9e44fe1b3ec660898357b7f85d3ec0c8e056","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-f265a41ae49164222966","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v02\n- v0b --supports--> v07\n- v07 --refutes--> v0a\n- v0b --refutes--> v00\n- v07 --supports--> v03\n- v04 --supports--> v06\n- v06 --supports--> v08\n- v08 --supports--> v0b\n- v02 --supports--> v09\n- v0c --supports--> v04\n- v05 --supports--> v0d\n- v05 --refutes--> v0c\n- v04 --supports--> v01\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"88246c9007b6b86824b252433686caf16630b793448c0f600ad6b426adeea34b","schema":"livegraph.boolean-propagation.v1","spec_hash":"f265a41ae491642229668052a259319bb59cb0566e4ab06020ceb8f9cba274bc","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-0c12b4324ce92f9cf7f0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v0c\n- v0b --refutes--> v0a\n- v07 --refutes--> v03\n- v01 --refutes--> v08\n- v03 --supports--> v04\n- v0d --refutes--> v02\n- v09 --supports--> v07\n- v02 --supports--> v09\n- v03 --refutes--> v06\n- v0b --refutes--> v00\n- v01 --supports--> v05\n- v06 --refutes--> v0b\n- v00 --supports--> v01\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8adc4f45633e9f0de6fb4d4f5926718526e23952f325d3427d44ee11ab18d5f1","schema":"livegraph.boolean-propagation.v1","spec_hash":"0c12b4324ce92f9cf7f01d7a28bd37abf8309952375e7f010f8f66786600370c","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-78d9e6598b432fcae9ad","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v0b\n- v03 --refutes--> v08\n- v0d --refutes--> v09\n- v09 --supports--> v05\n- v0c --refutes--> v0a\n- v08 --supports--> v00\n- v05 --supports--> v06\n- v02 --supports--> v04\n- v00 --supports--> v02\n- v05 --refutes--> v0c\n- v09 --supports--> v01\n- v0c --supports--> v03\n- v04 --supports--> v07\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"da69a425543c347cda9b880672713150b1739bd9434dec482b073d81978e446e","schema":"livegraph.boolean-propagation.v1","spec_hash":"78d9e6598b432fcae9adbb16e2a579e81c72e14b8488edbd026c7146d10f3023","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-da6d215838672a908232","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v00\n- v03 --refutes--> v09\n- v08 --refutes--> v0d\n- v0d --refutes--> v0b\n- v07 --refutes--> v08\n- v09 --supports--> v05\n- v03 --supports--> v0a\n- v09 --refutes--> v02\n- v04 --refutes--> v01\n- v07 --supports--> v0c\n- v06 --supports--> v07\n- v04 --supports--> v06\n- v0b --supports--> v03\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"87b5c52dfe94a1b6eeb88e4c98347e7ed87e57e08e9ec974f4a93d5b6f37858f","schema":"livegraph.boolean-propagation.v1","spec_hash":"da6d215838672a9082329b8278526983140c6fd2fb46c27fdb0a9cb3d2c5b226","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-ab4f2668700f46e970a2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v03\n- v01 --supports--> v02\n- v00 --supports--> v04\n- v02 --refutes--> v06\n- v0b --refutes--> v07\n- v0b --supports--> v0d\n- v07 --refutes--> v0a\n- v01 --supports--> v0c\n- v03 --supports--> v08\n- v04 --refutes--> v05\n- v08 --supports--> v09\n- v04 --refutes--> v01\n- v09 --supports--> v00\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5e78239ac76b7454494bebde9fe40429ae53523d248410592987abb7c01dc231","schema":"livegraph.boolean-propagation.v1","spec_hash":"ab4f2668700f46e970a259a4406f474312db00b7f4c1e6b6dbc1709ab5570485","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":9,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-ee34eae19a433a887872","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v06\n- v03 --supports--> v02\n- v0a --supports--> v03\n- v02 --refutes--> v0b\n- v00 --supports--> v05\n- v0c --supports--> v04\n- v00 --refutes--> v07\n- v04 --supports--> v0d\n- v0d --refutes--> v0a\n- v00 --supports--> v09\n- v07 --refutes--> v0c\n- v02 --refutes--> v01\n- v08 --refutes--> v00\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8cdf400daea67f06a05a3c6992f93318d69e3a610a36b711faffa645def22548","schema":"livegraph.boolean-propagation.v1","spec_hash":"ee34eae19a433a8878722b50e79e3b2a936fb8e11218f4d2cc9359197f7dc86c","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-c548d2c33ad35370d36c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v0b\n- v01 --refutes--> v0a\n- v00 --refutes--> v03\n- v0b --supports--> v09\n- v03 --refutes--> v08\n- v0a --refutes--> v02\n- v09 --refutes--> v01\n- v08 --refutes--> v0c\n- v0b --refutes--> v06\n- v07 --refutes--> v00\n- v0a --refutes--> v0d\n- v04 --refutes--> v05\n- v0a --supports--> v07\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1e47388db592c204b347f834edd51a3850122c9d08e8e5c2f6ea4173bb7a37b8","schema":"livegraph.boolean-propagation.v1","spec_hash":"c548d2c33ad35370d36cf041c74f50caaae1a2649f4f6936892b66e458d2eb01","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":9,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-42b74382679d34ce79c4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v08\n- v05 --supports--> v09\n- v03 --supports--> v0c\n- v03 --refutes--> v05\n- v0d --refutes--> v00\n- v0b --refutes--> v03\n- v07 --supports--> v04\n- v07 --refutes--> v06\n- v09 --supports--> v01\n- v02 --supports--> v07\n- v01 --supports--> v02\n- v06 --refutes--> v0d\n- v03 --supports--> v0a\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dac921219b8ca09bcfe301d24557e8d14655e576837cff7edd854df12feb42b9","schema":"livegraph.boolean-propagation.v1","spec_hash":"42b74382679d34ce79c4dd86a426636e9e95bd7e0cb7485d13674b4f45d00730","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-6da86fc5d1314af0454a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v0b\n- v06 --supports--> v02\n- v0c --supports--> v04\n- v09 --supports--> v0a\n- v0d --supports--> v05\n- v05 --supports--> v0c\n- v00 --refutes--> v08\n- v06 --supports--> v0d\n- v08 --supports--> v06\n- v09 --refutes--> v03\n- v0b --supports--> v01\n- v05 --refutes--> v07\n- v0b --refutes--> v00\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"07d0b4f8e2724c4bbff431e4747a29cce4cd8f32f0e91e415f8a2f5ae92a08f3","schema":"livegraph.boolean-propagation.v1","spec_hash":"6da86fc5d1314af0454a92a795504adb858c0b73b59a33907f7708c6d0121084","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-67b7e399e1052e287e9e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v05\n- v04 --supports--> v08\n- v00 --supports--> v06\n- v05 --supports--> v01\n- v00 --supports--> v0d\n- v0c --supports--> v04\n- v03 --supports--> v07\n- v05 --refutes--> v02\n- v0d --supports--> v0b\n- v0d --refutes--> v03\n- v01 --refutes--> v0c\n- v07 --supports--> v09\n- v03 --supports--> v0a\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c9bd30a42f0f30157d3ebe6025d44c929f2e1c80f9cd931c178d73d5ef9b1e5c","schema":"livegraph.boolean-propagation.v1","spec_hash":"67b7e399e1052e287e9e6a92598e0289621b5be9a25c82e17ec34cd4b5e74cfb","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-acce382ed9d336941dd8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v03\n- v02 --refutes--> v07\n- v09 --supports--> v0d\n- v09 --supports--> v06\n- v01 --supports--> v0a\n- v04 --refutes--> v00\n- v0a --refutes--> v02\n- v00 --refutes--> v0c\n- v09 --supports--> v08\n- v00 --refutes--> v05\n- v03 --supports--> v01\n- v02 --supports--> v0b\n- v06 --refutes--> v04\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1142c99a44fe875389ab8342362dd92bddd9ff1c9bc5cf57dcab7217811407af","schema":"livegraph.boolean-propagation.v1","spec_hash":"acce382ed9d336941dd86e2abc43451bb425302a8df2e227902f9f4651edc333","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":9,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-68ef6615f618160d16cd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v0d\n- v05 --refutes--> v07\n- v0f --refutes--> v08\n- v00 --supports--> v02\n- v0f --refutes--> v0a\n- v07 --supports--> v00\n- v0b --supports--> v05\n- v03 --supports--> v0c\n- v09 --supports--> v01\n- v01 --supports--> v0b\n- v02 --supports--> v0f\n- v0f --supports--> v04\n- v03 --supports--> v09\n- v07 --refutes--> v06\n- v07 --supports--> v0e\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7440a39edbbae5147be0b61d1798dfc5daf7a783f8397ed72151377849da8d84","schema":"livegraph.boolean-propagation.v1","spec_hash":"68ef6615f618160d16cdebc11bc01395a76693bdaa517ed9a962652d58061457","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-544e0489513e2938f71b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v06\n- v06 --refutes--> v0d\n- v09 --supports--> v0f\n- v0f --refutes--> v00\n- v03 --supports--> v04\n- v04 --refutes--> v09\n- v04 --supports--> v0b\n- v0a --supports--> v07\n- v0d --supports--> v03\n- v03 --supports--> v01\n- v00 --refutes--> v05\n- v00 --supports--> v0a\n- v04 --refutes--> v0c\n- v0f --supports--> v02\n- v07 --refutes--> v08\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c58068d2a82cc61eee839ac3806cc71cfa224e9b86331338c3646a238ebd4f53","schema":"livegraph.boolean-propagation.v1","spec_hash":"544e0489513e2938f71be0f3b495cb74761263932e1af0da212ef3ca5cfb1254","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-601004b57050ac110ccd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v01\n- v0a --supports--> v0c\n- v01 --supports--> v09\n- v0f --supports--> v08\n- v08 --supports--> v07\n- v04 --supports--> v0e\n- v00 --refutes--> v05\n- v0d --refutes--> v0b\n- v00 --supports--> v04\n- v01 --supports--> v03\n- v00 --refutes--> v06\n- v04 --refutes--> v0f\n- v03 --supports--> v0a\n- v0c --refutes--> v0d\n- v07 --refutes--> v02\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"10c7df2e0a16ee4d44373f784ebe33f20c130cc579a5a2839c6b660692eb425e","schema":"livegraph.boolean-propagation.v1","spec_hash":"601004b57050ac110ccd80ce722d5197cb224982eb555501973ff4cf1b8fc0f0","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-8e4569fdf455af4d716a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v0f\n- v02 --refutes--> v04\n- v06 --refutes--> v05\n- v0e --supports--> v0c\n- v01 --supports--> v02\n- v02 --refutes--> v0b\n- v08 --supports--> v03\n- v0e --supports--> v00\n- v05 --refutes--> v07\n- v04 --supports--> v0d\n- v01 --supports--> v09\n- v04 --refutes--> v0a\n- v03 --refutes--> v06\n- v0a --supports--> v08\n- v00 --supports--> v01\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"39ce94ff43348b06b14c9b29121312bb6593b7b600317574faf3ca91e3d9529f","schema":"livegraph.boolean-propagation.v1","spec_hash":"8e4569fdf455af4d716a07cf2adcf2bb89c2026a1ec36b525b8e21e56e09b1c8","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-2ec194a44ebda3fcd86a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v03\n- v02 --supports--> v06\n- v07 --supports--> v0a\n- v03 --refutes--> v05\n- v04 --supports--> v0b\n- v0c --supports--> v01\n- v09 --refutes--> v04\n- v03 --supports--> v00\n- v00 --refutes--> v09\n- v06 --supports--> v0c\n- v03 --supports--> v0d\n- v00 --supports--> v08\n- v04 --supports--> v0e\n- v01 --refutes--> v0f\n- v0e --supports--> v07\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"36dfd666059eda23667111c1a1bdeedf8144e6833a109d917400a25e39f8a919","schema":"livegraph.boolean-propagation.v1","spec_hash":"2ec194a44ebda3fcd86a3b82d478f891850fa48252b1f960c86329cc20b25aa1","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-0cd44d0e39fcd9069e40","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v02\n- v04 --refutes--> v0c\n- v02 --refutes--> v08\n- v08 --refutes--> v0b\n- v0c --supports--> v06\n- v0d --refutes--> v07\n- v0c --supports--> v05\n- v0d --supports--> v01\n- v03 --refutes--> v09\n- v01 --refutes--> v0a\n- v09 --supports--> v0d\n- v0e --supports--> v00\n- v01 --supports--> v0e\n- v0d --supports--> v0f\n- v06 --supports--> v03\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ad1ca1f25bb450e8868c691efde46abd52d8e0e0f32ccf27ecc55f1c044bf130","schema":"livegraph.boolean-propagation.v1","spec_hash":"0cd44d0e39fcd9069e4084866233e58bfcdb6a461bb171320ff7aa45c57ab490","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-559dceceaa186d4c2fab","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v03\n- v0b --refutes--> v0e\n- v01 --supports--> v04\n- v02 --refutes--> v0f\n- v0f --supports--> v06\n- v06 --refutes--> v08\n- v0e --supports--> v0c\n- v08 --supports--> v01\n- v08 --supports--> v09\n- v00 --refutes--> v0a\n- v0e --refutes--> v05\n- v04 --refutes--> v0b\n- v01 --supports--> v07\n- v0a --refutes--> v0d\n- v0a --refutes--> v02\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6b385a21fd566f78e1d70e6853ffcebfd45976caf6f7730539e99a7f182609fb","schema":"livegraph.boolean-propagation.v1","spec_hash":"559dceceaa186d4c2fabbab0c2bb06621e1e2c0bf680340e905d1f9e5165fcc6","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-da9b54d47392da0490cf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v01\n- v03 --refutes--> v04\n- v09 --supports--> v0c\n- v04 --refutes--> v06\n- v09 --supports--> v08\n- v01 --refutes--> v0d\n- v06 --supports--> v0b\n- v0f --supports--> v09\n- v0a --supports--> v03\n- v0d --refutes--> v0a\n- v08 --supports--> v0e\n- v0b --refutes--> v05\n- v0b --supports--> v07\n- v03 --supports--> v02\n- v0b --supports--> v00\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f262f234162021f5ce3c3c568080d832028044779e11ab6fc6b1faba0e0a2d19","schema":"livegraph.boolean-propagation.v1","spec_hash":"da9b54d47392da0490cfb0796620a8ba61682183a69b4f29d10656daa6f573ff","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-98182f70c92b2feb9ecb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v0b\n- v02 --supports--> v08\n- v0f --refutes--> v03\n- v0d --supports--> v09\n- v0a --supports--> v07\n- v00 --supports--> v0a\n- v0c --supports--> v02\n- v0c --refutes--> v04\n- v08 --refutes--> v01\n- v00 --supports--> v0e\n- v08 --supports--> v05\n- v09 --refutes--> v00\n- v07 --refutes--> v0c\n- v03 --supports--> v06\n- v03 --supports--> v0d\n\n[VERIFIED ROOT]\nv0f = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"90c291eb33c2f67f89ab653a59712c0d83eac8aac0caad8dfa340044b182b27a","schema":"livegraph.boolean-propagation.v1","spec_hash":"98182f70c92b2feb9ecbe1566b4a39c90b10a367eac550b92ed7472299c95023","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-2e5c4cc370f6d570989d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v0b\n- v0b --refutes--> v06\n- v08 --refutes--> v09\n- v0d --refutes--> v00\n- v03 --supports--> v04\n- v02 --supports--> v07\n- v07 --refutes--> v0d\n- v0b --supports--> v01\n- v00 --supports--> v0f\n- v04 --supports--> v0c\n- v0e --supports--> v08\n- v04 --supports--> v0e\n- v09 --refutes--> v02\n- v0b --supports--> v05\n- v0b --supports--> v0a\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0e9c35bd65b4f1c1be601860c53eb3ce29a2ee4e40becd3e07772c7aa1ab06d5","schema":"livegraph.boolean-propagation.v1","spec_hash":"2e5c4cc370f6d570989ddb5bb004eb21b85e73aa1450b2c77532a88012e39e67","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-262ea7d3faf52c66255f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v06\n- v05 --refutes--> v02\n- v0b --supports--> v0e\n- v07 --supports--> v08\n- v03 --refutes--> v0b\n- v06 --supports--> v09\n- v09 --refutes--> v0a\n- v0a --supports--> v04\n- v0b --refutes--> v0f\n- v05 --supports--> v00\n- v0a --refutes--> v0c\n- v0e --refutes--> v05\n- v04 --supports--> v07\n- v04 --refutes--> v0d\n- v09 --refutes--> v01\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"49cb77dd64b2b37ee1800860eae749c090a2d0ae29d835da3e9db41d5b82edf5","schema":"livegraph.boolean-propagation.v1","spec_hash":"262ea7d3faf52c66255fd6467f7ca4a85e9c6d0ac68e9980ec1162f21eabb2e7","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-4149337ed5971852bcac","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v0c\n- v0a --refutes--> v0b\n- v01 --refutes--> v0f\n- v00 --supports--> v09\n- v00 --refutes--> v08\n- v06 --refutes--> v07\n- v0d --supports--> v05\n- v03 --refutes--> v06\n- v01 --refutes--> v0e\n- v02 --supports--> v0a\n- v08 --supports--> v03\n- v01 --refutes--> v00\n- v07 --supports--> v02\n- v07 --refutes--> v04\n- v0b --supports--> v0d\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c0f9020796e4aeb247d53ad5077f5c708ccba69b89f423a9c9a2e487f7a42087","schema":"livegraph.boolean-propagation.v1","spec_hash":"4149337ed5971852bcac8b7f528c7390148ab04a8f0570885d5bef0c55fe433a","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":10,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-f0b7152018ab2b0ae05f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v07\n- v09 --refutes--> v05\n- v0b --refutes--> v03\n- v07 --refutes--> v0b\n- v01 --refutes--> v06\n- v06 --supports--> v0a\n- v0b --refutes--> v04\n- v0c --refutes--> v0e\n- v0e --refutes--> v09\n- v07 --refutes--> v00\n- v09 --supports--> v01\n- v04 --refutes--> v0d\n- v06 --refutes--> v08\n- v0c --supports--> v02\n- v0d --supports--> v0c\n\n[VERIFIED ROOT]\nv0f = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"62c4c9d61a9509f1d5929443306c73edc359b64259f3c9dc7da75c83c5672215","schema":"livegraph.boolean-propagation.v1","spec_hash":"f0b7152018ab2b0ae05fc11706746c31b428e26dc5c76b82cb69a81ef6e5fae1","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-4c0a87654dc2a17bae15","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v0e\n- v07 --supports--> v06\n- v02 --supports--> v03\n- v0c --supports--> v0b\n- v07 --supports--> v08\n- v05 --refutes--> v04\n- v0e --refutes--> v0a\n- v0b --refutes--> v09\n- v05 --refutes--> v0f\n- v0d --supports--> v02\n- v01 --refutes--> v05\n- v0b --refutes--> v07\n- v08 --supports--> v00\n- v0e --refutes--> v0d\n- v02 --supports--> v0c\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7e94163575e33dfab30d40265abab693ccea7ef823df76be52f8d745fbf20f8f","schema":"livegraph.boolean-propagation.v1","spec_hash":"4c0a87654dc2a17bae15b0ca2b32125047d3dab062fe285d93c1814265ea4477","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-cb8d4e9a32f845382a47","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v0f\n- v07 --supports--> v00\n- v0a --refutes--> v08\n- v0a --supports--> v04\n- v0f --supports--> v06\n- v06 --refutes--> v05\n- v06 --refutes--> v02\n- v0d --supports--> v01\n- v01 --refutes--> v09\n- v00 --refutes--> v0c\n- v0e --supports--> v07\n- v09 --supports--> v03\n- v06 --refutes--> v0e\n- v00 --refutes--> v0b\n- v09 --supports--> v0a\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5fa7ecd539d351e805b331cfb80831441ee9ad4e4c44188009e9195ee5d218b5","schema":"livegraph.boolean-propagation.v1","spec_hash":"cb8d4e9a32f845382a47f2d963249df0fc576ee99adfa2b186e2e765440f6305","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-701ecc6863436e623a17","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v02\n- v09 --supports--> v00\n- v0c --supports--> v0f\n- v01 --supports--> v04\n- v01 --refutes--> v08\n- v09 --supports--> v0b\n- v05 --supports--> v0c\n- v0f --supports--> v09\n- v06 --refutes--> v07\n- v01 --refutes--> v03\n- v0e --refutes--> v0a\n- v0f --supports--> v0d\n- v00 --supports--> v01\n- v04 --supports--> v0e\n- v06 --refutes--> v05\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"11d6d81878d15792cb17198b6d1b6033808f487054a402cd60e6feed21a006fd","schema":"livegraph.boolean-propagation.v1","spec_hash":"701ecc6863436e623a17afe834d0c803f91347697a92e2e874b1acfdd27a465d","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-f643d19e6379dfd29165","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v08\n- v08 --refutes--> v01\n- v0f --refutes--> v04\n- v09 --supports--> v03\n- v04 --supports--> v00\n- v00 --supports--> v05\n- v01 --refutes--> v07\n- v09 --supports--> v0f\n- v07 --refutes--> v0d\n- v0d --refutes--> v0b\n- v0d --supports--> v0c\n- v0b --refutes--> v09\n- v0e --refutes--> v02\n- v07 --refutes--> v06\n- v07 --refutes--> v0a\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b37b3a5ae3a38a5277bc2bcdff47b24e9abcd89fd2680879eb68dad3303ebd84","schema":"livegraph.boolean-propagation.v1","spec_hash":"f643d19e6379dfd291659b9c33e70fee6d9d97fcfa15f1bb9b7bb4f143f61320","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-53d32818086e46e22c27","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v06\n- v0f --refutes--> v09\n- v0c --refutes--> v07\n- v08 --refutes--> v0b\n- v08 --refutes--> v0e\n- v01 --supports--> v04\n- v05 --refutes--> v08\n- v07 --supports--> v0f\n- v00 --refutes--> v0d\n- v04 --refutes--> v0a\n- v0a --supports--> v05\n- v05 --refutes--> v03\n- v00 --supports--> v01\n- v0f --supports--> v02\n- v0e --supports--> v0c\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8504eb3403e1ceb50314a595870f2dfdc2aba3c4ffee576f789fa5956737d75c","schema":"livegraph.boolean-propagation.v1","spec_hash":"53d32818086e46e22c272469a5283f3d0a16ddbc7188cf1f6633a9b5aa28ee71","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-f301a546693e6fae30fc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v0f\n- v01 --refutes--> v06\n- v04 --supports--> v0a\n- v09 --refutes--> v0e\n- v08 --refutes--> v07\n- v01 --supports--> v04\n- v04 --supports--> v0c\n- v00 --supports--> v08\n- v05 --refutes--> v03\n- v03 --supports--> v01\n- v0a --supports--> v09\n- v00 --supports--> v0d\n- v02 --supports--> v00\n- v01 --supports--> v0b\n- v07 --refutes--> v05\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0c8638dc1e116575e0200b207b4e1a9a525410a2140955024e6a67b0ea619579","schema":"livegraph.boolean-propagation.v1","spec_hash":"f301a546693e6fae30fc643f3c419f65cc7c5eee976162bdc85b2d8e8f286db4","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":10,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-e97c8383c9d65c20dafa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v0b\n- v04 --refutes--> v02\n- v07 --supports--> v0e\n- v04 --refutes--> v0f\n- v0c --supports--> v07\n- v04 --refutes--> v09\n- v03 --supports--> v0c\n- v07 --supports--> v06\n- v05 --refutes--> v01\n- v03 --supports--> v00\n- v08 --refutes--> v0d\n- v02 --supports--> v08\n- v0b --supports--> v05\n- v07 --refutes--> v04\n- v0a --supports--> v03\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6325df0b5a17c46bbab254aaa659995bf52b0488884606e2b77e62ce3825243d","schema":"livegraph.boolean-propagation.v1","spec_hash":"e97c8383c9d65c20dafa55b2257267da5c635179860f8cb655bb6cb0ba599d30","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":10,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-dfcab4c2527b21f1c7ff","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v02\n- v0a --supports--> v0b\n- v0a --supports--> v0f\n- v0c --refutes--> v07\n- v05 --supports--> v01\n- v08 --supports--> v0d\n- v03 --supports--> v05\n- v03 --refutes--> v09\n- v08 --refutes--> v03\n- v04 --refutes--> v08\n- v01 --refutes--> v06\n- v0b --refutes--> v04\n- v06 --supports--> v0c\n- v03 --refutes--> v0e\n- v00 --refutes--> v0a\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"63e06286540a40a1f78ef3b915b0dd93b9192358c6ee395221900d6bb170066f","schema":"livegraph.boolean-propagation.v1","spec_hash":"dfcab4c2527b21f1c7ff6b95a267d3e7c4997709290444f4847f82140cc9b843","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":10,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-b123c2e4a81f252326c0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v05\n- v0c --refutes--> v07\n- v0d --refutes--> v0e\n- v0b --supports--> v01\n- v09 --supports--> v0f\n- v0d --supports--> v06\n- v04 --refutes--> v00\n- v00 --supports--> v02\n- v04 --refutes--> v0a\n- v0e --supports--> v0b\n- v00 --supports--> v08\n- v01 --refutes--> v0c\n- v05 --refutes--> v04\n- v04 --refutes--> v03\n- v00 --supports--> v09\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c83196d8f4433907c5e6f02b1e778219a1e385e98ed419328bde0e9cd9a77046","schema":"livegraph.boolean-propagation.v1","spec_hash":"b123c2e4a81f252326c03be8ebb003abbff6351f9a31f2e8b1bc7fd28f01db91","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-5b807cda7eb0911538a1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v0d\n- v0e --supports--> v02\n- v0b --supports--> v03\n- v0f --refutes--> v0a\n- v08 --refutes--> v0e\n- v07 --refutes--> v06\n- v0b --supports--> v01\n- v0e --refutes--> v05\n- v06 --supports--> v04\n- v06 --refutes--> v0c\n- v0e --supports--> v09\n- v01 --supports--> v0f\n- v07 --refutes--> v00\n- v0c --supports--> v0b\n- v02 --supports--> v07\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8223fb0fd57a582ae8393fa63f5181cae3d9f11cc74d347cc9748aef7db9a639","schema":"livegraph.boolean-propagation.v1","spec_hash":"5b807cda7eb0911538a1fec532af05350ebbf26ab29c9f234f9b34d453701a5d","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-d55e6d09950719184be4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v0e\n- v04 --supports--> v06\n- v01 --refutes--> v00\n- v0f --refutes--> v0c\n- v0e --refutes--> v02\n- v0e --supports--> v03\n- v0c --supports--> v07\n- v08 --supports--> v05\n- v0c --supports--> v0a\n- v00 --supports--> v08\n- v09 --refutes--> v01\n- v04 --supports--> v0f\n- v0a --supports--> v0d\n- v08 --refutes--> v04\n- v0d --refutes--> v0b\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5a313d2df67676ef164b3f1131568b667644a3fb30bfb8929319074c83a48565","schema":"livegraph.boolean-propagation.v1","spec_hash":"d55e6d09950719184be4d7c30a07e60d579bb0d2c2ead948491b946aa3fe77c0","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-aff0eaf88742ec34c992","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v0e\n- v0b --refutes--> v05\n- v03 --refutes--> v06\n- v04 --supports--> v0c\n- v02 --refutes--> v0b\n- v00 --refutes--> v0d\n- v00 --supports--> v07\n- v08 --supports--> v0f\n- v0c --supports--> v01\n- v0e --supports--> v08\n- v0b --supports--> v04\n- v09 --supports--> v03\n- v0d --supports--> v0a\n- v08 --supports--> v00\n- v0c --refutes--> v09\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"af5e5d48fc1a5f5f6ca4a8dbbd042dd63031c6e3406b15518f0c1170680e726b","schema":"livegraph.boolean-propagation.v1","spec_hash":"aff0eaf88742ec34c9926b6ae4b6849d3d5bfa0a9ad184c47ee9c89d945331ed","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-4bc322b68bab5030d57b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v03\n- v09 --refutes--> v02\n- v08 --refutes--> v06\n- v01 --supports--> v0c\n- v02 --refutes--> v0d\n- v04 --refutes--> v01\n- v0e --supports--> v04\n- v02 --refutes--> v0e\n- v0b --supports--> v08\n- v02 --supports--> v0f\n- v07 --refutes--> v00\n- v0c --supports--> v0b\n- v00 --supports--> v09\n- v04 --refutes--> v0a\n- v08 --supports--> v05\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"814927f7f4c469be9cdef5ab446d6b01ff999bdb64047a74424441f9cae14bfa","schema":"livegraph.boolean-propagation.v1","spec_hash":"4bc322b68bab5030d57bf52ed33b044cee673fa718642b81427e96c7af79ae72","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-d52c1de901b8330d061f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v0d\n- v01 --refutes--> v0c\n- v0c --supports--> v07\n- v09 --supports--> v05\n- v00 --supports--> v02\n- v0f --refutes--> v06\n- v09 --refutes--> v00\n- v0f --refutes--> v01\n- v02 --supports--> v0f\n- v02 --refutes--> v08\n- v0f --supports--> v04\n- v0b --refutes--> v0e\n- v0e --refutes--> v09\n- v03 --refutes--> v0a\n- v07 --refutes--> v03\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5cce727564c04ec9005a2cdab1a2591346417c0b1667c243f75e68b2e3523e04","schema":"livegraph.boolean-propagation.v1","spec_hash":"d52c1de901b8330d061fd4181c04708b9c1fe48aab8c00afbf32a59c60867c8d","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-8c8aca0c27cd968cd397","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v05\n- v03 --refutes--> v0d\n- v07 --supports--> v0f\n- v0a --refutes--> v07\n- v07 --supports--> v0c\n- v07 --refutes--> v00\n- v0f --supports--> v04\n- v0d --refutes--> v0b\n- v06 --supports--> v0a\n- v0f --supports--> v08\n- v02 --refutes--> v03\n- v09 --refutes--> v02\n- v0e --supports--> v09\n- v06 --supports--> v01\n- v0b --refutes--> v06\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2ae6d45bad101084b65bacec32b7c5fcd7df1a533d0457f81d5713a881628f2c","schema":"livegraph.boolean-propagation.v1","spec_hash":"8c8aca0c27cd968cd397c472e3335a6e2f5dd79eaefc7f282685748cac9aefce","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-3599976e155a9edc53d1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v05\n- v01 --supports--> v0a\n- v0b --refutes--> v09\n- v09 --supports--> v0f\n- v01 --supports--> v02\n- v0d --supports--> v03\n- v0a --supports--> v0b\n- v0a --supports--> v08\n- v00 --supports--> v0e\n- v04 --refutes--> v0d\n- v00 --supports--> v04\n- v0f --refutes--> v07\n- v07 --supports--> v00\n- v0c --supports--> v06\n- v0c --refutes--> v01\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f239f63f08cd42abb2cc5bf05314492e4c31e60f1b5931114f989a1f8de2862b","schema":"livegraph.boolean-propagation.v1","spec_hash":"3599976e155a9edc53d10706226001978fc2d323ecbe7e1f7db37d8dbba9bafe","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-87e6f2c964bae9a9c90c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v01\n- v00 --refutes--> v05\n- v08 --refutes--> v03\n- v03 --refutes--> v00\n- v0c --refutes--> v0a\n- v0f --refutes--> v09\n- v09 --refutes--> v0c\n- v0e --supports--> v02\n- v0c --refutes--> v0e\n- v00 --supports--> v0f\n- v0f --refutes--> v07\n- v04 --supports--> v0d\n- v0b --refutes--> v04\n- v06 --supports--> v0b\n- v0e --refutes--> v06\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e806fbc51cca9c2ea0cf4215ec99985c4545a80a92eccac3e60627cca6408b52","schema":"livegraph.boolean-propagation.v1","spec_hash":"87e6f2c964bae9a9c90cc64a63f556a3c56d20c4bb12017d0126ea01fe35f786","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-c448efd6b67b4903febb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v0f\n- v07 --supports--> v0c\n- v04 --refutes--> v09\n- v07 --supports--> v01\n- v0a --refutes--> v08\n- v04 --supports--> v0d\n- v0d --supports--> v05\n- v08 --supports--> v06\n- v03 --supports--> v0b\n- v0f --refutes--> v00\n- v06 --supports--> v03\n- v0f --refutes--> v04\n- v0b --supports--> v07\n- v07 --supports--> v02\n- v0d --refutes--> v0a\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"53fbd77b08e749aa04804b0d63a254fe2651f34fa01b949d9f3868430ba2f2c6","schema":"livegraph.boolean-propagation.v1","spec_hash":"c448efd6b67b4903febb808908641e53907f49cff94fbe38f85aef58b4dbd4d3","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-e9c298ebb9ddf9d16ec4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v08\n- v0e --refutes--> v09\n- v02 --refutes--> v00\n- v09 --supports--> v0a\n- v08 --refutes--> v0b\n- v02 --supports--> v05\n- v04 --supports--> v0e\n- v02 --refutes--> v0c\n- v00 --refutes--> v06\n- v03 --supports--> v04\n- v0b --supports--> v03\n- v09 --supports--> v0f\n- v04 --refutes--> v01\n- v06 --supports--> v07\n- v08 --supports--> v0d\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a1ebf847c6e206b74507f1b7a5d9c9d90ff26a2677b2d2223394ef35244faff4","schema":"livegraph.boolean-propagation.v1","spec_hash":"e9c298ebb9ddf9d16ec4923319bc3b6180df04471b20b3de0c2b9be13654bb1e","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-5cca0b012575f9612769","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v0c\n- v02 --supports--> v05\n- v0a --refutes--> v06\n- v0f --refutes--> v09\n- v05 --supports--> v00\n- v02 --supports--> v0d\n- v0c --refutes--> v04\n- v08 --supports--> v0f\n- v03 --refutes--> v01\n- v01 --refutes--> v02\n- v00 --refutes--> v0e\n- v0c --refutes--> v07\n- v0f --refutes--> v0a\n- v04 --refutes--> v08\n- v03 --refutes--> v0b\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"93f5b63a8721a71e0b34ce0da6a3c83f0879c42a8db004a480b4ed29512ae8f2","schema":"livegraph.boolean-propagation.v1","spec_hash":"5cca0b012575f96127693e8179001bfcde2a5752eeeffdb8a5832dba10f65a46","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-80660b86ec92e75b9ff7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v05\n- v0b --supports--> v01\n- v0a --refutes--> v09\n- v01 --refutes--> v00\n- v06 --supports--> v08\n- v05 --refutes--> v07\n- v02 --refutes--> v0b\n- v0c --supports--> v06\n- v00 --refutes--> v0f\n- v05 --refutes--> v02\n- v00 --supports--> v0d\n- v04 --refutes--> v0c\n- v08 --supports--> v0a\n- v0b --supports--> v03\n- v0a --supports--> v0e\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"50042663a5472050564be5a5762ab5ebd20baa7bf684d14cb8ca24fa2f2d9d0a","schema":"livegraph.boolean-propagation.v1","spec_hash":"80660b86ec92e75b9ff7f4d46ef9af0b49c71e97dc2e25473be555142cad1c55","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":10,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-88439aa7aab2d3855a25","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v02\n- v0d --refutes--> v01\n- v01 --refutes--> v0e\n- v08 --refutes--> v07\n- v02 --supports--> v0c\n- v0e --refutes--> v06\n- v0e --supports--> v0b\n- v06 --refutes--> v09\n- v04 --refutes--> v00\n- v08 --supports--> v0a\n- v02 --supports--> v08\n- v09 --refutes--> v04\n- v0d --refutes--> v0f\n- v04 --supports--> v03\n- v07 --refutes--> v0d\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3a62aa5e53a4a5f61b98115df4e1778a3d8bf9b17068f563c868d7e404b1ae45","schema":"livegraph.boolean-propagation.v1","spec_hash":"88439aa7aab2d3855a253ccb4e494d935ba3102c4899207ac55c753d51ff1142","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-9fe308645c80570830c4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v0c\n- v09 --refutes--> v0f\n- v01 --supports--> v06\n- v06 --refutes--> v02\n- v07 --supports--> v08\n- v0d --supports--> v00\n- v02 --supports--> v0b\n- v07 --refutes--> v0e\n- v0c --refutes--> v09\n- v04 --supports--> v0a\n- v0f --refutes--> v03\n- v0f --refutes--> v01\n- v04 --supports--> v0d\n- v07 --supports--> v05\n- v02 --supports--> v04\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f1f27ea525409ce46d61470f5e3eb7df91e37a5717bf326529ade21d8b120a18","schema":"livegraph.boolean-propagation.v1","spec_hash":"9fe308645c80570830c42f1760e0fe6447baf550a5a147afb3bf5cd534b41337","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-35ae059a966c7cf125e1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v0d\n- v0a --supports--> v09\n- v09 --supports--> v03\n- v0a --supports--> v08\n- v07 --supports--> v05\n- v05 --supports--> v00\n- v0b --supports--> v0c\n- v01 --refutes--> v0e\n- v0c --supports--> v0a\n- v05 --supports--> v02\n- v0f --refutes--> v04\n- v01 --refutes--> v07\n- v00 --supports--> v0f\n- v0b --supports--> v06\n- v03 --supports--> v01\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f27c52c704c4eac4f28e9e0ca80169190bcc61d39532abcd1bd85079736e75ad","schema":"livegraph.boolean-propagation.v1","spec_hash":"35ae059a966c7cf125e11a67c0a6d73ee74bea78c4cef5c1697e63c1ad9bb88c","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-7fdcdc73b19652449286","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v0c\n- v05 --refutes--> v03\n- v00 --supports--> v07\n- v00 --refutes--> v0b\n- v06 --supports--> v00\n- v01 --supports--> v06\n- v05 --refutes--> v08\n- v0e --supports--> v01\n- v00 --supports--> v09\n- v0c --refutes--> v0d\n- v0b --refutes--> v0f\n- v0b --refutes--> v02\n- v0a --refutes--> v04\n- v04 --refutes--> v0e\n- v05 --refutes--> v0a\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9d3b06db0622d98bb4e23e1401a9eee1c987a30a2d5f270ccafe9950e4d865f3","schema":"livegraph.boolean-propagation.v1","spec_hash":"7fdcdc73b196524492868857650b402d2a92cfbf85e2c1a13a9861caf931d054","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-309c8ce6a6b5b0ec1cc6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v0a\n- v03 --refutes--> v09\n- v06 --supports--> v0e\n- v0b --refutes--> v0f\n- v09 --refutes--> v02\n- v08 --refutes--> v0c\n- v02 --refutes--> v06\n- v04 --supports--> v0b\n- v09 --supports--> v01\n- v0f --refutes--> v08\n- v05 --refutes--> v03\n- v03 --supports--> v07\n- v0f --supports--> v00\n- v06 --refutes--> v04\n- v0c --refutes--> v0d\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"118d12963577a9f959207f9f537b594204fc4a91bd7f7f2cf339cb4b908aebbc","schema":"livegraph.boolean-propagation.v1","spec_hash":"309c8ce6a6b5b0ec1cc63e0164082d51754ce9f5bf320c007cd20bb494664b2f","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-56e9edfe7ca96c00c5bc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v02\n- v04 --refutes--> v07\n- v0a --supports--> v09\n- v09 --supports--> v04\n- v08 --refutes--> v0e\n- v0a --supports--> v06\n- v0a --refutes--> v0c\n- v0b --refutes--> v03\n- v07 --refutes--> v0d\n- v0e --supports--> v0b\n- v0b --refutes--> v00\n- v00 --supports--> v0a\n- v05 --refutes--> v01\n- v07 --refutes--> v0f\n- v01 --refutes--> v08\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"199b00f7f42c6d17d9f533ebedc2ea9230dbf5177f69cefb818a3d934078abad","schema":"livegraph.boolean-propagation.v1","spec_hash":"56e9edfe7ca96c00c5bcd9b2e234902a69c0a387b163977c76927ec5824061b7","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-d83bc5704e4fedfb7a05","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v0a\n- v0c --supports--> v0e\n- v09 --supports--> v02\n- v0b --refutes--> v04\n- v09 --refutes--> v08\n- v03 --supports--> v0c\n- v0e --refutes--> v0b\n- v0b --supports--> v0d\n- v06 --supports--> v00\n- v00 --supports--> v09\n- v05 --refutes--> v03\n- v07 --refutes--> v06\n- v05 --refutes--> v01\n- v08 --supports--> v05\n- v05 --refutes--> v0f\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a92a9a44e561e85e76f06956e98ad76c5b2d692053062af4b7bad3f8f8beeeec","schema":"livegraph.boolean-propagation.v1","spec_hash":"d83bc5704e4fedfb7a05a41f645d352985fbf3011120257148e42003978cad2c","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-c830cefae9ef9e1b4cfc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v06\n- v0b --supports--> v02\n- v02 --refutes--> v07\n- v0f --supports--> v03\n- v03 --refutes--> v0c\n- v03 --refutes--> v00\n- v0e --supports--> v0d\n- v08 --refutes--> v0a\n- v03 --supports--> v08\n- v05 --refutes--> v0b\n- v06 --supports--> v0f\n- v0b --refutes--> v09\n- v07 --supports--> v0e\n- v08 --supports--> v05\n- v0e --refutes--> v04\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e69dc76e4d74d07f42ed5ea12a3d4413e8ae61b3869d07da98294d33d41ea8fd","schema":"livegraph.boolean-propagation.v1","spec_hash":"c830cefae9ef9e1b4cfc9f122bd97b8d4afa5747c7aa3d7483662f883374a160","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-e302b79430eeb4bca8ad","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v09\n- v0a --supports--> v07\n- v01 --refutes--> v03\n- v0d --supports--> v02\n- v08 --refutes--> v0e\n- v02 --refutes--> v08\n- v06 --refutes--> v0a\n- v06 --refutes--> v04\n- v09 --refutes--> v06\n- v00 --supports--> v01\n- v03 --refutes--> v0b\n- v0a --refutes--> v05\n- v0a --refutes--> v00\n- v06 --supports--> v0f\n- v02 --refutes--> v0c\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1e28103302de4012f081ce161dd9727c1ec39b9bfa5362d3f8394fde14e37611","schema":"livegraph.boolean-propagation.v1","spec_hash":"e302b79430eeb4bca8adacb515152e95dcc291d860899ba0986814485a40d880","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-1c132f309a2b92b19141","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v05\n- v01 --supports--> v0b\n- v0e --supports--> v0c\n- v0e --refutes--> v02\n- v0d --refutes--> v06\n- v09 --refutes--> v00\n- v03 --supports--> v07\n- v07 --refutes--> v0e\n- v08 --refutes--> v0d\n- v02 --supports--> v08\n- v0f --refutes--> v01\n- v0d --refutes--> v0a\n- v01 --refutes--> v04\n- v04 --refutes--> v09\n- v0d --supports--> v0f\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5e2d81ab572eb15d01013a78eb16f2469e561b75ca8d3753cd6c2f5d09d8db1d","schema":"livegraph.boolean-propagation.v1","spec_hash":"1c132f309a2b92b1914179ec7460bee8b3b4e1277c5315cf3ab98118f8c2de3c","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-3ba049407119c7fac2a9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v08\n- v04 --refutes--> v00\n- v02 --supports--> v05\n- v05 --refutes--> v0f\n- v06 --refutes--> v07\n- v02 --refutes--> v0b\n- v06 --supports--> v01\n- v06 --supports--> v09\n- v03 --refutes--> v0e\n- v00 --supports--> v06\n- v04 --refutes--> v0d\n- v01 --refutes--> v03\n- v06 --refutes--> v0c\n- v0e --refutes--> v0a\n- v08 --supports--> v02\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"96cfbcc2cd3b90d60264853b7b4f560edc87337f91e75f30c6c66444ea789a9f","schema":"livegraph.boolean-propagation.v1","spec_hash":"3ba049407119c7fac2a993deb4a9ac903de05bdc3a32535e247e732340bfaef5","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-c68d459d0203899f93f1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v0f\n- v0c --supports--> v09\n- v0d --supports--> v0e\n- v0f --refutes--> v0b\n- v0e --supports--> v0a\n- v0c --supports--> v05\n- v0f --supports--> v07\n- v06 --supports--> v01\n- v08 --supports--> v0d\n- v06 --supports--> v00\n- v0a --supports--> v02\n- v0c --supports--> v03\n- v01 --supports--> v0c\n- v09 --refutes--> v08\n- v07 --supports--> v04\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"79e5f2d5ba10dcd70f23666185335eeca48e156557d81e03417ce9f65b041e74","schema":"livegraph.boolean-propagation.v1","spec_hash":"c68d459d0203899f93f1a4b4a0943af1fd1b5124bc4625aff407a7aad1f35f2b","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-32264c255f99c11357d2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v01\n- v03 --refutes--> v00\n- v04 --supports--> v03\n- v06 --supports--> v02\n- v06 --supports--> v07\n- v07 --supports--> v0c\n- v03 --refutes--> v0d\n- v0a --refutes--> v05\n- v01 --supports--> v0b\n- v0f --supports--> v0a\n- v0e --refutes--> v09\n- v0c --supports--> v0f\n- v0b --supports--> v06\n- v03 --refutes--> v08\n- v0e --refutes--> v04\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"90954b753e3cd94223db7495495025ffe4ff4ef529529166354967d94476525e","schema":"livegraph.boolean-propagation.v1","spec_hash":"32264c255f99c11357d2f02ae98d4e71c6b5ca773400cfca9ce38e5304855ffa","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-34ade07f1893a4050852","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v06\n- v0b --supports--> v03\n- v0d --supports--> v07\n- v09 --supports--> v0d\n- v03 --supports--> v0e\n- v07 --refutes--> v01\n- v0d --supports--> v08\n- v04 --supports--> v00\n- v00 --refutes--> v0f\n- v06 --supports--> v09\n- v03 --supports--> v0c\n- v0a --refutes--> v02\n- v0a --supports--> v0b\n- v07 --refutes--> v0a\n- v05 --refutes--> v04\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"87e632dbcb86f0afd84a394aaaf873646e5f8466f914087ad3688a4f7d81ef32","schema":"livegraph.boolean-propagation.v1","spec_hash":"34ade07f1893a4050852c6d72bf78e696f80db7e11b61a6c28e377316d6c795a","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":10,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-7d4f6ed476daa098b458","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v05\n- v01 --supports--> v06\n- v07 --supports--> v0f\n- v00 --refutes--> v0e\n- v0c --supports--> v02\n- v09 --supports--> v08\n- v0f --supports--> v03\n- v00 --supports--> v0b\n- v0c --supports--> v0a\n- v01 --supports--> v0d\n- v0e --refutes--> v09\n- v09 --refutes--> v0c\n- v03 --supports--> v00\n- v04 --supports--> v07\n- v06 --supports--> v04\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"73a0516a1ab11841dfd07b46509e74ebaadf4e599ac6b9ebf2236980513db73e","schema":"livegraph.boolean-propagation.v1","spec_hash":"7d4f6ed476daa098b458c4407be285d84ea4a5ded23679bdbbcfa476903307e9","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-66652e79c903fbf4aacc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v0c\n- v0f --refutes--> v05\n- v0e --refutes--> v00\n- v0f --refutes--> v03\n- v0a --refutes--> v0d\n- v0b --supports--> v0a\n- v06 --supports--> v0e\n- v00 --refutes--> v07\n- v00 --refutes--> v09\n- v01 --supports--> v06\n- v07 --supports--> v0f\n- v00 --refutes--> v04\n- v02 --refutes--> v08\n- v02 --refutes--> v0b\n- v0d --supports--> v01\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"24ff2abb01f212bc2aac032b41088f43b1f86d25710a9810dd4da3db4499a2b0","schema":"livegraph.boolean-propagation.v1","spec_hash":"66652e79c903fbf4aacc616fbe1e083e41d43c68796a265e5d8821812e5136a8","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-78d16767cff3de539054","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v03\n- v0d --refutes--> v02\n- v0a --refutes--> v01\n- v09 --supports--> v0f\n- v09 --supports--> v04\n- v02 --supports--> v0b\n- v06 --refutes--> v08\n- v06 --supports--> v0d\n- v0d --supports--> v05\n- v01 --refutes--> v00\n- v0f --refutes--> v0e\n- v0b --refutes--> v0c\n- v07 --refutes--> v0a\n- v01 --supports--> v06\n- v0e --refutes--> v07\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"382254e74cda12735743e2a048abe2e074ccd19be80f066a3434fba14bd5495b","schema":"livegraph.boolean-propagation.v1","spec_hash":"78d16767cff3de539054e7de64ae6a1299811eac5915a52469d8b8c86f496b20","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-3ef83e4cff6025a7df4e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v07\n- v0f --refutes--> v0d\n- v02 --refutes--> v06\n- v0a --refutes--> v03\n- v04 --refutes--> v0f\n- v02 --supports--> v09\n- v0d --refutes--> v01\n- v05 --supports--> v0b\n- v0f --supports--> v0e\n- v07 --supports--> v0a\n- v0d --supports--> v08\n- v0b --refutes--> v00\n- v0b --refutes--> v0c\n- v00 --refutes--> v04\n- v0a --refutes--> v02\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ca87ae805ba15635f5147b0cb99cf1fedf9c3e9c7246a30ec1227e7705472ebc","schema":"livegraph.boolean-propagation.v1","spec_hash":"3ef83e4cff6025a7df4e107a599316ac15a854089d13ab9b1dc950a1f3e65f31","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-8792063c9319f962fe7b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v0c\n- v06 --refutes--> v01\n- v06 --refutes--> v08\n- v06 --supports--> v0f\n- v06 --supports--> v02\n- v0e --supports--> v0a\n- v04 --refutes--> v06\n- v0a --supports--> v0b\n- v05 --supports--> v0d\n- v05 --refutes--> v03\n- v04 --refutes--> v07\n- v0c --refutes--> v05\n- v0f --refutes--> v0e\n- v00 --refutes--> v09\n- v0b --refutes--> v00\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"47e12b41922030247900ae2635dfd72c4cd8c5640821cb2efcff899a9948a11d","schema":"livegraph.boolean-propagation.v1","spec_hash":"8792063c9319f962fe7be6c200b87d0c63337fbb3e69c498fadbcbaed4748abf","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-5784b5442cd00e1fa81b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v02\n- v0c --refutes--> v00\n- v00 --refutes--> v0e\n- v02 --refutes--> v0a\n- v05 --refutes--> v03\n- v04 --supports--> v0b\n- v03 --refutes--> v0f\n- v0a --refutes--> v08\n- v07 --supports--> v01\n- v0c --refutes--> v09\n- v00 --refutes--> v04\n- v02 --supports--> v0d\n- v0a --supports--> v0c\n- v06 --refutes--> v05\n- v0f --supports--> v07\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2a9fba915a783aaf1ed23c9c692847924a7f6b607cdf3e71d79347443b3e01e7","schema":"livegraph.boolean-propagation.v1","spec_hash":"5784b5442cd00e1fa81b9b6c785b9bc7097cae907db681f012870f02222c7798","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-629308b6ec6120541b88","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v00\n- v06 --supports--> v0f\n- v0e --supports--> v09\n- v0a --refutes--> v07\n- v0e --supports--> v0c\n- v01 --supports--> v0d\n- v03 --supports--> v05\n- v02 --supports--> v01\n- v03 --supports--> v08\n- v04 --refutes--> v0a\n- v0d --refutes--> v04\n- v0d --refutes--> v0b\n- v07 --supports--> v03\n- v0e --refutes--> v02\n- v0f --refutes--> v0e\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1d31e226ad7db79126a8d54d9e08ac3161d72c7c7ad15c31e894739f6dad45a1","schema":"livegraph.boolean-propagation.v1","spec_hash":"629308b6ec6120541b88be2649d85fa16eac6c1af462cdc1400127c4d72db2c0","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-9c6c962cf5ce8edb5724","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v00\n- v08 --refutes--> v09\n- v07 --supports--> v0d\n- v06 --supports--> v0b\n- v0a --refutes--> v0f\n- v08 --refutes--> v0e\n- v01 --refutes--> v02\n- v06 --refutes--> v04\n- v0f --supports--> v01\n- v0d --supports--> v0c\n- v0d --refutes--> v0a\n- v0e --refutes--> v06\n- v07 --refutes--> v03\n- v05 --refutes--> v07\n- v00 --refutes--> v05\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e567c828eb8d8eee9062dad89da5431ddf53635b36fd3acaf6e400db9dbffb15","schema":"livegraph.boolean-propagation.v1","spec_hash":"9c6c962cf5ce8edb5724f7f4590018873168e9eb2a47d5ee590ab0695467ea28","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-9fe39ca37c0bf245b94c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v0a\n- v08 --refutes--> v06\n- v0e --refutes--> v0d\n- v0d --refutes--> v02\n- v08 --refutes--> v03\n- v03 --refutes--> v0b\n- v07 --refutes--> v0e\n- v0d --supports--> v09\n- v03 --refutes--> v07\n- v0a --refutes--> v0c\n- v0f --supports--> v01\n- v0a --supports--> v00\n- v00 --supports--> v0f\n- v03 --refutes--> v05\n- v04 --supports--> v08\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bf3f217f30ce5947fa378f08774c41731077c4ac1ea0d3be9cba565272930f02","schema":"livegraph.boolean-propagation.v1","spec_hash":"9fe39ca37c0bf245b94c20f0b7d33fefbe75dabbd90c79ba90b5c1b3081281c5","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-6a7e3575e984b0fc3d44","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v06\n- v0c --refutes--> v07\n- v00 --refutes--> v0d\n- v00 --refutes--> v0f\n- v0a --supports--> v02\n- v0b --supports--> v05\n- v04 --supports--> v0c\n- v0f --refutes--> v04\n- v06 --supports--> v0b\n- v05 --refutes--> v0a\n- v07 --supports--> v09\n- v0b --supports--> v0e\n- v08 --supports--> v01\n- v00 --refutes--> v03\n- v0a --refutes--> v00\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d82cafe286029a7c5e563fcbc74eaa33dbafa03ec60a30ae81969ac30a00dade","schema":"livegraph.boolean-propagation.v1","spec_hash":"6a7e3575e984b0fc3d443b7ad842491f3470f482e1300d8d6dc75a76d2709ef4","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-74d19e1a1a66cfd9f9ff","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v09\n- v04 --refutes--> v0b\n- v06 --supports--> v05\n- v06 --refutes--> v00\n- v0c --refutes--> v0a\n- v0e --supports--> v06\n- v06 --refutes--> v03\n- v08 --supports--> v01\n- v0f --refutes--> v08\n- v09 --refutes--> v0f\n- v08 --refutes--> v0e\n- v0a --refutes--> v02\n- v0c --refutes--> v07\n- v08 --supports--> v0d\n- v0b --refutes--> v0c\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7e9ada772f5dd9d9667e5cacd92dd8a5893e3e07045e84110fa4896b2d193765","schema":"livegraph.boolean-propagation.v1","spec_hash":"74d19e1a1a66cfd9f9ff57bb0e71c26b6ba371a68dd9c61434b771f8f4526c6f","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-e7527caf424c13ccaa6d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v0e\n- v01 --supports--> v05\n- v06 --refutes--> v0f\n- v0b --refutes--> v0d\n- v02 --supports--> v0b\n- v0e --supports--> v08\n- v03 --supports--> v06\n- v0b --supports--> v00\n- v0c --refutes--> v02\n- v06 --refutes--> v01\n- v02 --supports--> v09\n- v01 --refutes--> v07\n- v04 --refutes--> v0a\n- v0a --supports--> v03\n- v0e --supports--> v0c\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ec829919bacbd0b50b27ce3c958cc4b9b3997897294c50a969132731cd522ce7","schema":"livegraph.boolean-propagation.v1","spec_hash":"e7527caf424c13ccaa6d755d86478272f8d73b60e0c83b0e73901a55175e95c0","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-9bb6a89e384b58be54cb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v02\n- v0f --supports--> v07\n- v08 --supports--> v00\n- v05 --refutes--> v0f\n- v0c --refutes--> v06\n- v0e --supports--> v0c\n- v03 --supports--> v0d\n- v06 --supports--> v03\n- v03 --refutes--> v08\n- v0f --refutes--> v04\n- v00 --supports--> v05\n- v01 --supports--> v0e\n- v00 --refutes--> v0a\n- v09 --refutes--> v01\n- v03 --refutes--> v0b\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fc4709a14547ce1d73572acdff34a465510ab4f3c2237b9d430bf1ea023c80ab","schema":"livegraph.boolean-propagation.v1","spec_hash":"9bb6a89e384b58be54cbff6f7c3576e8b79eff5835f2b88930ab2e0468075dc3","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-ae4a94c655aecbca7efd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v03\n- v06 --supports--> v0e\n- v02 --supports--> v06\n- v06 --refutes--> v07\n- v03 --supports--> v0c\n- v05 --refutes--> v00\n- v0d --supports--> v02\n- v01 --refutes--> v0b\n- v07 --refutes--> v08\n- v06 --refutes--> v0f\n- v09 --refutes--> v05\n- v06 --supports--> v0a\n- v08 --supports--> v01\n- v03 --refutes--> v0d\n- v03 --refutes--> v04\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"daed366d00ce514d0ceebab78387990f9ab1757488fbfab2c28b7f358943d171","schema":"livegraph.boolean-propagation.v1","spec_hash":"ae4a94c655aecbca7efdab83173ff3ad76dce57d9a8734eda82b9e71391ac849","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":10,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-35ca0da8102d289249fc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v03\n- v0e --supports--> v0b\n- v02 --refutes--> v06\n- v0f --supports--> v07\n- v06 --supports--> v09\n- v00 --refutes--> v08\n- v00 --supports--> v01\n- v0e --refutes--> v0d\n- v09 --supports--> v04\n- v02 --refutes--> v0c\n- v0a --refutes--> v00\n- v0b --supports--> v0a\n- v04 --supports--> v0f\n- v0a --refutes--> v05\n- v07 --refutes--> v0e\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"65e12f835cec9a22962bd3ba74c8fe5f0ab9c9f89c73184d7e559f35328b7bb9","schema":"livegraph.boolean-propagation.v1","spec_hash":"35ca0da8102d289249fcae7334ecf8ffeb60cfbffda48e13119ad0badcbdca21","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-a3a044b164913c89ce68","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v0f\n- v03 --supports--> v02\n- v0b --supports--> v07\n- v08 --refutes--> v03\n- v02 --refutes--> v0c\n- v02 --refutes--> v04\n- v09 --supports--> v00\n- v06 --supports--> v01\n- v00 --refutes--> v05\n- v06 --supports--> v0a\n- v07 --supports--> v08\n- v0f --supports--> v09\n- v04 --refutes--> v06\n- v0f --refutes--> v0e\n- v03 --refutes--> v0d\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"feb57631dae797e883bde4018bb30a1b8f2a0ca981e9085c4d8c631e8b6dbe58","schema":"livegraph.boolean-propagation.v1","spec_hash":"a3a044b164913c89ce687b975f240569151cb41d898b32b98e09c1967870a8a8","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-4dbb73c874917bd28e94","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v07\n- v0e --refutes--> v05\n- v00 --refutes--> v04\n- v0a --refutes--> v0d\n- v02 --refutes--> v0c\n- v06 --refutes--> v01\n- v0d --supports--> v0e\n- v04 --refutes--> v02\n- v0e --supports--> v0b\n- v0a --refutes--> v09\n- v06 --supports--> v03\n- v03 --supports--> v0a\n- v04 --supports--> v08\n- v0a --supports--> v0f\n- v05 --refutes--> v00\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c6dba582fbf8358eeac62bd2bc38a4c998acfbe16f5a744a64436cb2a613b0fe","schema":"livegraph.boolean-propagation.v1","spec_hash":"4dbb73c874917bd28e9478c004cd474dc835aac91964c5a54c3107dfc0561555","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-1c459125dc28b48b8295","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v0c\n- v08 --supports--> v0e\n- v09 --supports--> v0d\n- v0f --supports--> v05\n- v0d --refutes--> v03\n- v0a --supports--> v0f\n- v00 --supports--> v02\n- v05 --supports--> v0b\n- v06 --refutes--> v08\n- v0d --refutes--> v01\n- v06 --refutes--> v07\n- v0d --supports--> v06\n- v00 --refutes--> v04\n- v0e --supports--> v0a\n- v0b --supports--> v00\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c46718a213a53bf78ba5c77bb613a07b43266627b719de5b7cb503ecae12e351","schema":"livegraph.boolean-propagation.v1","spec_hash":"1c459125dc28b48b82956b53b020eea6a8f291a504c43cb7ebed6fd6f7ce98ef","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-28db3d058e05667f2026","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v06\n- v0f --refutes--> v07\n- v05 --refutes--> v01\n- v0b --supports--> v0e\n- v08 --refutes--> v0c\n- v01 --supports--> v0a\n- v05 --supports--> v0d\n- v08 --supports--> v04\n- v0f --supports--> v05\n- v00 --refutes--> v02\n- v0a --refutes--> v0b\n- v03 --refutes--> v00\n- v0b --refutes--> v09\n- v0e --supports--> v08\n- v02 --supports--> v0f\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"35f8e6ea25739ee741c16009b9eb052941a6306b7586c33dfebcc11293880de0","schema":"livegraph.boolean-propagation.v1","spec_hash":"28db3d058e05667f202642a8df053d8ecc943a2aa69792ef53a96d433285ae66","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-9c376c8cd41f7d332afa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v03\n- v0c --refutes--> v00\n- v03 --supports--> v06\n- v05 --refutes--> v0d\n- v0b --refutes--> v0f\n- v0c --refutes--> v09\n- v0c --supports--> v0a\n- v07 --refutes--> v05\n- v02 --supports--> v0b\n- v01 --supports--> v0e\n- v06 --supports--> v07\n- v0a --refutes--> v01\n- v05 --refutes--> v0c\n- v0e --refutes--> v08\n- v0e --refutes--> v04\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"eadd24d8fe52084ac63e419c551ad38c217075d06bc23410f39ee360318f072c","schema":"livegraph.boolean-propagation.v1","spec_hash":"9c376c8cd41f7d332afaf01d3305186983234490b040fe00cc9c1413aa1ae307","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-2e8d770df17c3a15ed33","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v0b\n- v05 --refutes--> v0a\n- v0f --refutes--> v08\n- v0c --refutes--> v03\n- v0c --refutes--> v07\n- v0d --refutes--> v0c\n- v00 --refutes--> v0f\n- v0e --supports--> v02\n- v0d --supports--> v01\n- v0c --supports--> v06\n- v0a --supports--> v0d\n- v0b --refutes--> v05\n- v04 --refutes--> v00\n- v06 --supports--> v0e\n- v0e --refutes--> v09\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f3bfa10f1cb3cc013daec6610ef21a9dd24c33e57274fe022cdc36be4aaaa682","schema":"livegraph.boolean-propagation.v1","spec_hash":"2e8d770df17c3a15ed3321789f0bbbb5e5808ab410d99216612326b8490b4350","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-07e180524f54a038798f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v0d\n- v09 --refutes--> v03\n- v07 --supports--> v06\n- v04 --supports--> v09\n- v07 --refutes--> v0e\n- v05 --supports--> v0a\n- v03 --refutes--> v0f\n- v02 --supports--> v01\n- v0b --refutes--> v07\n- v0e --refutes--> v04\n- v01 --supports--> v0c\n- v02 --refutes--> v00\n- v0d --refutes--> v02\n- v03 --supports--> v08\n- v01 --supports--> v05\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f808c6925ec97d24ff3362fd267142815101230a7ec926a6c9d71a09a2d648da","schema":"livegraph.boolean-propagation.v1","spec_hash":"07e180524f54a038798f1175ca33ce3ef02d3965e5ca301024c0e88f635e3729","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-b4d69040e92d065ec55b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v0d\n- v0c --supports--> v0b\n- v09 --refutes--> v0e\n- v05 --supports--> v00\n- v08 --supports--> v09\n- v00 --supports--> v0f\n- v03 --supports--> v04\n- v01 --refutes--> v0c\n- v0c --supports--> v07\n- v0c --refutes--> v08\n- v0f --refutes--> v06\n- v09 --refutes--> v0a\n- v0e --supports--> v05\n- v03 --supports--> v01\n- v02 --supports--> v03\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6c2975766366dbc62d543a89b7d4b559c088165870c740a5133a832f49341f65","schema":"livegraph.boolean-propagation.v1","spec_hash":"b4d69040e92d065ec55bf7407660071759412ad95d18ffca6cbf579d8f0c029b","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-f49cd56f21b44b0b7a9b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v0b\n- v07 --refutes--> v0c\n- v01 --refutes--> v06\n- v06 --supports--> v0a\n- v08 --supports--> v02\n- v06 --refutes--> v05\n- v00 --refutes--> v08\n- v03 --supports--> v07\n- v09 --supports--> v03\n- v08 --supports--> v09\n- v0b --supports--> v00\n- v07 --refutes--> v0e\n- v0c --supports--> v01\n- v03 --supports--> v0d\n- v08 --supports--> v0f\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8654a097bd99beb1bdce7e25dd0a3e02584c0ea12ad1c9b7e47751437b71e5b3","schema":"livegraph.boolean-propagation.v1","spec_hash":"f49cd56f21b44b0b7a9bffd6b6296d3788bc6a230c89edb178dd041d7d08f081","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":10,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-49e418eab92203419226","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v09\n- v0b --supports--> v03\n- v01 --supports--> v0a\n- v02 --refutes--> v0c\n- v0c --refutes--> v05\n- v0b --refutes--> v0f\n- v01 --refutes--> v04\n- v01 --refutes--> v0e\n- v0c --refutes--> v0b\n- v0d --refutes--> v06\n- v0b --supports--> v07\n- v06 --supports--> v01\n- v09 --supports--> v08\n- v00 --refutes--> v0d\n- v0f --refutes--> v00\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a5ab21cd9efab4b26cbf9fd4d2175c2cb2f2ecf2b9e11752c77221440c1ffd51","schema":"livegraph.boolean-propagation.v1","spec_hash":"49e418eab9220341922608da0ba2e04156973e34a27714a1d9fc6e3772c42ee1","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-70b6526e5ecb0da42a60","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v09\n- v01 --supports--> v08\n- v0e --refutes--> v0a\n- v0f --refutes--> v05\n- v0d --supports--> v0b\n- v02 --refutes--> v0d\n- v01 --supports--> v03\n- v01 --supports--> v00\n- v0f --supports--> v01\n- v0c --supports--> v07\n- v09 --refutes--> v02\n- v00 --supports--> v0e\n- v0e --refutes--> v06\n- v09 --supports--> v04\n- v07 --supports--> v0f\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1a09942d5877f72680672bf8b7f678f7f42a95cec89de33a72744023b47975d7","schema":"livegraph.boolean-propagation.v1","spec_hash":"70b6526e5ecb0da42a607439a6e7cae7d7e12a8f6d2349287096b64dcdf80b37","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-70d4df627e14e3f909b9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v07\n- v01 --refutes--> v03\n- v0b --supports--> v0c\n- v0d --supports--> v02\n- v07 --refutes--> v09\n- v03 --refutes--> v05\n- v0e --supports--> v0a\n- v02 --supports--> v08\n- v0f --supports--> v06\n- v07 --supports--> v01\n- v02 --refutes--> v00\n- v03 --supports--> v0e\n- v0f --supports--> v0b\n- v02 --refutes--> v04\n- v0a --supports--> v0d\n\n[VERIFIED ROOT]\nv0f = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"75d8b43f11ef29fb14285e1bdac1602a6aabe520cbf41957f890ec261e43519e","schema":"livegraph.boolean-propagation.v1","spec_hash":"70d4df627e14e3f909b988d8c96a1b4c153e094fe2e8351232d97401f0aba2e1","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-db158f70792c8f2d8bf3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v05\n- v01 --refutes--> v08\n- v04 --supports--> v0a\n- v0b --supports--> v01\n- v0d --supports--> v06\n- v05 --refutes--> v0d\n- v09 --supports--> v07\n- v01 --refutes--> v0c\n- v04 --refutes--> v09\n- v06 --refutes--> v0f\n- v0b --supports--> v02\n- v0b --refutes--> v0e\n- v08 --supports--> v04\n- v09 --refutes--> v03\n- v00 --supports--> v0b\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bd9f9d6308a4fd9bb60d96e3fded3685ac479aedde4bf630b6c1cca816f620ba","schema":"livegraph.boolean-propagation.v1","spec_hash":"db158f70792c8f2d8bf32feaabf82d0f02024ece95a21085c137cc4a8d241155","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":10,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-8f5789a27a97fe86612e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v0e\n- v02 --refutes--> v05\n- v04 --supports--> v03\n- v09 --refutes--> v0c\n- v02 --supports--> v00\n- v0e --refutes--> v06\n- v0f --refutes--> v0b\n- v0c --supports--> v07\n- v00 --refutes--> v0d\n- v03 --supports--> v0a\n- v0d --refutes--> v09\n- v0a --supports--> v02\n- v0d --supports--> v08\n- v0e --supports--> v04\n- v0d --refutes--> v01\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5e2e0df8a5f349544e410366222c518b12abef0a4167d0f6513a0377972a4cd1","schema":"livegraph.boolean-propagation.v1","spec_hash":"8f5789a27a97fe86612e86389d94bb6188185e9edd3baad91d04391b5f2b33ba","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-f2bc6c2992e8ddbde062","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v0c\n- v05 --refutes--> v0a\n- v0f --supports--> v04\n- v0d --supports--> v09\n- v08 --refutes--> v06\n- v03 --supports--> v02\n- v08 --supports--> v01\n- v04 --supports--> v0d\n- v09 --supports--> v05\n- v08 --refutes--> v00\n- v02 --supports--> v0b\n- v0b --refutes--> v0f\n- v06 --refutes--> v03\n- v0d --refutes--> v07\n- v05 --refutes--> v0e\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"863eea6851408e2c366482c8ca4ad9a388139de653026dbd1d9548e2108247b6","schema":"livegraph.boolean-propagation.v1","spec_hash":"f2bc6c2992e8ddbde062cab6bf26bdc356ae5dc4d1a4e5d8d7fa51e498f8c533","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-59a2eca3a87c8e5233a5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v00\n- v00 --refutes--> v03\n- v04 --refutes--> v08\n- v09 --refutes--> v07\n- v00 --refutes--> v06\n- v08 --refutes--> v0c\n- v0f --supports--> v05\n- v03 --refutes--> v04\n- v01 --refutes--> v0f\n- v08 --refutes--> v0a\n- v02 --supports--> v01\n- v0c --supports--> v09\n- v04 --refutes--> v0b\n- v09 --supports--> v02\n- v01 --supports--> v0e\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5093a00aa26c7f43ecb7592c6cac1d3b154d1b67e1a012a011be31fb66c519b9","schema":"livegraph.boolean-propagation.v1","spec_hash":"59a2eca3a87c8e5233a5d7d84f71c688e6121972c5d0e00f093e256f29f87b58","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-7903f64d687dbe7c220e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v0b\n- v0a --refutes--> v04\n- v06 --refutes--> v0f\n- v02 --refutes--> v06\n- v05 --supports--> v0d\n- v05 --refutes--> v00\n- v0a --refutes--> v05\n- v0e --refutes--> v0c\n- v00 --supports--> v03\n- v0f --refutes--> v0e\n- v0e --supports--> v09\n- v05 --refutes--> v08\n- v0b --supports--> v0a\n- v03 --supports--> v02\n- v05 --refutes--> v07\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b67979015c73033b41247a467249ac92dd3222fe214da6937cf08d6de6e3a3ad","schema":"livegraph.boolean-propagation.v1","spec_hash":"7903f64d687dbe7c220e4dee413285c8e860d00ddd3dfac7b547ba65dd48e37a","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-a259a93419cf1954b6e6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v03\n- v0e --supports--> v0a\n- v0a --refutes--> v01\n- v0f --refutes--> v00\n- v02 --refutes--> v0f\n- v01 --supports--> v0d\n- v0f --refutes--> v04\n- v0f --supports--> v08\n- v0c --refutes--> v0e\n- v0b --refutes--> v02\n- v08 --supports--> v05\n- v01 --supports--> v09\n- v0f --refutes--> v07\n- v03 --supports--> v06\n- v05 --supports--> v0c\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7d6f14f2637e8758613ddbb7efcc83c2c53b6a30749fb465936a45b03bbaf9bf","schema":"livegraph.boolean-propagation.v1","spec_hash":"a259a93419cf1954b6e6a5523ae16fecf1e82251a400e05c87d3dc391986935a","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-daffbc97140e7483e786","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v00\n- v0a --supports--> v03\n- v0c --refutes--> v05\n- v00 --refutes--> v08\n- v0a --refutes--> v02\n- v0a --refutes--> v0f\n- v0e --supports--> v06\n- v09 --refutes--> v0c\n- v06 --supports--> v07\n- v08 --refutes--> v0a\n- v0d --refutes--> v01\n- v09 --supports--> v04\n- v01 --refutes--> v0e\n- v03 --supports--> v09\n- v00 --supports--> v0b\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5a303749b671287a7d98d72485e947f5f3ee1c755b47706db6dd8f29fae351ff","schema":"livegraph.boolean-propagation.v1","spec_hash":"daffbc97140e7483e786b24ef5cd528b5ad008b74a154d12eaa3772ce4f772e0","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-86e7911d4f50a07bd3dc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v05\n- v04 --supports--> v02\n- v0a --supports--> v08\n- v04 --supports--> v0e\n- v00 --supports--> v0d\n- v07 --refutes--> v0a\n- v08 --supports--> v03\n- v04 --refutes--> v09\n- v0f --supports--> v0b\n- v01 --refutes--> v07\n- v08 --supports--> v06\n- v0d --refutes--> v01\n- v07 --supports--> v0c\n- v0b --refutes--> v04\n- v06 --supports--> v0f\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fcd2388a2fdfb8dafb0b8c62b3a894af2d5f4f1acc0fb0f68749b3a41863c343","schema":"livegraph.boolean-propagation.v1","spec_hash":"86e7911d4f50a07bd3dc686b17894746961c09d8d08695725ca365d168a4da46","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-78d9bc9ae9dddc99140e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v09\n- v07 --supports--> v0c\n- v06 --supports--> v01\n- v0b --refutes--> v00\n- v0a --refutes--> v06\n- v07 --refutes--> v0a\n- v07 --refutes--> v04\n- v07 --refutes--> v08\n- v03 --supports--> v0f\n- v05 --refutes--> v07\n- v0e --supports--> v0b\n- v01 --refutes--> v0e\n- v0f --supports--> v02\n- v00 --refutes--> v03\n- v03 --supports--> v0d\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9f53b4b833ec8e80f78abeca1dea7d8f7109448f94c608442e21bd92b7c401f4","schema":"livegraph.boolean-propagation.v1","spec_hash":"78d9bc9ae9dddc99140eab27c798aa24c3d4db62ba16d1a44045940a34e73bc7","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-94245950010a67502dbf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v01\n- v07 --supports--> v09\n- v00 --refutes--> v0f\n- v00 --refutes--> v08\n- v04 --supports--> v00\n- v00 --refutes--> v0a\n- v00 --supports--> v0e\n- v0c --refutes--> v07\n- v04 --refutes--> v0d\n- v01 --refutes--> v0c\n- v08 --supports--> v06\n- v02 --supports--> v0b\n- v01 --refutes--> v05\n- v09 --supports--> v02\n- v06 --refutes--> v03\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4399bbb665339dff4e4dd642926e59b06828a9ccafd3808a1411d58e09fbd470","schema":"livegraph.boolean-propagation.v1","spec_hash":"94245950010a67502dbf7cc2d894e3a7bf86244d295561dec07fb818395d2ed4","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":10,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-7d520e43527dd43cf5a0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v09\n- v02 --refutes--> v01\n- v0f --supports--> v02\n- v07 --refutes--> v0b\n- v04 --supports--> v06\n- v03 --refutes--> v00\n- v0f --supports--> v0c\n- v0b --supports--> v0e\n- v0a --refutes--> v04\n- v01 --supports--> v0d\n- v0f --refutes--> v08\n- v03 --refutes--> v05\n- v01 --refutes--> v03\n- v05 --refutes--> v07\n- v06 --refutes--> v0f\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8abb47c89909a263999149408c57038cecc76c7711e8ce6d055297b66b478820","schema":"livegraph.boolean-propagation.v1","spec_hash":"7d520e43527dd43cf5a04397949d7a3b55b3b328c97a6abf2fe6376915f9180b","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-7c96cc53411e38c69dea","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v05\n- v0c --supports--> v0d\n- v0f --refutes--> v0b\n- v01 --supports--> v07\n- v0f --supports--> v03\n- v0c --supports--> v04\n- v02 --supports--> v09\n- v0f --supports--> v08\n- v00 --refutes--> v0a\n- v06 --refutes--> v01\n- v03 --supports--> v0e\n- v05 --supports--> v0f\n- v0a --supports--> v02\n- v0c --refutes--> v06\n- v0e --refutes--> v0c\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"95f21553e5fcb7255f2fbbc86decdb9028c15cdea1041fae70cdd9fb4ed52aa7","schema":"livegraph.boolean-propagation.v1","spec_hash":"7c96cc53411e38c69dea1b4d882cb86c1470b5539e24c31cefdfe576f9dd2df3","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-c8046a2f341edbcb7eb6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v06\n- v09 --refutes--> v02\n- v0b --supports--> v04\n- v06 --refutes--> v00\n- v01 --supports--> v05\n- v0c --refutes--> v01\n- v0a --refutes--> v08\n- v03 --supports--> v0c\n- v00 --refutes--> v0a\n- v09 --refutes--> v0e\n- v0a --supports--> v0f\n- v0b --supports--> v07\n- v05 --supports--> v0d\n- v0a --supports--> v0b\n- v0b --supports--> v03\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"19b5c544291c31b224b74b809e1a47120175e3509c5ea0421dab4426e358a759","schema":"livegraph.boolean-propagation.v1","spec_hash":"c8046a2f341edbcb7eb6a75a525f6679fe74d8a020a8208a9325803dc0622319","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":10,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-dfd45fd33afd22be630b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v08\n- v0d --refutes--> v09\n- v05 --supports--> v0e\n- v0f --supports--> v07\n- v0c --refutes--> v05\n- v09 --refutes--> v00\n- v04 --refutes--> v02\n- v04 --supports--> v0a\n- v0a --refutes--> v03\n- v05 --refutes--> v01\n- v0d --refutes--> v0b\n- v0e --supports--> v0f\n- v0e --supports--> v06\n- v07 --refutes--> v0d\n- v0a --refutes--> v0c\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4856c5a39700058b5b91419b11e9818cca140fe11b4771c653cc185b24fc8efa","schema":"livegraph.boolean-propagation.v1","spec_hash":"dfd45fd33afd22be630b405df9e75db782dcc883510c7225967cfe75f4043b0b","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-9971420ee4db355763c4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v0c\n- v05 --refutes--> v0b\n- v0b --refutes--> v06\n- v0b --refutes--> v0d\n- v04 --refutes--> v0e\n- v0e --refutes--> v05\n- v0f --supports--> v08\n- v09 --refutes--> v02\n- v07 --supports--> v01\n- v02 --supports--> v00\n- v07 --refutes--> v03\n- v08 --supports--> v09\n- v0b --supports--> v07\n- v02 --supports--> v0a\n- v01 --refutes--> v0f\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"eeb05db9ec3267feff8c61534a35591b9563816e2552fbaf6136c67e9b16b66b","schema":"livegraph.boolean-propagation.v1","spec_hash":"9971420ee4db355763c4ae4e56d1c75a8bc000dbe5ed7ae4012d5e494f9a7c15","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-150d8ff22934ebb7ccfa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v01\n- v00 --supports--> v0c\n- v0b --supports--> v07\n- v02 --supports--> v06\n- v05 --supports--> v0d\n- v03 --supports--> v00\n- v02 --refutes--> v0f\n- v02 --refutes--> v0e\n- v03 --supports--> v08\n- v0c --supports--> v05\n- v04 --refutes--> v0a\n- v05 --refutes--> v09\n- v0b --refutes--> v04\n- v01 --supports--> v03\n- v0d --supports--> v02\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"64573e4cfc71bdc946427470dd501400ddbeee8d0f1ef4484c9be10535c43cf3","schema":"livegraph.boolean-propagation.v1","spec_hash":"150d8ff22934ebb7ccfaf6305971e5d7fc5d0d56d21afb1a73b030f168d8013a","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":10,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-7aeaabe3a4b067fa68ca","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v04\n- v0c --refutes--> v0e\n- v0d --refutes--> v0f\n- v0d --refutes--> v09\n- v00 --supports--> v0b\n- v06 --refutes--> v02\n- v01 --refutes--> v0a\n- v0b --supports--> v0c\n- v09 --refutes--> v07\n- v01 --refutes--> v06\n- v03 --supports--> v08\n- v00 --supports--> v05\n- v02 --refutes--> v03\n- v0e --refutes--> v01\n- v09 --refutes--> v00\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3229fe68f8d1c36c0c53f19d6aad1ef970f6cc968f9c2da16ff505d1b1a17f7e","schema":"livegraph.boolean-propagation.v1","spec_hash":"7aeaabe3a4b067fa68ca6b81bec17281fe180a87b7f12672acc3da3ead7beadc","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-ec6d04457a4dc51f9914","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v0f\n- v05 --supports--> v03\n- v0a --supports--> v09\n- v02 --refutes--> v06\n- v09 --refutes--> v05\n- v00 --supports--> v0b\n- v02 --supports--> v0d\n- v03 --supports--> v07\n- v04 --refutes--> v01\n- v0b --supports--> v0c\n- v0b --supports--> v0a\n- v0f --refutes--> v00\n- v04 --supports--> v08\n- v08 --refutes--> v02\n- v05 --refutes--> v0e\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3fadf0043bbda75c162c3fcd826b9e44df40eafe2d57399fc6f19ed64bcb9417","schema":"livegraph.boolean-propagation.v1","spec_hash":"ec6d04457a4dc51f9914699058a4aa4442089fddfdd6053352ac4087ce67db41","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-5b0a8735da8dc792fd75","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v0a\n- v0a --refutes--> v0b\n- v08 --refutes--> v01\n- v0a --supports--> v03\n- v0f --supports--> v04\n- v02 --refutes--> v0d\n- v06 --supports--> v02\n- v04 --supports--> v0e\n- v04 --supports--> v0c\n- v05 --refutes--> v08\n- v0a --supports--> v07\n- v03 --refutes--> v09\n- v09 --refutes--> v06\n- v00 --supports--> v05\n- v02 --refutes--> v0f\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b54235619826addb77df22ba73279fabdbe0eb472bf90cc10d96aaf8d336068c","schema":"livegraph.boolean-propagation.v1","spec_hash":"5b0a8735da8dc792fd7574a1808c9ed539f2dd2cf30377cb420b3eb0e51a66b8","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-17fbffb4ee38929e26b9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v0b\n- v01 --supports--> v0d\n- v0b --supports--> v03\n- v07 --refutes--> v05\n- v08 --supports--> v09\n- v01 --refutes--> v0a\n- v05 --refutes--> v01\n- v00 --supports--> v08\n- v06 --supports--> v0f\n- v07 --supports--> v04\n- v0c --supports--> v02\n- v02 --supports--> v0e\n- v01 --supports--> v06\n- v03 --refutes--> v07\n- v08 --refutes--> v0c\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f77d43f1fa646a156ce6702a8c0a9f0b3bc61ebd888f8cecbd127c8d06d9712d","schema":"livegraph.boolean-propagation.v1","spec_hash":"17fbffb4ee38929e26b90d075c92c72c5536aefc9d4c49deebfc2e70e00f45ef","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":10,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-ca312e3b7051eb6e4e7c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v04\n- v03 --refutes--> v0b\n- v07 --supports--> v08\n- v0e --supports--> v03\n- v04 --supports--> v05\n- v04 --refutes--> v00\n- v0c --refutes--> v01\n- v08 --refutes--> v02\n- v04 --refutes--> v07\n- v03 --supports--> v09\n- v02 --supports--> v0c\n- v0c --supports--> v0a\n- v0a --refutes--> v0d\n- v03 --supports--> v06\n- v0f --supports--> v0e\n\n[VERIFIED ROOT]\nv0f = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1ff2afab9d1f1c5037f57b803334e2f07835096202bde9e8e4967171dca67a88","schema":"livegraph.boolean-propagation.v1","spec_hash":"ca312e3b7051eb6e4e7c64dee7ed68929c9adc8cb639f3ab2e3a8382fdbdd46c","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-a5cbb9eb9f3d0183fb5c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v01\n- v06 --refutes--> v07\n- v05 --refutes--> v04\n- v04 --refutes--> v03\n- v04 --supports--> v0f\n- v0e --refutes--> v00\n- v09 --refutes--> v0b\n- v07 --supports--> v0d\n- v08 --refutes--> v05\n- v05 --supports--> v0c\n- v0d --refutes--> v0e\n- v0e --supports--> v0a\n- v04 --supports--> v02\n- v00 --supports--> v08\n- v0b --refutes--> v06\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d7ae9a3638ee11e74c8c1933f5de18a0b1b70408acc17c42da98346ce272a16b","schema":"livegraph.boolean-propagation.v1","spec_hash":"a5cbb9eb9f3d0183fb5c142832bb2ab5a4d60cb5a65b99ba733fd0d816c4a5e0","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-a6d2a3b69472a35fef9a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v04\n- v0b --refutes--> v05\n- v04 --refutes--> v07\n- v0d --supports--> v0e\n- v0c --supports--> v08\n- v08 --supports--> v06\n- v05 --supports--> v02\n- v09 --supports--> v01\n- v04 --supports--> v03\n- v01 --supports--> v0f\n- v0b --supports--> v00\n- v0d --refutes--> v0c\n- v01 --refutes--> v0d\n- v0a --refutes--> v0b\n- v05 --refutes--> v09\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1cc6df2cc241713bb559e3a2e5f6ddb11889248d557bb08f2f99b704cfec795a","schema":"livegraph.boolean-propagation.v1","spec_hash":"a6d2a3b69472a35fef9a8199d5c11cae0af3dfaae7664b4a05ed0525c02da765","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":10,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-0fc0d73bcb6112d10e84","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v07\n- v06 --refutes--> v0b\n- v0b --refutes--> v0d\n- v02 --refutes--> v03\n- v02 --refutes--> v0e\n- v07 --refutes--> v0a\n- v0d --supports--> v05\n- v07 --supports--> v02\n- v01 --refutes--> v00\n- v03 --supports--> v0f\n- v00 --refutes--> v0c\n- v0f --refutes--> v08\n- v0b --refutes--> v04\n- v0d --supports--> v01\n- v0d --refutes--> v09\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"622cf0dbadd4215b8758bf682079832e0a6ca792597b07015d5d187675f564e8","schema":"livegraph.boolean-propagation.v1","spec_hash":"0fc0d73bcb6112d10e8402f9e389169f61f9100f6d60dea88f0ad5a9c16dfdc0","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-d54dfa85dfee01e2cec8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v06\n- v0d --supports--> v0c\n- v05 --refutes--> v03\n- v04 --supports--> v0d\n- v01 --refutes--> v07\n- v07 --supports--> v09\n- v09 --supports--> v02\n- v0b --refutes--> v0e\n- v08 --refutes--> v0b\n- v05 --refutes--> v00\n- v05 --refutes--> v0a\n- v06 --supports--> v08\n- v05 --supports--> v04\n- v0b --refutes--> v05\n- v01 --supports--> v0f\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c15506912925cb326c4be8715c0fcca37b7ed833b058458e01bbd4b7afe6b184","schema":"livegraph.boolean-propagation.v1","spec_hash":"d54dfa85dfee01e2cec8be685cfecf315e814052df7faaaa619219a7a335f292","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":10,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-b46c7aac7b437d0a18ab","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v02\n- v0e --refutes--> v08\n- v07 --refutes--> v0e\n- v05 --supports--> v07\n- v0d --refutes--> v01\n- v0d --supports--> v03\n- v06 --supports--> v04\n- v0f --supports--> v0a\n- v08 --refutes--> v0b\n- v08 --supports--> v09\n- v0d --refutes--> v0c\n- v04 --supports--> v05\n- v0c --refutes--> v06\n- v0b --supports--> v00\n- v0f --supports--> v0d\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"eaad16d4eda10039357c636fa91085010e959720e5a42bc75d53e7e1a1868069","schema":"livegraph.boolean-propagation.v1","spec_hash":"b46c7aac7b437d0a18abc665219ade8a7e6c65fa5b6880b38c5bbf19ca63b52b","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-b6777d9eed178ee88826","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v06\n- v04 --supports--> v0e\n- v02 --refutes--> v0c\n- v02 --supports--> v07\n- v05 --refutes--> v01\n- v0f --refutes--> v05\n- v07 --refutes--> v04\n- v0e --refutes--> v08\n- v0b --refutes--> v00\n- v00 --supports--> v03\n- v09 --supports--> v0b\n- v00 --supports--> v0f\n- v06 --refutes--> v0a\n- v0e --supports--> v09\n- v09 --supports--> v0d\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"86f73c0ce6e4f82e1b70de5b59df803adb5e7c5eb354313580f79454cd84d5cf","schema":"livegraph.boolean-propagation.v1","spec_hash":"b6777d9eed178ee8882678f01791bda1ba319072c93cc050ac67f7a00df695f4","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-c16e22302603f677986f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v0d\n- v0f --refutes--> v09\n- v08 --supports--> v0a\n- v07 --refutes--> v08\n- v0c --refutes--> v00\n- v05 --refutes--> v0b\n- v0c --supports--> v02\n- v0a --supports--> v0c\n- v00 --refutes--> v0f\n- v04 --refutes--> v07\n- v08 --refutes--> v03\n- v00 --refutes--> v0e\n- v09 --supports--> v06\n- v0b --refutes--> v01\n- v09 --refutes--> v05\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9be1e7c7100f0a4f513a611d6bcae868f543f1708134f0e67caf816b00e673ba","schema":"livegraph.boolean-propagation.v1","spec_hash":"c16e22302603f677986fffecdab91db70441861c7abc33feb88f18ca6c8115b9","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-5e983917050339853ad6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v04\n- v0d --supports--> v05\n- v08 --refutes--> v0a\n- v0b --supports--> v00\n- v0f --refutes--> v03\n- v0a --supports--> v02\n- v0f --supports--> v07\n- v0e --refutes--> v0d\n- v0c --supports--> v0f\n- v00 --supports--> v08\n- v0a --refutes--> v01\n- v03 --supports--> v0e\n- v0c --refutes--> v06\n- v0d --supports--> v09\n- v02 --supports--> v0c\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a8afe44643576d24d61201b6bcfc97d22a4f9fdbd793749eabbb750fcdcc9116","schema":"livegraph.boolean-propagation.v1","spec_hash":"5e983917050339853ad61b0b5725b250d0fda92f5f904593f70e0b45ed94146c","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-7c62ff3ba63686df357d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v0b\n- v0c --refutes--> v05\n- v05 --supports--> v0a\n- v01 --refutes--> v0c\n- v08 --refutes--> v02\n- v0e --supports--> v04\n- v00 --refutes--> v0e\n- v02 --refutes--> v06\n- v07 --refutes--> v09\n- v07 --refutes--> v01\n- v06 --supports--> v00\n- v00 --supports--> v03\n- v0e --supports--> v0d\n- v0a --supports--> v08\n- v06 --refutes--> v0f\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"571c822d972df675812d638c4870b88ff6a599f969b19bc969340c8c747c64ae","schema":"livegraph.boolean-propagation.v1","spec_hash":"7c62ff3ba63686df357dc722233adf798d690c48dcb830cd382337d7c12f3791","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-94cbb3ed2433dcbd7e03","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v0d\n- v00 --supports--> v09\n- v0b --refutes--> v03\n- v0e --supports--> v00\n- v0a --supports--> v0e\n- v03 --supports--> v01\n- v0c --refutes--> v02\n- v06 --supports--> v0f\n- v03 --refutes--> v05\n- v0f --refutes--> v0a\n- v0e --refutes--> v08\n- v0a --refutes--> v04\n- v09 --refutes--> v07\n- v09 --supports--> v0b\n- v02 --refutes--> v06\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e9071a0923544f38c4ffd004264463f6f7b1038bba522f71f00677300ff02669","schema":"livegraph.boolean-propagation.v1","spec_hash":"94cbb3ed2433dcbd7e0333bf725a844043ac4b64ce80f164bf32c215a3d4e87b","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-ae45b1d1132d3743d4f9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v0d\n- v0e --supports--> v07\n- v04 --refutes--> v0f\n- v0c --supports--> v02\n- v05 --supports--> v06\n- v05 --supports--> v01\n- v01 --refutes--> v0c\n- v00 --refutes--> v09\n- v04 --supports--> v0a\n- v00 --supports--> v0b\n- v04 --supports--> v05\n- v07 --supports--> v03\n- v03 --supports--> v04\n- v08 --refutes--> v0e\n- v02 --refutes--> v00\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c6629f22fee48199a04f597b46337fbf90d707e7c38d59dfa14d5633280f5e13","schema":"livegraph.boolean-propagation.v1","spec_hash":"ae45b1d1132d3743d4f99f36be4e21c8a38ce530cd373008b37d694a501ff97a","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-e4e670868044d0cc0bfe","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v0b\n- v07 --refutes--> v0a\n- v06 --refutes--> v0c\n- v0b --supports--> v08\n- v05 --refutes--> v04\n- v0a --refutes--> v00\n- v07 --refutes--> v02\n- v0e --refutes--> v05\n- v05 --refutes--> v0f\n- v00 --refutes--> v09\n- v0b --refutes--> v03\n- v0b --refutes--> v07\n- v0a --refutes--> v01\n- v04 --supports--> v06\n- v09 --supports--> v0e\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"00e4b6e60737aafbfb21710e763a8a94a7e8cf4026a076c6c4fd1f24e936e553","schema":"livegraph.boolean-propagation.v1","spec_hash":"e4e670868044d0cc0bfe7252ea019dc49329750b7e3fb731ee1a0b0cb3865c37","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-9f8f8ba1704e8f9a12a6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v0f\n- v0d --supports--> v0c\n- v03 --supports--> v0a\n- v04 --refutes--> v05\n- v02 --refutes--> v09\n- v0a --refutes--> v06\n- v06 --refutes--> v00\n- v0e --supports--> v07\n- v05 --supports--> v0d\n- v0e --supports--> v0b\n- v0e --refutes--> v08\n- v02 --refutes--> v0e\n- v01 --supports--> v03\n- v0d --refutes--> v02\n- v08 --refutes--> v01\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a6a235928d42828671a7ddf0fb29e37d55ce24b919ca1b351c53d7c9af891424","schema":"livegraph.boolean-propagation.v1","spec_hash":"9f8f8ba1704e8f9a12a69a108243af5c969ccb93ff6460947623ec38348ed637","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-69c95435ad02c1fd1f5b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v0a\n- v0e --refutes--> v02\n- v01 --refutes--> v03\n- v09 --refutes--> v08\n- v0e --refutes--> v01\n- v0d --supports--> v07\n- v01 --supports--> v06\n- v0d --refutes--> v0f\n- v0b --supports--> v09\n- v08 --supports--> v00\n- v0d --supports--> v04\n- v0a --supports--> v0e\n- v0c --supports--> v0d\n- v0d --refutes--> v05\n- v00 --refutes--> v0c\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4f809ab0d67f32b244d1490262a99f8c10e8b91e444b8bc03426429bb35c1e74","schema":"livegraph.boolean-propagation.v1","spec_hash":"69c95435ad02c1fd1f5b0c64a10921d07bc588f4485578785aabc7495a1b4fc8","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-14ebd4c68ee5e4955d24","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v05\n- v07 --supports--> v00\n- v0a --refutes--> v0c\n- v0c --refutes--> v08\n- v0c --refutes--> v06\n- v0e --refutes--> v0b\n- v0d --supports--> v02\n- v00 --supports--> v0e\n- v03 --supports--> v0a\n- v0c --refutes--> v01\n- v05 --supports--> v04\n- v02 --refutes--> v03\n- v0e --supports--> v09\n- v07 --refutes--> v0f\n- v05 --refutes--> v0d\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ee5258cf3aaed4adfedccd351a46242a6659ce33c229800a4245d3907bdcba27","schema":"livegraph.boolean-propagation.v1","spec_hash":"14ebd4c68ee5e4955d24f721f51308a76fa10fa2afa71a7555029d9dfcb04328","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-af07ef5e51d7b3146ac8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v03\n- v08 --refutes--> v0f\n- v04 --refutes--> v0d\n- v01 --supports--> v07\n- v00 --refutes--> v02\n- v06 --supports--> v09\n- v0a --supports--> v08\n- v03 --supports--> v00\n- v0e --refutes--> v0a\n- v05 --supports--> v06\n- v0e --refutes--> v0b\n- v02 --refutes--> v04\n- v06 --refutes--> v01\n- v04 --refutes--> v0e\n- v01 --refutes--> v0c\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e5cf474f337bd5d773728f3850a2774cbb480eea3a37671691abee25e9ab435e","schema":"livegraph.boolean-propagation.v1","spec_hash":"af07ef5e51d7b3146ac861ccc5640d7535726a0e761624aed69d6be4c327bdb3","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-bdf15678cbedcbedd6d5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v0f\n- v06 --supports--> v03\n- v04 --refutes--> v0e\n- v05 --refutes--> v06\n- v07 --refutes--> v0d\n- v0c --supports--> v0a\n- v09 --refutes--> v04\n- v0c --supports--> v07\n- v03 --refutes--> v0c\n- v04 --refutes--> v08\n- v0c --refutes--> v00\n- v01 --refutes--> v02\n- v04 --supports--> v01\n- v0f --refutes--> v0b\n- v0b --refutes--> v05\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"da75ee9b30b799276d827c5a82dbe149f8fcd0c419e71915dfe504a23f4ecbbb","schema":"livegraph.boolean-propagation.v1","spec_hash":"bdf15678cbedcbedd6d53021c8faeaaa4e15a5464d9075a135395b6c4285cb70","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-0a247a6f7d4b0cadb8db","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v09\n- v0b --refutes--> v0c\n- v05 --supports--> v07\n- v0a --refutes--> v06\n- v03 --refutes--> v01\n- v00 --refutes--> v0a\n- v0f --supports--> v03\n- v0f --supports--> v02\n- v06 --supports--> v0b\n- v0c --supports--> v0e\n- v08 --supports--> v00\n- v03 --refutes--> v0d\n- v0e --refutes--> v0f\n- v05 --supports--> v08\n- v03 --supports--> v04\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e7c48089ddea769c3114bc46e60ce16ba86190356794398a71407729495f23ac","schema":"livegraph.boolean-propagation.v1","spec_hash":"0a247a6f7d4b0cadb8db5ef8d541d216f3fdedd4b53849885ad5a883930a9cc6","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-1615ff93f78b43497553","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v01\n- v09 --supports--> v05\n- v03 --supports--> v0f\n- v0a --refutes--> v06\n- v06 --supports--> v04\n- v04 --supports--> v00\n- v00 --refutes--> v08\n- v0e --refutes--> v0d\n- v00 --supports--> v02\n- v0e --refutes--> v09\n- v09 --refutes--> v0b\n- v02 --supports--> v0e\n- v0f --refutes--> v0a\n- v09 --supports--> v07\n- v0a --supports--> v0c\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a7361cf9f758a948f55c70e9d84cc15ebe24a04d036b2bca3e994a386ca33a7c","schema":"livegraph.boolean-propagation.v1","spec_hash":"1615ff93f78b434975532e2510864394db6f45e4cb2d3e8fc728a629f1a5ebef","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-66af620b50577b02fb63","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v08\n- v01 --refutes--> v06\n- v0c --supports--> v05\n- v02 --supports--> v09\n- v05 --refutes--> v04\n- v01 --refutes--> v0f\n- v05 --supports--> v0d\n- v09 --refutes--> v0c\n- v0b --supports--> v01\n- v01 --supports--> v0e\n- v0a --supports--> v02\n- v08 --refutes--> v0b\n- v0d --supports--> v07\n- v08 --supports--> v03\n- v0d --supports--> v00\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ed9457c6d11d4afea05884b92fd7efa59bd6011b3514194ba02bd639937ba726","schema":"livegraph.boolean-propagation.v1","spec_hash":"66af620b50577b02fb63cde9b5a475e8215cd399b44186ab038d8ac66f44649b","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":10,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-73e2c6044aca213e38ca","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v04\n- v00 --supports--> v09\n- v0b --supports--> v05\n- v0e --refutes--> v03\n- v09 --refutes--> v06\n- v01 --refutes--> v0d\n- v09 --refutes--> v01\n- v0f --supports--> v0a\n- v01 --refutes--> v0b\n- v0f --refutes--> v0e\n- v04 --refutes--> v0c\n- v01 --refutes--> v02\n- v0c --supports--> v0f\n- v03 --refutes--> v07\n- v08 --supports--> v00\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e627469c2cfdbc2180b598a75b4192ffd4bb39574b3380a4c01638513c521ecc","schema":"livegraph.boolean-propagation.v1","spec_hash":"73e2c6044aca213e38ca343e2d697070a55782958270450eff5707f9dc09f0ea","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-16e1fb813789caa55d99","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v06\n- v09 --supports--> v04\n- v0f --refutes--> v0a\n- v0a --refutes--> v08\n- v0e --refutes--> v05\n- v0c --supports--> v0e\n- v05 --refutes--> v07\n- v0c --refutes--> v0d\n- v03 --refutes--> v09\n- v01 --refutes--> v03\n- v08 --supports--> v0c\n- v08 --supports--> v0b\n- v04 --refutes--> v0f\n- v03 --supports--> v00\n- v08 --refutes--> v02\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"732794a1a84f19d858fcb424201c01275ad4e554a6bfe9dccdbe32aa9c243f3d","schema":"livegraph.boolean-propagation.v1","spec_hash":"16e1fb813789caa55d99ef42f15bc5595dd0524baa50993485c64c2eb9ac8f8d","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-666e2844a631e61b6740","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v07\n- v0b --refutes--> v00\n- v0c --refutes--> v06\n- v08 --refutes--> v0c\n- v0f --supports--> v05\n- v04 --supports--> v0f\n- v03 --supports--> v0d\n- v0c --refutes--> v04\n- v07 --refutes--> v09\n- v07 --refutes--> v02\n- v09 --refutes--> v03\n- v09 --refutes--> v0a\n- v0d --refutes--> v0b\n- v0b --supports--> v0e\n- v0d --supports--> v01\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3b3b780bf7cb05e66e7adbaf6c21fd40ade0007643134c8b5366fd8152151e6c","schema":"livegraph.boolean-propagation.v1","spec_hash":"666e2844a631e61b674046f4d8a348824e5efaafb53dcf2a00024fdae0b36fee","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-176bd6c136301369d947","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v06\n- v02 --refutes--> v00\n- v0d --refutes--> v03\n- v04 --supports--> v0c\n- v0e --refutes--> v0b\n- v02 --supports--> v01\n- v0b --supports--> v0d\n- v0a --supports--> v08\n- v01 --refutes--> v04\n- v07 --supports--> v0e\n- v08 --supports--> v02\n- v05 --refutes--> v0f\n- v0a --supports--> v09\n- v04 --refutes--> v05\n- v05 --supports--> v07\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3ff0ebd4f9e41dad7ff8a691d93d2a323df558f71ed374bf6b5e5f7858c4b1bc","schema":"livegraph.boolean-propagation.v1","spec_hash":"176bd6c136301369d9473d5b3e812c65a85efa004e39888dbb8114c38172c7b9","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-be1c1c136bf1c466e3b3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v01\n- v02 --refutes--> v0c\n- v00 --supports--> v08\n- v0e --refutes--> v05\n- v09 --refutes--> v00\n- v04 --supports--> v0e\n- v07 --supports--> v09\n- v0f --supports--> v0b\n- v0d --supports--> v03\n- v02 --supports--> v0f\n- v0f --supports--> v04\n- v0e --supports--> v07\n- v03 --supports--> v02\n- v09 --supports--> v0a\n- v0e --refutes--> v06\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"caae9bbc96788cd944f1e697ecdecba30283a2b55f73d74f757365c531a897fe","schema":"livegraph.boolean-propagation.v1","spec_hash":"be1c1c136bf1c466e3b3cd983ebdd8ddc14db92a86d5f6ceed7ae537fdef51e6","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-000b5c0c6bec46d19042","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v0d\n- v01 --refutes--> v02\n- v0a --refutes--> v0f\n- v0e --supports--> v0b\n- v0b --refutes--> v06\n- v00 --supports--> v0e\n- v03 --refutes--> v08\n- v05 --refutes--> v03\n- v04 --refutes--> v01\n- v09 --refutes--> v07\n- v01 --supports--> v00\n- v0b --supports--> v09\n- v07 --refutes--> v0a\n- v03 --supports--> v04\n- v01 --supports--> v0c\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e3d8707e600de00ca49b458412d070612bf35bc67386d641b00bd31820f90da2","schema":"livegraph.boolean-propagation.v1","spec_hash":"000b5c0c6bec46d19042e9282ba2c8d475e85e5f9736b56a43e3f0e6eeff8e0f","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-0690bc31d08b0bc6c416","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v03\n- v0b --refutes--> v07\n- v03 --refutes--> v08\n- v06 --supports--> v02\n- v0a --supports--> v0b\n- v03 --supports--> v0d\n- v04 --supports--> v05\n- v09 --supports--> v0a\n- v0c --supports--> v01\n- v08 --refutes--> v0f\n- v08 --supports--> v0c\n- v08 --refutes--> v0e\n- v04 --supports--> v09\n- v06 --supports--> v00\n- v02 --supports--> v04\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"64fa733d00495ce522bdedd9dcd506dde812867933efdcf96090d0624d4c3989","schema":"livegraph.boolean-propagation.v1","spec_hash":"0690bc31d08b0bc6c416469ee9446e10952f03cd5f7f11619430e0faf0771761","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":10,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-34c6202c86d62867b068","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v04\n- v0d --refutes--> v00\n- v06 --supports--> v02\n- v0b --refutes--> v09\n- v00 --supports--> v0f\n- v08 --supports--> v06\n- v0f --supports--> v08\n- v07 --refutes--> v0e\n- v0e --supports--> v0b\n- v06 --supports--> v0c\n- v0b --refutes--> v01\n- v00 --supports--> v05\n- v09 --refutes--> v0d\n- v0d --refutes--> v0a\n- v0c --supports--> v03\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"50a32bdc09e52128c5b86fca5d3e5683a5ae99d34ec9dade0a1e94418652a252","schema":"livegraph.boolean-propagation.v1","spec_hash":"34c6202c86d62867b0683845e95afea8cfbda520a2dbf721fd7de66c44c2c146","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-36d2ef313a2a6b788fb6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v01\n- v0e --refutes--> v0a\n- v05 --refutes--> v0f\n- v0a --supports--> v02\n- v0c --refutes--> v0e\n- v09 --supports--> v0b\n- v08 --supports--> v07\n- v0b --refutes--> v05\n- v09 --supports--> v0d\n- v0a --refutes--> v06\n- v0a --refutes--> v00\n- v04 --supports--> v08\n- v01 --supports--> v09\n- v0f --supports--> v0c\n- v08 --refutes--> v03\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6d167946c857fc0f8549ae47a4a76239ff18bc37b7b1cb5343c8e0542b102d33","schema":"livegraph.boolean-propagation.v1","spec_hash":"36d2ef313a2a6b788fb6edd2a8eab4466b7502d6a212fcfe078898806353b3c3","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-c0cfc29f21ef45e51516","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v01\n- v00 --refutes--> v06\n- v06 --refutes--> v0e\n- v0c --refutes--> v07\n- v0d --refutes--> v0f\n- v04 --supports--> v0a\n- v00 --supports--> v02\n- v0a --refutes--> v00\n- v06 --refutes--> v03\n- v08 --refutes--> v0d\n- v0d --supports--> v09\n- v01 --supports--> v0c\n- v0c --supports--> v0b\n- v0d --refutes--> v04\n- v03 --refutes--> v05\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cbdf2bbab53802b1997252340987a6535ca35d661d1dc4c47ecd369454502419","schema":"livegraph.boolean-propagation.v1","spec_hash":"c0cfc29f21ef45e5151625207564103c5540ea9f5d9ad38fd77113b4c7921903","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":10,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-ab7d56cd898006e7ce2e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v07\n- v06 --supports--> v04\n- v0f --supports--> v01\n- v07 --refutes--> v0a\n- v07 --supports--> v05\n- v04 --supports--> v0e\n- v00 --supports--> v08\n- v0f --supports--> v0d\n- v05 --supports--> v00\n- v06 --refutes--> v03\n- v0e --refutes--> v09\n- v01 --supports--> v0c\n- v09 --supports--> v0f\n- v04 --supports--> v0b\n- v0e --refutes--> v02\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"50a7d3b2bcbbe8c5f2ab167f6fbdc8c2ca679ebf6317b9bd7191788f622be8f9","schema":"livegraph.boolean-propagation.v1","spec_hash":"ab7d56cd898006e7ce2e785196d4a429f24aa243e6afd953068ddf16bf6b545e","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-3ddfc09ad8bf8cbd16c5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v0e\n- v0f --refutes--> v09\n- v0c --supports--> v07\n- v02 --supports--> v0a\n- v06 --refutes--> v00\n- v09 --refutes--> v03\n- v08 --refutes--> v0c\n- v05 --supports--> v06\n- v02 --refutes--> v0b\n- v06 --supports--> v08\n- v07 --refutes--> v01\n- v03 --refutes--> v05\n- v07 --supports--> v02\n- v0f --refutes--> v0d\n- v0f --supports--> v04\n\n[VERIFIED ROOT]\nv0f = FALSE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3e6a7642eb9036086b622d05c333a7567ef43145163558adcce5ff3ca3615a98","schema":"livegraph.boolean-propagation.v1","spec_hash":"3ddfc09ad8bf8cbd16c5d6cfdc4f82df33534a85f9aad3264668cb33f5877f5d","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-5f6955db0c1dd9b685b5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v06\n- v04 --refutes--> v0d\n- v0a --refutes--> v00\n- v02 --supports--> v0a\n- v07 --supports--> v0c\n- v09 --supports--> v08\n- v08 --supports--> v07\n- v0d --refutes--> v02\n- v0a --refutes--> v03\n- v09 --supports--> v0f\n- v0c --supports--> v0b\n- v03 --supports--> v0e\n- v0e --supports--> v09\n- v07 --refutes--> v05\n- v0c --refutes--> v01\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"27cd867ed3f8aca28ad3664d7d8d85f86977e6d6d54eb5609cb373ac938a38a9","schema":"livegraph.boolean-propagation.v1","spec_hash":"5f6955db0c1dd9b685b57e55e0ca121d21759c59cef2f10b884f03da5ad3c379","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-6b619f598c9d33b99e18","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v09\n- v0d --supports--> v0a\n- v0d --refutes--> v0c\n- v0f --supports--> v07\n- v0a --supports--> v0b\n- v07 --refutes--> v06\n- v02 --refutes--> v00\n- v04 --supports--> v02\n- v07 --refutes--> v05\n- v0a --refutes--> v0e\n- v03 --refutes--> v0f\n- v01 --supports--> v0d\n- v00 --supports--> v01\n- v0b --refutes--> v03\n- v03 --supports--> v08\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6737607287f04ae2b776bb0d6dee0f674a4928476279b5b4d85f1a37d98e06c7","schema":"livegraph.boolean-propagation.v1","spec_hash":"6b619f598c9d33b99e18950a256c6e2694993f4a3062973e741de46ffb8b9b15","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-f237416336822248f5de","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v04\n- v03 --refutes--> v06\n- v0a --supports--> v00\n- v08 --supports--> v0d\n- v02 --refutes--> v03\n- v0b --refutes--> v02\n- v04 --supports--> v07\n- v05 --supports--> v08\n- v04 --refutes--> v0f\n- v0e --supports--> v09\n- v03 --supports--> v01\n- v05 --supports--> v0c\n- v08 --refutes--> v0e\n- v07 --refutes--> v05\n- v03 --refutes--> v0a\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f8f994b3c2303a5e3ad2ea9402e2f6ce09f3c95ef5566855e17cf4ed5d222159","schema":"livegraph.boolean-propagation.v1","spec_hash":"f237416336822248f5dee9008f9b5178868d0cea1df096b627bea33a674e4193","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-8ecfcaf7c02873e7c335","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v02\n- v05 --refutes--> v0d\n- v09 --refutes--> v03\n- v0f --supports--> v0a\n- v03 --supports--> v0e\n- v0f --refutes--> v0c\n- v04 --supports--> v07\n- v01 --refutes--> v05\n- v07 --supports--> v0b\n- v0e --refutes--> v0f\n- v0d --supports--> v09\n- v03 --supports--> v06\n- v00 --refutes--> v04\n- v09 --refutes--> v08\n- v07 --refutes--> v01\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e7a2f71a9e34308b3bbc653dac54c9c4cec1b1dc35c0e1a6278e3b31c8322aa4","schema":"livegraph.boolean-propagation.v1","spec_hash":"8ecfcaf7c02873e7c33592d08804ec4ae08baaf2fa719fe7d34d1901a2e68a71","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-0f91f90fff19582a3dd4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v03\n- v03 --supports--> v08\n- v0d --supports--> v05\n- v0f --refutes--> v02\n- v04 --supports--> v06\n- v0b --supports--> v0c\n- v0c --refutes--> v0d\n- v03 --supports--> v00\n- v04 --refutes--> v0b\n- v08 --refutes--> v0f\n- v01 --supports--> v04\n- v0e --refutes--> v09\n- v07 --refutes--> v0e\n- v0c --refutes--> v0a\n- v0e --refutes--> v01\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1a1f39823c698a383287eafb4f46d632cf56da457e407f6274f4fe05bb32a046","schema":"livegraph.boolean-propagation.v1","spec_hash":"0f91f90fff19582a3dd4836866c710e32610ed43a33d0336efb21bedbf4742e1","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-fab5e08603ae9c83bb9c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v0f\n- v00 --refutes--> v0e\n- v0b --refutes--> v0c\n- v0a --supports--> v04\n- v0e --refutes--> v08\n- v04 --refutes--> v07\n- v0a --supports--> v09\n- v0e --supports--> v0b\n- v07 --refutes--> v0d\n- v03 --refutes--> v06\n- v01 --supports--> v03\n- v0c --refutes--> v01\n- v0d --refutes--> v00\n- v03 --refutes--> v05\n- v0a --refutes--> v02\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ce0aa3cbeb725217dfbf9e48a346cb56e1fa5aa9f6772b163aa12c145f10deb4","schema":"livegraph.boolean-propagation.v1","spec_hash":"fab5e08603ae9c83bb9cd9692d7757046978ebb098c4c286d7c31c899ce35bf3","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-6f9954358407db87f068","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v01\n- v0b --refutes--> v00\n- v03 --supports--> v0a\n- v0c --refutes--> v03\n- v0b --supports--> v0f\n- v03 --refutes--> v08\n- v09 --supports--> v0c\n- v04 --refutes--> v0b\n- v01 --supports--> v0d\n- v08 --refutes--> v04\n- v09 --supports--> v02\n- v05 --supports--> v07\n- v08 --supports--> v0e\n- v0f --supports--> v06\n- v05 --refutes--> v09\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1438e6c88e050d10516a31f74356354fd2c3963a8eac0398ae5f2700fd767168","schema":"livegraph.boolean-propagation.v1","spec_hash":"6f9954358407db87f0683f0e42d676c9f91ce765beb40954b719828074229421","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-7eb4a09c71b89fd6fa64","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v05\n- v00 --refutes--> v01\n- v02 --refutes--> v0e\n- v0a --supports--> v0f\n- v0e --supports--> v04\n- v0e --supports--> v08\n- v01 --supports--> v03\n- v0f --refutes--> v0b\n- v0b --refutes--> v00\n- v03 --supports--> v07\n- v03 --supports--> v0d\n- v0e --supports--> v09\n- v07 --refutes--> v0c\n- v08 --refutes--> v0a\n- v07 --refutes--> v06\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bf41020f92b536ef1bd05926b1fa6c4ae617205a3232236bc9f46e19732c7bcb","schema":"livegraph.boolean-propagation.v1","spec_hash":"7eb4a09c71b89fd6fa64d6d83dde3b7a01641da0fbc92410eb8f34357d875b87","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-3d924162b57c5348f73a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v07\n- v0c --supports--> v0a\n- v0b --refutes--> v01\n- v09 --supports--> v0c\n- v09 --refutes--> v00\n- v02 --refutes--> v03\n- v0c --supports--> v05\n- v04 --supports--> v09\n- v0f --refutes--> v06\n- v05 --supports--> v0d\n- v04 --refutes--> v0e\n- v02 --refutes--> v0f\n- v05 --refutes--> v08\n- v07 --refutes--> v0b\n- v0d --supports--> v02\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7147370fbefc6acfb8bb888c96c4b517a9aee8d3322ccc66b37d5737cff7ae65","schema":"livegraph.boolean-propagation.v1","spec_hash":"3d924162b57c5348f73ab419e851fd9e160a8742c3857a7517564b99241ddd67","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-b0f6292641784260e534","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v01\n- v07 --supports--> v0b\n- v0b --supports--> v03\n- v0d --refutes--> v09\n- v0c --refutes--> v0e\n- v09 --supports--> v08\n- v04 --refutes--> v0d\n- v01 --supports--> v06\n- v05 --refutes--> v0a\n- v06 --supports--> v07\n- v06 --refutes--> v00\n- v09 --supports--> v05\n- v09 --refutes--> v0f\n- v0d --supports--> v02\n- v05 --refutes--> v0c\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0737a9983b5d9782555bff7368dd1e43a66e1d2980a69a7b0e53e2f4e195856e","schema":"livegraph.boolean-propagation.v1","spec_hash":"b0f6292641784260e53437f5bf805b2d4a19a569d8ddbeee951ef5ea8808a7ad","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-5a72a7d6eefafcd20268","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v01\n- v06 --refutes--> v0c\n- v04 --refutes--> v02\n- v07 --refutes--> v03\n- v03 --supports--> v09\n- v0a --supports--> v05\n- v06 --refutes--> v04\n- v06 --supports--> v00\n- v0a --supports--> v06\n- v02 --refutes--> v07\n- v0e --refutes--> v0d\n- v09 --refutes--> v0f\n- v08 --refutes--> v0e\n- v0f --refutes--> v08\n- v04 --supports--> v0b\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"33e09fabcec3a872c68cae8c69d981ce45fff47b4d2d216b62c421950e06bf4e","schema":"livegraph.boolean-propagation.v1","spec_hash":"5a72a7d6eefafcd2026868be04ca1c612adfe5859706e486fcb40b07869772e6","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-9c2d8cb43f3b8f6b2dde","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v07\n- v0d --refutes--> v0f\n- v06 --supports--> v0b\n- v0f --supports--> v08\n- v06 --refutes--> v0d\n- v0a --supports--> v02\n- v01 --refutes--> v0a\n- v05 --supports--> v00\n- v0e --supports--> v09\n- v0a --supports--> v0c\n- v0a --supports--> v03\n- v04 --supports--> v01\n- v0e --supports--> v06\n- v02 --refutes--> v05\n- v07 --supports--> v04\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b0fb4d4d7c202bb7296f0fd37231ae15eb5207398c5bef8991666581fe195ae3","schema":"livegraph.boolean-propagation.v1","spec_hash":"9c2d8cb43f3b8f6b2dde3f24c0974a098d6e0205d3e4b8f4d74e268e976d63d9","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-deb22d486a4d2b48117b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v09\n- v0a --supports--> v03\n- v00 --refutes--> v04\n- v02 --supports--> v0a\n- v05 --refutes--> v0e\n- v06 --supports--> v00\n- v09 --supports--> v07\n- v03 --supports--> v05\n- v0c --refutes--> v08\n- v0b --refutes--> v0f\n- v0e --supports--> v0b\n- v0c --supports--> v06\n- v00 --supports--> v0d\n- v0c --supports--> v01\n- v09 --refutes--> v0c\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"84c1d7a1b3eca2d8d090ab4c532659a7fe02dd3d9882470cb50f0d6c2872a772","schema":"livegraph.boolean-propagation.v1","spec_hash":"deb22d486a4d2b48117b5080e996cdf2736712f7cb162d0b2892fba82c48bddd","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-017eb26212eb09accdd1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v09\n- v00 --refutes--> v0e\n- v05 --supports--> v01\n- v03 --refutes--> v0d\n- v09 --supports--> v03\n- v09 --supports--> v0c\n- v06 --refutes--> v05\n- v0a --supports--> v06\n- v06 --refutes--> v07\n- v00 --supports--> v04\n- v0d --supports--> v0b\n- v04 --refutes--> v02\n- v01 --refutes--> v00\n- v0b --supports--> v08\n- v01 --supports--> v0f\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b3322e5cacfc6a6c6d6dd5ac1b624413e8a0574a81136b7f81433a39f81536f6","schema":"livegraph.boolean-propagation.v1","spec_hash":"017eb26212eb09accdd15b4a09725cfd16b4f4ee1369b9f6e05aff75d932defa","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":10,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-4d3e01d432229ce26ce0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v05\n- v0e --refutes--> v0a\n- v0a --supports--> v0b\n- v0a --refutes--> v0c\n- v06 --refutes--> v04\n- v06 --refutes--> v0d\n- v07 --refutes--> v06\n- v0a --supports--> v0f\n- v02 --supports--> v00\n- v0e --supports--> v09\n- v00 --refutes--> v08\n- v08 --refutes--> v01\n- v03 --refutes--> v0e\n- v04 --refutes--> v02\n- v0c --refutes--> v07\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9b58a7a0a4bf9b92ec607f7ecf7e6847d6645114df7c87fe7a5b2e9772b878c5","schema":"livegraph.boolean-propagation.v1","spec_hash":"4d3e01d432229ce26ce0451a7b9c8652b4ddee93f003e3262d8d44ce35d8e2c3","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-324be3676613ad632e8b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v04\n- v00 --refutes--> v09\n- v0b --supports--> v0a\n- v08 --supports--> v0c\n- v0a --refutes--> v05\n- v0b --refutes--> v0f\n- v0c --supports--> v0d\n- v0d --supports--> v02\n- v03 --supports--> v0b\n- v0d --refutes--> v01\n- v09 --supports--> v0e\n- v07 --refutes--> v08\n- v0d --supports--> v00\n- v04 --refutes--> v07\n- v0a --supports--> v06\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"da017e8f9084eb4110f51a59af81b524fc7cd441803521d37d5624e11de19edd","schema":"livegraph.boolean-propagation.v1","spec_hash":"324be3676613ad632e8b8a5a05f23a344f3928ee3b85d97ae35b0b981418d144","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":10,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-7654eb3ba02c29a8927b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v04\n- v0c --supports--> v0a\n- v0b --supports--> v07\n- v00 --refutes--> v05\n- v00 --supports--> v09\n- v0f --refutes--> v02\n- v03 --refutes--> v0e\n- v03 --supports--> v06\n- v04 --refutes--> v03\n- v0b --refutes--> v0c\n- v05 --refutes--> v01\n- v01 --supports--> v0f\n- v00 --supports--> v08\n- v05 --supports--> v0d\n- v06 --supports--> v0b\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e26bd5c51544a6af75a5194cd5dd12b03e02631e2fedb3bd5b22c66dcd70002a","schema":"livegraph.boolean-propagation.v1","spec_hash":"7654eb3ba02c29a8927bfe87d5792e515a1c414995a29d2bce584d54945585a6","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-066b195eba32074644cd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v00\n- v07 --refutes--> v09\n- v0e --supports--> v02\n- v0e --refutes--> v08\n- v0d --refutes--> v03\n- v00 --supports--> v04\n- v06 --supports--> v01\n- v05 --refutes--> v0c\n- v00 --supports--> v07\n- v0e --supports--> v06\n- v0a --refutes--> v0e\n- v0f --refutes--> v0a\n- v0c --refutes--> v0d\n- v0f --refutes--> v0b\n- v09 --supports--> v0f\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"763bba353843e6becfeabaead6af6ed25426403dcddbf96b74bd59e86c0a67e7","schema":"livegraph.boolean-propagation.v1","spec_hash":"066b195eba32074644cd56e5682de0a40154edef55fc841a292624639e68c967","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-cb6107e0e325d63d5da4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v0d\n- v0e --refutes--> v08\n- v02 --supports--> v04\n- v02 --refutes--> v03\n- v0c --refutes--> v02\n- v0d --refutes--> v09\n- v0a --supports--> v0e\n- v04 --refutes--> v07\n- v06 --refutes--> v0f\n- v0f --supports--> v0a\n- v0d --refutes--> v06\n- v0d --supports--> v05\n- v06 --supports--> v0b\n- v08 --supports--> v01\n- v0a --supports--> v00\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"42c5d5491495fb5d623dc11c941a2ed083c2b6373d3e69e74856aabe07c0dcc5","schema":"livegraph.boolean-propagation.v1","spec_hash":"cb6107e0e325d63d5da43be809bb5a5ab848c685aff58e7a0b7ab0dac1a4d668","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-e207ff22899a74288ce0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v09\n- v09 --refutes--> v02\n- v0a --refutes--> v01\n- v04 --refutes--> v0b\n- v0d --refutes--> v0a\n- v0a --supports--> v00\n- v0f --refutes--> v0d\n- v0b --refutes--> v05\n- v0a --supports--> v0c\n- v0f --refutes--> v03\n- v07 --refutes--> v0e\n- v02 --supports--> v07\n- v01 --refutes--> v08\n- v02 --refutes--> v06\n- v01 --supports--> v04\n\n[VERIFIED ROOT]\nv0f = FALSE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"86b9a2ad1663148b2c264f706fc200829c8936638fdead28b833c9874a86f156","schema":"livegraph.boolean-propagation.v1","spec_hash":"e207ff22899a74288ce077293c2ae60f1f2253a84e328b1286f2a06901861411","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-055c7e2f09e6a14c55dd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v0e\n- v0d --supports--> v03\n- v09 --supports--> v0c\n- v0f --refutes--> v06\n- v0c --refutes--> v0d\n- v07 --supports--> v04\n- v08 --supports--> v09\n- v04 --refutes--> v0f\n- v05 --supports--> v07\n- v05 --supports--> v0b\n- v02 --supports--> v00\n- v0f --supports--> v01\n- v02 --refutes--> v05\n- v0d --refutes--> v0a\n- v03 --supports--> v02\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"165bdbb96f644f1b3c38f48fd3558741202ee11fb2bceaf278c2e1735845b969","schema":"livegraph.boolean-propagation.v1","spec_hash":"055c7e2f09e6a14c55dd8166e842c1d32ae94d96e3b261eaff188f38c0384802","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-253f8fb53da85456bc95","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v00\n- v06 --supports--> v0b\n- v0d --supports--> v03\n- v08 --refutes--> v04\n- v01 --supports--> v0e\n- v04 --supports--> v02\n- v09 --refutes--> v06\n- v01 --supports--> v05\n- v09 --supports--> v07\n- v08 --supports--> v0f\n- v0e --refutes--> v0c\n- v0b --supports--> v08\n- v04 --supports--> v0a\n- v02 --supports--> v01\n- v0d --supports--> v09\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b476afbbd00f06d6934dbbff653ffb8e410e79db4e416582a256fdd68897fdcb","schema":"livegraph.boolean-propagation.v1","spec_hash":"253f8fb53da85456bc95bbdc7a052980c853c8697342bd845233136f2fea8e92","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-a67bd703a9794896387a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v0b\n- v0d --supports--> v08\n- v0c --supports--> v0f\n- v0c --refutes--> v0a\n- v04 --refutes--> v0d\n- v0b --refutes--> v07\n- v05 --supports--> v01\n- v00 --supports--> v0c\n- v05 --supports--> v09\n- v0d --supports--> v05\n- v07 --refutes--> v04\n- v09 --refutes--> v00\n- v02 --supports--> v06\n- v09 --supports--> v0e\n- v00 --supports--> v03\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"525022022dfc8392e17bcd6f6a4fd3a66c7b7563704b0161c80b5d5d4e82d3e2","schema":"livegraph.boolean-propagation.v1","spec_hash":"a67bd703a9794896387a05fe04d33ee164af519c33c834bd4fa1dd9887f3123d","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":10,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-e8e18ba67dbc014ceefa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v05\n- v04 --refutes--> v0f\n- v0c --refutes--> v0d\n- v0f --refutes--> v0c\n- v06 --refutes--> v01\n- v0c --supports--> v0b\n- v02 --refutes--> v03\n- v0f --refutes--> v00\n- v0c --supports--> v07\n- v02 --supports--> v0e\n- v0d --supports--> v02\n- v05 --supports--> v08\n- v08 --supports--> v09\n- v09 --supports--> v06\n- v0f --refutes--> v0a\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"24dd6a274e2588cf18c8670c1aa4a272054ff22714fb07c1eec2e27fb9704adc","schema":"livegraph.boolean-propagation.v1","spec_hash":"e8e18ba67dbc014ceefaa13bdafa59a9ddebfc7a2a67c2a65617c9878c267d60","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-5bfd9266f05bc229abb8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v0e\n- v0c --supports--> v05\n- v08 --refutes--> v04\n- v0b --supports--> v03\n- v0f --supports--> v02\n- v03 --refutes--> v0a\n- v06 --supports--> v0f\n- v0e --refutes--> v01\n- v01 --supports--> v08\n- v0f --supports--> v07\n- v08 --supports--> v06\n- v02 --refutes--> v0b\n- v0f --refutes--> v0d\n- v08 --supports--> v00\n- v06 --refutes--> v09\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3a297608d84a746619f9d88c131bb5a28948ab381768e1a858c746074bb71ab5","schema":"livegraph.boolean-propagation.v1","spec_hash":"5bfd9266f05bc229abb8ef4f7c6cbfa429ccd840a6b2b1bb488ed0e8e69e47ca","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-b00ffb3564f4bf9bb3f9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v0c\n- v06 --refutes--> v0d\n- v03 --refutes--> v0f\n- v05 --supports--> v06\n- v0d --supports--> v04\n- v09 --refutes--> v0a\n- v00 --supports--> v0e\n- v07 --supports--> v00\n- v09 --supports--> v08\n- v0c --refutes--> v03\n- v0b --refutes--> v09\n- v0d --refutes--> v07\n- v0e --refutes--> v0b\n- v0c --refutes--> v02\n- v0d --refutes--> v01\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6b795caea5c584f7390674caba4fbba189e684431bb088f3efeb02d24d9c4b70","schema":"livegraph.boolean-propagation.v1","spec_hash":"b00ffb3564f4bf9bb3f9bdc975b999868ad54e752a0e083f4b5ecba60749e1da","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-6ced3be4430d170b8c05","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v0b\n- v0e --supports--> v00\n- v0f --refutes--> v0d\n- v0a --refutes--> v0e\n- v0a --refutes--> v05\n- v07 --refutes--> v09\n- v0f --supports--> v02\n- v01 --refutes--> v07\n- v0f --refutes--> v06\n- v0a --refutes--> v04\n- v06 --refutes--> v08\n- v03 --refutes--> v01\n- v09 --refutes--> v0c\n- v0c --supports--> v0a\n- v00 --refutes--> v0f\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"37c8339919e8f88d5992dba3b4b7818747ac84e3a3420b0bed0f123399c9c071","schema":"livegraph.boolean-propagation.v1","spec_hash":"6ced3be4430d170b8c057380142c126581944ae419e74acac8bc33f8a3710adc","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-c0e20a8040b225dc5ebe","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v09\n- v05 --refutes--> v08\n- v0d --refutes--> v0f\n- v0f --supports--> v0e\n- v08 --supports--> v01\n- v04 --refutes--> v0b\n- v0e --supports--> v04\n- v00 --refutes--> v07\n- v05 --supports--> v02\n- v0f --refutes--> v06\n- v01 --supports--> v0c\n- v0b --refutes--> v0a\n- v0c --refutes--> v0d\n- v0c --supports--> v03\n- v0b --refutes--> v00\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0188efb0e752b290b660a53e7f2d5f5752b9223ef73694134627cf4a62bf975f","schema":"livegraph.boolean-propagation.v1","spec_hash":"c0e20a8040b225dc5ebea053e5c7b619e94918b23d35fecda0606b68a099c567","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-96112fe07bda64a987ee","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v0a\n- v0f --refutes--> v0c\n- v05 --supports--> v0e\n- v09 --refutes--> v0d\n- v06 --supports--> v09\n- v05 --refutes--> v00\n- v07 --supports--> v02\n- v05 --supports--> v03\n- v0e --supports--> v08\n- v09 --refutes--> v0f\n- v01 --refutes--> v0b\n- v0c --refutes--> v04\n- v07 --supports--> v05\n- v01 --supports--> v06\n- v0a --refutes--> v01\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2f56a7368a30650b2d4c2d1b17162abce8dfb4b79cb13787a591f2d9e3526789","schema":"livegraph.boolean-propagation.v1","spec_hash":"96112fe07bda64a987ee1cb1c4c1cda746f6388872514aecbbd1bacef5e7a684","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-973b89b6e14755d430ea","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v0f\n- v0c --refutes--> v0b\n- v0b --refutes--> v03\n- v0b --refutes--> v07\n- v0e --refutes--> v0d\n- v06 --supports--> v0a\n- v0f --supports--> v04\n- v02 --supports--> v0e\n- v0e --supports--> v06\n- v08 --supports--> v05\n- v08 --refutes--> v00\n- v03 --refutes--> v08\n- v04 --refutes--> v0c\n- v08 --refutes--> v01\n- v04 --supports--> v09\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1df5ba4ffaf89752b59b0f990fd2a38c957b3cf934ffc9590780b25dfa18a86b","schema":"livegraph.boolean-propagation.v1","spec_hash":"973b89b6e14755d430eae9e1628ad5eb77ea9235c7213c44f61245f91195b00d","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-80f01c861a7e581a4770","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v01\n- v05 --supports--> v00\n- v0d --supports--> v03\n- v0b --supports--> v0f\n- v0c --supports--> v0b\n- v0f --refutes--> v09\n- v05 --refutes--> v04\n- v0d --refutes--> v06\n- v02 --refutes--> v08\n- v03 --supports--> v0e\n- v09 --supports--> v0d\n- v0e --supports--> v05\n- v05 --refutes--> v0a\n- v0a --supports--> v07\n- v02 --supports--> v0c\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f94bb9305ad5f9af4a6d809ac6640a8b84d390dcacc5d854cd596ee23e1bdf94","schema":"livegraph.boolean-propagation.v1","spec_hash":"80f01c861a7e581a4770454277259e47e10525ba41ee68451d1603d4d5943097","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-de6d16bab47be9e32a99","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v0a\n- v06 --refutes--> v0b\n- v02 --refutes--> v0f\n- v03 --refutes--> v05\n- v03 --supports--> v08\n- v0d --refutes--> v06\n- v07 --supports--> v0c\n- v0c --supports--> v00\n- v0e --refutes--> v0d\n- v0b --supports--> v04\n- v02 --refutes--> v07\n- v0b --refutes--> v03\n- v05 --supports--> v02\n- v05 --supports--> v09\n- v0d --refutes--> v01\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0434d9c5aadd566bbda82cf48519847c134256563d00e545d0e5b0a144cd2d02","schema":"livegraph.boolean-propagation.v1","spec_hash":"de6d16bab47be9e32a99740bffd15c2ce00fb81a3dd13e02b46f11977d8faa8a","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-b1b13eda18a7fea8b8f5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v08\n- v05 --refutes--> v04\n- v09 --supports--> v05\n- v02 --refutes--> v06\n- v0d --supports--> v0f\n- v0a --refutes--> v0d\n- v0c --refutes--> v01\n- v01 --supports--> v03\n- v06 --refutes--> v00\n- v05 --refutes--> v02\n- v0d --supports--> v0e\n- v06 --refutes--> v0a\n- v08 --supports--> v09\n- v07 --refutes--> v0b\n- v01 --supports--> v07\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"442794ce10f9219c261e72ffdcd1ba4df73c0d26eecfacb9ebf9c7fe24b7c475","schema":"livegraph.boolean-propagation.v1","spec_hash":"b1b13eda18a7fea8b8f58edcfe5936f44dcf6094bae235e7779c38b232650b67","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":10,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-456d0bfa91ea31b936ca","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v03\n- v00 --supports--> v0c\n- v0d --refutes--> v09\n- v08 --supports--> v0e\n- v01 --refutes--> v00\n- v06 --supports--> v01\n- v00 --refutes--> v0b\n- v09 --refutes--> v05\n- v03 --supports--> v0f\n- v0a --refutes--> v02\n- v04 --supports--> v06\n- v00 --refutes--> v0d\n- v08 --refutes--> v0a\n- v09 --refutes--> v08\n- v03 --refutes--> v04\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c801c12a2a1d4da00daa3ada076bd41da0f669c9cc3e061d4e6617847f5c56c2","schema":"livegraph.boolean-propagation.v1","spec_hash":"456d0bfa91ea31b936ca667d7f1216e6034cff946ff58169d013c3b21200a7ea","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":10,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-ab28e560f736503f4fec","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v04\n- v0c --refutes--> v0a\n- v03 --supports--> v0b\n- v00 --supports--> v02\n- v0a --refutes--> v09\n- v04 --refutes--> v00\n- v03 --supports--> v01\n- v02 --refutes--> v0f\n- v03 --supports--> v07\n- v03 --refutes--> v0d\n- v02 --supports--> v08\n- v0f --supports--> v0c\n- v0d --supports--> v0e\n- v04 --supports--> v05\n- v0e --supports--> v06\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6b2998c041b22e5534b63f56e8e3700121f9f23c87f9e708b37161c171eb1d6c","schema":"livegraph.boolean-propagation.v1","spec_hash":"ab28e560f736503f4fecd92f427cbf8f1d766196063fdcec3eb2302dd449625b","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-61118886939848a5b148","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v0e\n- v05 --supports--> v01\n- v03 --refutes--> v0b\n- v05 --supports--> v02\n- v0d --refutes--> v00\n- v0f --refutes--> v04\n- v0f --refutes--> v07\n- v00 --refutes--> v0f\n- v06 --supports--> v05\n- v08 --supports--> v0d\n- v03 --supports--> v0c\n- v04 --supports--> v06\n- v0d --refutes--> v09\n- v0a --refutes--> v03\n- v02 --supports--> v0a\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"67127e72f3e9c0b0a9769f0a15ac5c4fcbf93c38ab84e709a0ebfd906ddbcb90","schema":"livegraph.boolean-propagation.v1","spec_hash":"61118886939848a5b148aa8789835d6701abcc31596d1923c0e93644de464671","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-b17925e65f39be02f81d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v07\n- v0b --refutes--> v05\n- v06 --supports--> v0e\n- v02 --refutes--> v04\n- v08 --supports--> v02\n- v08 --refutes--> v09\n- v0b --supports--> v0f\n- v03 --refutes--> v0c\n- v0e --refutes--> v0b\n- v0c --supports--> v08\n- v0b --refutes--> v03\n- v02 --supports--> v0a\n- v00 --refutes--> v01\n- v04 --supports--> v00\n- v01 --refutes--> v0d\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2f984e105304b53f2fd1b9b2b5fc8a6628ba21c740b2a732fe58f25c129ea5f1","schema":"livegraph.boolean-propagation.v1","spec_hash":"b17925e65f39be02f81da3c91e4841af0c6d7394848af2aa313da187ea0bda15","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-3e76d860c6dee7b5eabb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v0a\n- v06 --supports--> v00\n- v0e --supports--> v01\n- v04 --supports--> v0b\n- v05 --refutes--> v08\n- v04 --supports--> v05\n- v0e --supports--> v02\n- v0f --supports--> v09\n- v06 --refutes--> v0e\n- v07 --supports--> v06\n- v0c --supports--> v04\n- v09 --supports--> v0c\n- v0d --supports--> v07\n- v0e --refutes--> v0f\n- v03 --refutes--> v0d\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"eab32a2763c26bcbce092add8f7be74dfb130063dce3b507cecd2a048e72024d","schema":"livegraph.boolean-propagation.v1","spec_hash":"3e76d860c6dee7b5eabb196d9068327ce29f98d913318d228cef676620940934","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-d007ce57385aa4d275a4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v03\n- v0f --refutes--> v06\n- v0d --supports--> v08\n- v0d --refutes--> v0e\n- v07 --refutes--> v0d\n- v05 --refutes--> v00\n- v0c --refutes--> v0b\n- v05 --supports--> v01\n- v05 --supports--> v04\n- v00 --supports--> v09\n- v09 --supports--> v0c\n- v06 --supports--> v0a\n- v06 --refutes--> v05\n- v0b --supports--> v07\n- v0e --supports--> v02\n\n[VERIFIED ROOT]\nv0f = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cbc4eab9c173a0490700499a77e3646d800dd1af2c146add51193632c9ec824c","schema":"livegraph.boolean-propagation.v1","spec_hash":"d007ce57385aa4d275a4380eacce40edb9d2dd9e116d583135e56613c819204f","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-9cfa0baeafce82a77c13","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v06\n- v0b --refutes--> v02\n- v07 --supports--> v0c\n- v0c --refutes--> v0e\n- v0c --refutes--> v00\n- v06 --refutes--> v07\n- v06 --refutes--> v05\n- v0b --refutes--> v0a\n- v04 --refutes--> v0d\n- v0d --supports--> v0b\n- v0f --supports--> v04\n- v0f --refutes--> v03\n- v0c --refutes--> v09\n- v01 --supports--> v08\n- v09 --refutes--> v0f\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5d7123223b8963e54babd8cfa97dc1a65da6a52e10102273e6ba7f125edfdee3","schema":"livegraph.boolean-propagation.v1","spec_hash":"9cfa0baeafce82a77c1347ade272ebb7a539a470a4ca179df1c66b9de08961f7","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-37456b361cbe26d45cf6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v06\n- v05 --refutes--> v0a\n- v05 --refutes--> v03\n- v0b --supports--> v00\n- v04 --supports--> v05\n- v06 --refutes--> v02\n- v0d --refutes--> v0e\n- v0b --refutes--> v0f\n- v0e --refutes--> v09\n- v09 --supports--> v04\n- v05 --supports--> v0b\n- v02 --supports--> v08\n- v0b --refutes--> v07\n- v0c --supports--> v0d\n- v0e --refutes--> v01\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a535d8ae6f5ec737b52a6f74d2316c47336b18e61567c8eeaa43a7fceda6aa31","schema":"livegraph.boolean-propagation.v1","spec_hash":"37456b361cbe26d45cf6a47e746414c1a4f2608cd639491f103faa51dc61bd7c","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-0dfa7d152994b654970a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v0c\n- v01 --refutes--> v04\n- v07 --supports--> v05\n- v0c --supports--> v09\n- v08 --refutes--> v00\n- v05 --supports--> v02\n- v02 --supports--> v0e\n- v0a --refutes--> v0b\n- v01 --supports--> v0a\n- v0e --supports--> v03\n- v08 --refutes--> v01\n- v01 --supports--> v06\n- v03 --refutes--> v08\n- v01 --refutes--> v0f\n- v0c --supports--> v07\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e5c2bda40640f8d7aa93ec9bfe33ce387c811d5d4116d5c7fbf90b98cf06810c","schema":"livegraph.boolean-propagation.v1","spec_hash":"0dfa7d152994b654970a60e476332de2512aa4240078cd94e68070b6a7d13b40","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-a0c224cb3352849e2f0e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v0b\n- v0c --refutes--> v03\n- v00 --supports--> v08\n- v0c --supports--> v09\n- v01 --supports--> v02\n- v09 --refutes--> v0a\n- v0d --supports--> v04\n- v0f --refutes--> v0c\n- v08 --supports--> v0e\n- v0d --refutes--> v0f\n- v00 --supports--> v06\n- v0b --supports--> v05\n- v01 --supports--> v0d\n- v05 --refutes--> v01\n- v05 --refutes--> v07\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c4276ec0ada4b7dc34d9156f38f8983cc1d08df84cc3acc58aabeca8191374a8","schema":"livegraph.boolean-propagation.v1","spec_hash":"a0c224cb3352849e2f0eb57fb14027d8f5dba69a6c1ea424204e91402bbc8945","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-8bf00eeacc914b4630af","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v09\n- v0c --refutes--> v01\n- v09 --refutes--> v0e\n- v05 --supports--> v08\n- v08 --refutes--> v0b\n- v0a --refutes--> v04\n- v0b --refutes--> v02\n- v04 --supports--> v05\n- v01 --refutes--> v0f\n- v01 --refutes--> v03\n- v0b --supports--> v0d\n- v05 --supports--> v00\n- v06 --refutes--> v0a\n- v0b --refutes--> v0c\n- v07 --refutes--> v06\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7511995beab0582e2ba7c6bd2074625fa6e91d5b45d54acc2828b2fcaa0e566f","schema":"livegraph.boolean-propagation.v1","spec_hash":"8bf00eeacc914b4630afd687ac27e75829c58bc0eb31e3d519d176bc84933e67","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":10,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-a5a7558bdae849f7133d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v0c\n- v01 --supports--> v06\n- v09 --refutes--> v00\n- v02 --refutes--> v09\n- v0c --refutes--> v0d\n- v01 --supports--> v0b\n- v08 --refutes--> v0a\n- v03 --refutes--> v08\n- v09 --supports--> v03\n- v0d --supports--> v04\n- v08 --refutes--> v0f\n- v0f --refutes--> v05\n- v02 --refutes--> v07\n- v0d --refutes--> v02\n- v05 --supports--> v01\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4c94147859c07d39d258453375dc17a4c95b05b5bb2ab950045578539f2183cb","schema":"livegraph.boolean-propagation.v1","spec_hash":"a5a7558bdae849f7133d494014092def771ea539b906baa0401d541dfd6691c5","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-b278288b3fede369c78b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v09\n- v0f --supports--> v00\n- v06 --supports--> v0b\n- v09 --refutes--> v0c\n- v02 --refutes--> v0f\n- v00 --supports--> v0a\n- v07 --refutes--> v05\n- v0b --refutes--> v02\n- v0e --refutes--> v06\n- v06 --refutes--> v08\n- v0a --refutes--> v07\n- v06 --refutes--> v03\n- v0e --supports--> v01\n- v04 --supports--> v0e\n- v02 --refutes--> v0d\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e9f8149dcebb2c7928ddd1049cb3fe51f278716c1cec12c23c0d170110863d7f","schema":"livegraph.boolean-propagation.v1","spec_hash":"b278288b3fede369c78b66ab640db7e8abc9c742acd2b48b24c65f7040317ed6","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-25dbdda50d3f909c9842","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v05\n- v0a --supports--> v0d\n- v0e --refutes--> v07\n- v0d --refutes--> v01\n- v0e --refutes--> v0c\n- v0f --refutes--> v0b\n- v09 --supports--> v03\n- v04 --refutes--> v08\n- v0a --supports--> v00\n- v0b --refutes--> v0a\n- v0f --supports--> v02\n- v0e --supports--> v04\n- v09 --refutes--> v0e\n- v01 --refutes--> v06\n- v05 --refutes--> v09\n\n[VERIFIED ROOT]\nv0f = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2565a915ca3c651ca6ce3abe61aa068cf350d559dbb4e661dc09126ad75bc42f","schema":"livegraph.boolean-propagation.v1","spec_hash":"25dbdda50d3f909c98423b75c5b3c3e0ed49a8dd2f83fa6aff3de2fe0c07d2aa","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-dfa7d8258c78ea86f535","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v04\n- v02 --refutes--> v03\n- v0e --supports--> v00\n- v01 --refutes--> v0e\n- v07 --supports--> v0f\n- v05 --supports--> v09\n- v0a --refutes--> v08\n- v0b --refutes--> v0c\n- v0b --refutes--> v06\n- v08 --refutes--> v02\n- v09 --refutes--> v01\n- v02 --refutes--> v07\n- v0b --refutes--> v05\n- v0f --refutes--> v0b\n- v0e --refutes--> v0d\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9f3ee217001018f19888c00d7ecfbd6dc874ead4a305fb7036b07e37c1a4adef","schema":"livegraph.boolean-propagation.v1","spec_hash":"dfa7d8258c78ea86f5354f15765acfa6241df71b9057fc853791037021cd4c4c","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-9130d3832fc6648a551e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v0c\n- v0b --supports--> v08\n- v0c --supports--> v0d\n- v04 --refutes--> v02\n- v08 --supports--> v01\n- v0a --refutes--> v03\n- v0f --refutes--> v04\n- v07 --supports--> v0f\n- v02 --refutes--> v0b\n- v0f --refutes--> v05\n- v0a --supports--> v07\n- v0d --supports--> v0a\n- v07 --refutes--> v00\n- v04 --refutes--> v0e\n- v08 --refutes--> v06\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"66ff97715f93c4456a4ef5d4c4ed63cb4cce590bf7287ca14c19fb4ae843d15e","schema":"livegraph.boolean-propagation.v1","spec_hash":"9130d3832fc6648a551e88eb25b3ae1f7590312ca2927a77d16a57708e8c5103","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-482566beab9e8f5de646","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v07\n- v0a --refutes--> v03\n- v0e --supports--> v0c\n- v09 --supports--> v0e\n- v04 --supports--> v01\n- v00 --supports--> v09\n- v0e --supports--> v0b\n- v01 --supports--> v00\n- v03 --refutes--> v02\n- v0f --supports--> v05\n- v0e --supports--> v0d\n- v0f --refutes--> v04\n- v09 --supports--> v06\n- v07 --refutes--> v08\n- v08 --supports--> v0f\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bc1739dabc1a02a8594b0eac85f2a92b59595040a252715ff4fe4678a4ca8681","schema":"livegraph.boolean-propagation.v1","spec_hash":"482566beab9e8f5de6466bb43279937c11f268326719a8d4c10476388636817f","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-c23137c71e45423a9017","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v01\n- v0f --supports--> v03\n- v09 --refutes--> v0b\n- v0b --refutes--> v0d\n- v08 --refutes--> v0f\n- v0b --refutes--> v06\n- v07 --refutes--> v05\n- v0c --supports--> v08\n- v01 --supports--> v09\n- v0d --refutes--> v02\n- v08 --refutes--> v0e\n- v0c --supports--> v04\n- v07 --supports--> v0c\n- v0f --supports--> v0a\n- v02 --supports--> v07\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"662e8e1919f0dea3a4e0afc3221369e210e7513e9ac512d9b9e7f4a133913439","schema":"livegraph.boolean-propagation.v1","spec_hash":"c23137c71e45423a901767816f8259cdafafa376a70ee84aa54e88f3d45e8a47","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-20092752f054fc3d54cd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v0c\n- v06 --refutes--> v0e\n- v0a --supports--> v01\n- v05 --refutes--> v07\n- v0e --refutes--> v09\n- v07 --supports--> v08\n- v0c --supports--> v0a\n- v04 --refutes--> v00\n- v01 --refutes--> v02\n- v04 --supports--> v0d\n- v09 --supports--> v05\n- v09 --supports--> v0b\n- v09 --supports--> v0f\n- v0e --supports--> v03\n- v01 --refutes--> v04\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6e11e93af2166360f9ed48c95346a63e3e1036656562d84b4a1657d01eb2aca0","schema":"livegraph.boolean-propagation.v1","spec_hash":"20092752f054fc3d54cdf97e51053a7c9a1785cc244fc4e735d179e4d3902b26","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-fbf210c28ff70bffd938","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v04\n- v0d --refutes--> v07\n- v07 --supports--> v0a\n- v0b --refutes--> v0d\n- v05 --refutes--> v02\n- v0e --supports--> v08\n- v08 --refutes--> v03\n- v0b --refutes--> v09\n- v01 --supports--> v05\n- v08 --refutes--> v00\n- v0a --refutes--> v0e\n- v0e --refutes--> v0c\n- v04 --supports--> v01\n- v01 --refutes--> v0f\n- v04 --refutes--> v06\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"52083ee301c20cfae89d0366d54142027e7b04602d196adc11484d2e583a698c","schema":"livegraph.boolean-propagation.v1","spec_hash":"fbf210c28ff70bffd93864e9b515a0db60f6ecc85ea6290b978f0804479024f9","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-7db5d35555a1c587e44b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v0c\n- v02 --supports--> v05\n- v08 --refutes--> v00\n- v07 --supports--> v0e\n- v0b --supports--> v03\n- v09 --supports--> v07\n- v0e --refutes--> v0d\n- v0d --supports--> v04\n- v0b --supports--> v06\n- v04 --supports--> v08\n- v00 --supports--> v02\n- v0c --refutes--> v09\n- v08 --supports--> v01\n- v0c --refutes--> v0a\n- v0e --supports--> v0f\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0050ba3de9411bba2c2e0386a28a6b5789a7772a7b48bf6b6442721652350c9d","schema":"livegraph.boolean-propagation.v1","spec_hash":"7db5d35555a1c587e44b3db6d069abb758b0940d4823fdb9b1b08201e4af0a4a","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-4c8e410a21f50aa39ded","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v0f\n- v0a --refutes--> v05\n- v08 --refutes--> v09\n- v0f --refutes--> v0a\n- v03 --supports--> v0c\n- v0f --refutes--> v00\n- v0e --refutes--> v06\n- v0e --refutes--> v04\n- v0b --refutes--> v0e\n- v0e --refutes--> v01\n- v03 --refutes--> v08\n- v05 --supports--> v02\n- v09 --supports--> v0d\n- v0e --supports--> v07\n- v02 --supports--> v03\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"168138dd4a7a5e60e24ea82fb59553056e60715958d6cc39b9df5284d8f4ba99","schema":"livegraph.boolean-propagation.v1","spec_hash":"4c8e410a21f50aa39dedf42975071ec3f35041478762d52517c29baf111c19ab","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-73195a5a76503a185dde","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v0d\n- v04 --supports--> v0a\n- v0c --refutes--> v07\n- v07 --supports--> v08\n- v04 --supports--> v0f\n- v06 --supports--> v02\n- v0e --refutes--> v0c\n- v06 --supports--> v05\n- v02 --refutes--> v01\n- v02 --refutes--> v0b\n- v08 --supports--> v04\n- v03 --supports--> v0e\n- v01 --supports--> v03\n- v04 --supports--> v00\n- v0a --refutes--> v09\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8fd8a4bf285ca171e39ba4306ea436d6b569a76b7491869640d46563ab5fbefd","schema":"livegraph.boolean-propagation.v1","spec_hash":"73195a5a76503a185dde5b28963f48297215c9ee32ab304daa8cc37a91ad33a7","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-c49d799695ae4a66d7c3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v04\n- v05 --refutes--> v09\n- v0c --supports--> v0a\n- v06 --refutes--> v0d\n- v01 --supports--> v07\n- v0b --refutes--> v00\n- v02 --supports--> v05\n- v0c --supports--> v0e\n- v07 --refutes--> v08\n- v0d --supports--> v02\n- v0b --refutes--> v06\n- v09 --supports--> v0f\n- v08 --supports--> v0c\n- v0d --supports--> v03\n- v0c --supports--> v0b\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8bdfe79f030ef8298b96874a8c003eb81e34b0ad098b80e3dc745a6b884a495d","schema":"livegraph.boolean-propagation.v1","spec_hash":"c49d799695ae4a66d7c39efe226e282c6692e7553b89ff128b89175f8686e98f","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":10,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-f020cae80a929cdc5b94","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v02\n- v01 --refutes--> v03\n- v00 --supports--> v04\n- v0b --refutes--> v0d\n- v03 --refutes--> v07\n- v0d --supports--> v09\n- v0c --supports--> v00\n- v0a --refutes--> v01\n- v04 --refutes--> v0f\n- v0c --refutes--> v05\n- v0e --supports--> v0b\n- v03 --supports--> v0c\n- v0a --refutes--> v06\n- v0f --refutes--> v0e\n- v0d --supports--> v08\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"eab48d99a53b1ed4970297cafb71719595e56443aedd6e05630dc4af5d79e419","schema":"livegraph.boolean-propagation.v1","spec_hash":"f020cae80a929cdc5b943421e1bcb0f06a1200654b48f5abd9a5b855e6dc65de","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-3cfce08e873e2393d34a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v04\n- v08 --refutes--> v00\n- v08 --refutes--> v06\n- v08 --refutes--> v02\n- v05 --supports--> v0d\n- v01 --refutes--> v09\n- v03 --supports--> v0e\n- v04 --refutes--> v05\n- v01 --refutes--> v07\n- v0c --refutes--> v0b\n- v05 --supports--> v08\n- v08 --refutes--> v0c\n- v0b --refutes--> v0a\n- v0f --refutes--> v03\n- v07 --supports--> v0f\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"da9badd40aa8386242557d5711fea4c611108916337a711d8602950c1c236d05","schema":"livegraph.boolean-propagation.v1","spec_hash":"3cfce08e873e2393d34ac822511c2ebd5de01c580841693f254b8be7c8ff6465","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-901f69c3aa0c17952702","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v05\n- v04 --refutes--> v0a\n- v09 --supports--> v0d\n- v0e --supports--> v06\n- v0e --refutes--> v0f\n- v0c --supports--> v0b\n- v01 --refutes--> v0c\n- v00 --refutes--> v0e\n- v0b --refutes--> v04\n- v0b --refutes--> v07\n- v05 --supports--> v08\n- v0f --refutes--> v03\n- v0f --supports--> v02\n- v05 --supports--> v00\n- v0f --refutes--> v09\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fff9f95d0bf71eb4c68e6c0705bd57c7c60e22fdff9e63abb247a4a7735a59a9","schema":"livegraph.boolean-propagation.v1","spec_hash":"901f69c3aa0c179527025d1656779d43efd9619492aa48797d732b0dc85cdf70","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-b4c11d7a3ec32b59513b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v0f\n- v0c --supports--> v01\n- v0e --refutes--> v03\n- v0e --supports--> v04\n- v03 --refutes--> v09\n- v01 --supports--> v07\n- v0d --supports--> v02\n- v00 --supports--> v05\n- v00 --refutes--> v0d\n- v09 --supports--> v0b\n- v01 --supports--> v00\n- v09 --supports--> v08\n- v0d --supports--> v06\n- v0a --refutes--> v0c\n- v06 --supports--> v0e\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"16e36a19b36b8b08665ccfcc3863062d5c8dac0298fa0863a59f622ca6a996bd","schema":"livegraph.boolean-propagation.v1","spec_hash":"b4c11d7a3ec32b59513bd6215c0a2262220d32ad3c1efd05ca6821c9bf380eee","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-5e6b9d83f07cba1bc2f2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v07\n- v00 --supports--> v0e\n- v01 --refutes--> v04\n- v05 --supports--> v0b\n- v0d --refutes--> v01\n- v07 --refutes--> v08\n- v0c --refutes--> v00\n- v04 --supports--> v02\n- v06 --supports--> v0f\n- v0f --refutes--> v09\n- v04 --supports--> v05\n- v0d --supports--> v03\n- v0f --refutes--> v0d\n- v05 --refutes--> v0c\n- v07 --refutes--> v06\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0d6c32eba22f2d5e421be97994cbbf3d894751506636837a5e05220d170410b2","schema":"livegraph.boolean-propagation.v1","spec_hash":"5e6b9d83f07cba1bc2f21683d1bc135685856d058e84b69414e86c197f14b70f","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-02572708fbee7daeb5dc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v0e\n- v0e --supports--> v06\n- v07 --supports--> v01\n- v0e --supports--> v00\n- v04 --supports--> v09\n- v08 --refutes--> v0a\n- v02 --supports--> v07\n- v0b --supports--> v05\n- v06 --refutes--> v0b\n- v0c --supports--> v0f\n- v0c --refutes--> v08\n- v0b --refutes--> v04\n- v09 --refutes--> v0c\n- v0b --supports--> v03\n- v01 --refutes--> v0d\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ad4eb88b4854d8bb1dc1bb100e6c83fa61a0038ff7b01c01394cd07c42dfe88e","schema":"livegraph.boolean-propagation.v1","spec_hash":"02572708fbee7daeb5dc5e3dc6f4741b6d4e05f287a7983191fefb032b4b7f7e","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-b9801ae4bfe85d504cd5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v00\n- v0a --supports--> v02\n- v0b --refutes--> v07\n- v00 --supports--> v05\n- v02 --refutes--> v0d\n- v07 --supports--> v0a\n- v0d --refutes--> v0f\n- v02 --supports--> v08\n- v0d --supports--> v04\n- v03 --supports--> v0e\n- v05 --supports--> v0b\n- v0f --refutes--> v03\n- v05 --supports--> v06\n- v00 --supports--> v09\n- v00 --supports--> v01\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"96990e7bf4af770f1317dffdfb7e752be7cfbee47a4839da6fb81ca5379035ba","schema":"livegraph.boolean-propagation.v1","spec_hash":"b9801ae4bfe85d504cd5e37ffbcbed9ae496efdd8e2e504038f94236e88d68ad","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-f498290e94cdb368f924","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v02\n- v07 --refutes--> v03\n- v01 --supports--> v0b\n- v0a --refutes--> v09\n- v03 --supports--> v0d\n- v0a --refutes--> v05\n- v01 --refutes--> v0f\n- v0f --refutes--> v06\n- v04 --supports--> v0a\n- v0d --supports--> v04\n- v09 --supports--> v0e\n- v02 --refutes--> v01\n- v0f --refutes--> v0c\n- v02 --refutes--> v08\n- v09 --supports--> v00\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"542ba8431353da803c8e66b81c2cb983798ff2bf704f4ca845d19e423e8bb960","schema":"livegraph.boolean-propagation.v1","spec_hash":"f498290e94cdb368f9247e82920cf769d4c5ceaba19e0a5a914635de5c7d9a52","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-7ddd895234f95cf466d8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v0c\n- v03 --supports--> v08\n- v02 --supports--> v0b\n- v07 --supports--> v09\n- v07 --supports--> v01\n- v06 --refutes--> v00\n- v06 --refutes--> v0e\n- v05 --supports--> v0a\n- v08 --supports--> v0f\n- v0c --supports--> v05\n- v08 --supports--> v04\n- v02 --refutes--> v07\n- v0f --supports--> v0d\n- v0a --refutes--> v02\n- v00 --supports--> v03\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3eb83b2a73b39113c47277026107477711b2cfe2cd0e4c10322eb316965a8b25","schema":"livegraph.boolean-propagation.v1","spec_hash":"7ddd895234f95cf466d860bd741ccc77238015e55b579d7fc6189d448a53965d","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-149410db054c5f7933d4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v03\n- v0b --supports--> v0a\n- v03 --supports--> v0d\n- v06 --supports--> v07\n- v02 --supports--> v09\n- v0e --refutes--> v00\n- v0e --supports--> v0c\n- v0a --supports--> v0f\n- v04 --refutes--> v0b\n- v04 --refutes--> v05\n- v09 --supports--> v01\n- v03 --supports--> v04\n- v0f --supports--> v0e\n- v0c --refutes--> v06\n- v0f --refutes--> v08\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4e2381986c947c36bbe6cf824a1303be9f46115e75212ffa30c22f03701a641d","schema":"livegraph.boolean-propagation.v1","spec_hash":"149410db054c5f7933d418b53e8cdadc258294a6bc9948c2eea0dc0e7c45932c","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-9c27a6c4af44ba5752fb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v0c\n- v0f --refutes--> v0e\n- v07 --supports--> v03\n- v0a --supports--> v06\n- v00 --refutes--> v04\n- v0e --refutes--> v08\n- v08 --refutes--> v0d\n- v09 --refutes--> v07\n- v00 --refutes--> v05\n- v0a --refutes--> v09\n- v03 --refutes--> v0b\n- v0d --refutes--> v01\n- v02 --supports--> v0a\n- v03 --supports--> v0f\n- v00 --supports--> v02\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d9050f766575e338b5af8db711953f93b1853a401add5e25884895057c3068f5","schema":"livegraph.boolean-propagation.v1","spec_hash":"9c27a6c4af44ba5752fb5a4b568aab7149fa4a9c499f13f1c3dcf62eacaa0cd0","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-615699be368307ebc134","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v0a\n- v03 --refutes--> v04\n- v0a --refutes--> v02\n- v0d --supports--> v08\n- v02 --refutes--> v0e\n- v02 --refutes--> v0b\n- v0e --refutes--> v01\n- v06 --refutes--> v07\n- v00 --supports--> v0f\n- v00 --supports--> v09\n- v04 --refutes--> v00\n- v0d --supports--> v06\n- v00 --supports--> v0c\n- v06 --refutes--> v05\n- v0c --refutes--> v0d\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1809f7f7ed80ee160f91ca4d4f277e22794ec20db7aea501826f219ae7cb23f1","schema":"livegraph.boolean-propagation.v1","spec_hash":"615699be368307ebc134693afd728ee808330aebdf673d4f1bb6661e6f906253","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-696f65351da4dcbec474","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v04\n- v05 --supports--> v0e\n- v0d --refutes--> v05\n- v01 --supports--> v0b\n- v02 --supports--> v07\n- v00 --refutes--> v0a\n- v07 --refutes--> v06\n- v09 --refutes--> v0d\n- v07 --refutes--> v0c\n- v0e --refutes--> v01\n- v07 --supports--> v09\n- v07 --supports--> v08\n- v0a --refutes--> v02\n- v0e --supports--> v03\n- v0f --supports--> v00\n\n[VERIFIED ROOT]\nv0f = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d17d04bd013da065f617dbfe395e28763f315c4d856b11c16ed44e8e36246a7c","schema":"livegraph.boolean-propagation.v1","spec_hash":"696f65351da4dcbec4741ae7d6d700b11eb2eef17ccdbe2f7d9eb4eea7976b59","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-4714f0e0c257be9c07c2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v0f\n- v03 --supports--> v06\n- v0c --supports--> v0b\n- v0e --supports--> v07\n- v09 --supports--> v05\n- v0c --refutes--> v03\n- v06 --refutes--> v00\n- v02 --refutes--> v0e\n- v04 --supports--> v02\n- v00 --refutes--> v04\n- v08 --refutes--> v0d\n- v05 --refutes--> v08\n- v08 --refutes--> v0c\n- v06 --refutes--> v0a\n- v08 --refutes--> v01\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b6f1847007ad927d9e88d316f9ce61745848c1e31c3dfc28bc5c5e77d9bc9787","schema":"livegraph.boolean-propagation.v1","spec_hash":"4714f0e0c257be9c07c2c969094036fd2508ca159f8aa4cd2be0933e9335af9f","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-9d2f24f13db3c3460486","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v0f\n- v0c --supports--> v06\n- v03 --refutes--> v0a\n- v06 --supports--> v0b\n- v05 --supports--> v0d\n- v06 --supports--> v01\n- v0d --refutes--> v0c\n- v0e --supports--> v00\n- v0b --refutes--> v04\n- v0c --refutes--> v07\n- v08 --refutes--> v0e\n- v0d --refutes--> v09\n- v00 --supports--> v02\n- v02 --supports--> v05\n- v03 --supports--> v08\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"88b7c2b9aad1902e1c1eb255db16de092e6812a5f0da1a8896816eefa79c5093","schema":"livegraph.boolean-propagation.v1","spec_hash":"9d2f24f13db3c3460486f0e24ed1687199ab294721bb65919ecfc22d2c3ebeff","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-3209fefe57374b60beab","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v0b\n- v01 --refutes--> v04\n- v0d --refutes--> v0a\n- v0b --refutes--> v01\n- v03 --supports--> v0c\n- v06 --supports--> v0f\n- v03 --supports--> v00\n- v0d --supports--> v0e\n- v01 --refutes--> v08\n- v00 --refutes--> v05\n- v06 --refutes--> v03\n- v05 --refutes--> v0d\n- v02 --refutes--> v06\n- v02 --supports--> v09\n- v08 --refutes--> v02\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"76cfc14b356a3c9da1cf8c9a2a62c7e62e09dc007e81a42636eca2f7bd503544","schema":"livegraph.boolean-propagation.v1","spec_hash":"3209fefe57374b60beab8eb4d26e85df0c358c2f0e3d1b3d8887f8021de4c8c2","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":10,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-881838db9cc3f79966c0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v00\n- v0f --refutes--> v0c\n- v0f --supports--> v05\n- v0b --supports--> v01\n- v04 --supports--> v09\n- v0a --supports--> v06\n- v04 --refutes--> v0b\n- v06 --supports--> v02\n- v01 --refutes--> v08\n- v0a --supports--> v03\n- v0c --supports--> v04\n- v02 --supports--> v0f\n- v0e --refutes--> v0a\n- v0d --supports--> v0e\n- v02 --supports--> v07\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b854623d0e905d43d966af41229b4ab214d67de1c6c605f68936bfb536aa750e","schema":"livegraph.boolean-propagation.v1","spec_hash":"881838db9cc3f79966c0728de093d32866ff4cb4559e57b0be87e998a8896bbc","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-c8aea9ff44e5b1c75d6a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v09\n- v06 --supports--> v0a\n- v07 --supports--> v05\n- v08 --refutes--> v0d\n- v0f --supports--> v06\n- v04 --supports--> v00\n- v09 --refutes--> v07\n- v07 --supports--> v04\n- v00 --refutes--> v03\n- v09 --supports--> v0c\n- v0b --refutes--> v02\n- v00 --refutes--> v08\n- v02 --refutes--> v0e\n- v0d --supports--> v0b\n- v09 --supports--> v01\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0b84a6fc430758ed207f4bcc0ab9962b0d34e6d16c9d6d6bc18bac93a8457e0c","schema":"livegraph.boolean-propagation.v1","spec_hash":"c8aea9ff44e5b1c75d6aaa20eeab0bacf209b0a75547705ce31cfd5209f9496e","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-f76c0486b7b2834a7a54","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v02\n- v09 --supports--> v03\n- v0d --refutes--> v00\n- v0d --refutes--> v04\n- v0d --refutes--> v05\n- v07 --supports--> v0a\n- v0c --refutes--> v0b\n- v0f --refutes--> v07\n- v01 --refutes--> v09\n- v06 --refutes--> v08\n- v05 --supports--> v0f\n- v0c --refutes--> v0d\n- v0e --refutes--> v0c\n- v07 --refutes--> v01\n- v03 --supports--> v06\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f716cb39a9070fa3ab214573d34206cb3293beee7fe2d50418a167dd564c7368","schema":"livegraph.boolean-propagation.v1","spec_hash":"f76c0486b7b2834a7a54bd727531f5839bb8c671eef651b5b5015b189a5ca075","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-f0385247387d53b0deaa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v05\n- v05 --refutes--> v03\n- v0b --refutes--> v00\n- v02 --supports--> v0d\n- v07 --refutes--> v08\n- v00 --supports--> v0e\n- v0d --supports--> v0b\n- v0d --refutes--> v04\n- v03 --refutes--> v01\n- v00 --supports--> v07\n- v06 --supports--> v09\n- v05 --supports--> v0a\n- v0f --refutes--> v06\n- v07 --supports--> v0f\n- v03 --refutes--> v0c\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a161cd2b9ab3f4e9ebcfa28bf504f76b010d1e3c4eb400bbd0906b72e81a600a","schema":"livegraph.boolean-propagation.v1","spec_hash":"f0385247387d53b0deaa5f541b3c382279d7b106552fb8d6a976d5b906816c56","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-c42c875822a286bad896","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v08\n- v02 --supports--> v0b\n- v0a --supports--> v02\n- v00 --refutes--> v06\n- v09 --supports--> v0d\n- v0c --supports--> v0e\n- v0f --refutes--> v07\n- v00 --supports--> v01\n- v03 --supports--> v0a\n- v02 --refutes--> v00\n- v03 --supports--> v05\n- v07 --supports--> v0c\n- v00 --refutes--> v09\n- v04 --supports--> v0f\n- v0e --supports--> v03\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a9a5789fabfbeb3fa3dd3a9ff9ae7c573712a21f9654125e960ca743a74e1a32","schema":"livegraph.boolean-propagation.v1","spec_hash":"c42c875822a286bad8968473b363e6f2c5d92fb039a88664fbb9393579b0e639","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-8ed31d96fe5802a2217c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v07\n- v02 --supports--> v06\n- v0d --refutes--> v0b\n- v01 --refutes--> v09\n- v02 --supports--> v0d\n- v04 --supports--> v03\n- v05 --supports--> v0f\n- v03 --refutes--> v00\n- v0b --refutes--> v0a\n- v05 --supports--> v0c\n- v07 --supports--> v01\n- v04 --supports--> v0e\n- v0b --refutes--> v08\n- v00 --refutes--> v02\n- v01 --supports--> v04\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d33def11b65f4705f88cb38f19d38c5c655549862cd275f7e6d43e31d8d406f6","schema":"livegraph.boolean-propagation.v1","spec_hash":"8ed31d96fe5802a2217c3a8b6908a03028084b061d3b20ce5423f20d805b308b","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-7e137feab956e4b1b47b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v0c\n- v03 --refutes--> v01\n- v05 --refutes--> v06\n- v07 --supports--> v0f\n- v0f --supports--> v0b\n- v00 --refutes--> v0e\n- v0b --supports--> v02\n- v09 --supports--> v07\n- v0b --supports--> v03\n- v0e --refutes--> v05\n- v0a --supports--> v00\n- v05 --refutes--> v08\n- v04 --supports--> v09\n- v05 --supports--> v04\n- v04 --supports--> v0d\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"50f57a653d9b87a694b8b5f29397029d110bc713d56447a567cfe9069e35c2b0","schema":"livegraph.boolean-propagation.v1","spec_hash":"7e137feab956e4b1b47bfc80608dbff2131ac9213d101f98024efb1c8b359fae","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-b7d4cf15b0609bcd17b9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v00\n- v0c --refutes--> v05\n- v00 --refutes--> v08\n- v07 --refutes--> v03\n- v01 --supports--> v0a\n- v04 --refutes--> v09\n- v07 --refutes--> v0d\n- v08 --supports--> v0f\n- v0c --refutes--> v04\n- v0d --refutes--> v0c\n- v0c --supports--> v0e\n- v09 --supports--> v06\n- v0a --supports--> v07\n- v0d --refutes--> v02\n- v0a --supports--> v0b\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ada65c803d2462778e362cf02f05e4330ef7b8439635a1059c0bcef7c8a98a53","schema":"livegraph.boolean-propagation.v1","spec_hash":"b7d4cf15b0609bcd17b9c846f2833fa8814ca231964b3ddfa32e06e396bfb0e5","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":10,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-0721b3db5ab3476925e4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v0d\n- v09 --refutes--> v06\n- v0a --refutes--> v0b\n- v05 --supports--> v01\n- v08 --refutes--> v04\n- v08 --supports--> v0c\n- v02 --refutes--> v0e\n- v0a --refutes--> v02\n- v00 --refutes--> v0f\n- v07 --supports--> v00\n- v03 --refutes--> v05\n- v02 --supports--> v07\n- v06 --supports--> v08\n- v01 --refutes--> v09\n- v00 --supports--> v03\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d09f89d298ef7e8442003a894359fdfeb6058f4cff64e533c1dd2eb95e1e50ae","schema":"livegraph.boolean-propagation.v1","spec_hash":"0721b3db5ab3476925e467524782104df4af63715634e6b5e199c81eaefe7b9d","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":10,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-11bdf62084e11106cf69","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v07\n- v00 --refutes--> v01\n- v0d --refutes--> v0e\n- v06 --refutes--> v04\n- v06 --refutes--> v0b\n- v0e --supports--> v09\n- v01 --supports--> v0d\n- v0f --supports--> v03\n- v0d --refutes--> v08\n- v0f --supports--> v05\n- v05 --refutes--> v02\n- v0c --supports--> v00\n- v0e --refutes--> v0a\n- v09 --supports--> v06\n- v0b --refutes--> v0f\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e9182b5d006e9542ec64a430e2fad5162c7c263c1f7b6fb55e77a2f7eaf6e085","schema":"livegraph.boolean-propagation.v1","spec_hash":"11bdf62084e11106cf69506ec83e7b92d64c7d8af1be45a5e13827e3f202d99b","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-e17938fa08b177f15317","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v02\n- v00 --refutes--> v01\n- v02 --supports--> v0d\n- v02 --supports--> v04\n- v02 --supports--> v0c\n- v09 --supports--> v0f\n- v07 --supports--> v08\n- v07 --supports--> v0e\n- v05 --refutes--> v07\n- v0a --supports--> v05\n- v03 --refutes--> v0a\n- v05 --refutes--> v0b\n- v0d --supports--> v09\n- v00 --supports--> v06\n- v06 --refutes--> v03\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3fa104c814dfd8c80b3327c27d04731c374f95f13901cf56fa1f5d8aa8f9c328","schema":"livegraph.boolean-propagation.v1","spec_hash":"e17938fa08b177f15317896e2ce74bd02d52a7bdfc6da9e655ee0217cdf90e59","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-4dab7e79af8abac67bff","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v09\n- v05 --supports--> v0c\n- v01 --refutes--> v0f\n- v01 --supports--> v07\n- v0e --refutes--> v08\n- v03 --refutes--> v04\n- v06 --refutes--> v02\n- v0c --supports--> v0a\n- v0e --supports--> v03\n- v0e --refutes--> v00\n- v0d --refutes--> v06\n- v07 --refutes--> v0e\n- v0a --supports--> v01\n- v0d --refutes--> v0b\n- v02 --refutes--> v05\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"75de1de0dde6a73a33e84cf57d1135e2b3441f617e31b7c308cf444636147f45","schema":"livegraph.boolean-propagation.v1","spec_hash":"4dab7e79af8abac67bffd5cb1bf5ec05e4f067f7087eae92de8f3d257b5ba24b","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-85f38759d635cb8fecd2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v0c\n- v00 --refutes--> v02\n- v0e --supports--> v03\n- v04 --refutes--> v0d\n- v0c --supports--> v08\n- v03 --refutes--> v09\n- v01 --supports--> v0f\n- v03 --refutes--> v0b\n- v04 --supports--> v06\n- v05 --refutes--> v0e\n- v00 --supports--> v01\n- v0a --supports--> v00\n- v08 --supports--> v0a\n- v01 --refutes--> v07\n- v04 --refutes--> v05\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f9e86ab16b82a49baf1e6ca4d862571999ea839695a3c005c950093920f34fa0","schema":"livegraph.boolean-propagation.v1","spec_hash":"85f38759d635cb8fecd250b7fe0306f095a4388e790c2178a7f56baa749579e5","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-1e993df5cd2e78f61b7d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v01\n- v00 --supports--> v0e\n- v09 --refutes--> v0b\n- v0d --supports--> v03\n- v02 --supports--> v0a\n- v0e --supports--> v0c\n- v02 --supports--> v07\n- v08 --supports--> v02\n- v03 --supports--> v0f\n- v07 --refutes--> v0d\n- v01 --supports--> v09\n- v07 --refutes--> v05\n- v0d --refutes--> v06\n- v0e --supports--> v08\n- v03 --refutes--> v04\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a3bce7c125741a3ddb366f15d7811151db02d4a472ca8308a0766f28d71880e9","schema":"livegraph.boolean-propagation.v1","spec_hash":"1e993df5cd2e78f61b7d1ee8557c2a69e38915f2525428e5ef6cd21de0239eff","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-ec817d54e0b21a893a70","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v09\n- v06 --refutes--> v05\n- v0b --supports--> v08\n- v00 --supports--> v0e\n- v04 --refutes--> v07\n- v02 --refutes--> v0f\n- v08 --refutes--> v02\n- v00 --refutes--> v0c\n- v04 --refutes--> v01\n- v05 --supports--> v0d\n- v0c --refutes--> v0b\n- v08 --refutes--> v0a\n- v09 --refutes--> v03\n- v02 --supports--> v04\n- v03 --supports--> v06\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ca9c9f920e35e7d1f469a42cdbeb7e0187284fa4370a28c2977439e11a0fc7b3","schema":"livegraph.boolean-propagation.v1","spec_hash":"ec817d54e0b21a893a70cd0a4a950441fcab6ffd5e883454d337dc89aea7d060","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":10,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-c8e055a1e0ae39e0f0e2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v08\n- v0d --refutes--> v06\n- v03 --refutes--> v0e\n- v07 --refutes--> v0d\n- v03 --refutes--> v07\n- v06 --supports--> v01\n- v06 --supports--> v0b\n- v04 --supports--> v03\n- v0f --refutes--> v0a\n- v09 --supports--> v02\n- v0b --refutes--> v0f\n- v0c --supports--> v09\n- v06 --supports--> v00\n- v09 --supports--> v05\n- v05 --supports--> v04\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4263bb1f99478be74c1d8d504e676b55b7f3645e5f2100798f53f95e9ebeda73","schema":"livegraph.boolean-propagation.v1","spec_hash":"c8e055a1e0ae39e0f0e2024753f57c5c564fac79d52d547dacf08574be16ba41","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-481aba4ede2815218c6e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v0e\n- v0c --supports--> v03\n- v0e --supports--> v02\n- v03 --refutes--> v09\n- v09 --refutes--> v06\n- v02 --supports--> v05\n- v0a --refutes--> v04\n- v0c --refutes--> v08\n- v02 --supports--> v0c\n- v0a --supports--> v0f\n- v03 --refutes--> v07\n- v06 --refutes--> v0d\n- v0c --refutes--> v0b\n- v0f --refutes--> v00\n- v0d --supports--> v0a\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1183e121ab29fd7acc569b19d080576dd8cf04741ec1e1c3405a57b94bb9fc00","schema":"livegraph.boolean-propagation.v1","spec_hash":"481aba4ede2815218c6e19a94e69b2d808d2b2c914d5d5b63bcb431e0ca02d35","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-25c2b1bf46f50c4650c7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v0c\n- v04 --refutes--> v03\n- v08 --refutes--> v01\n- v01 --refutes--> v0b\n- v09 --supports--> v0a\n- v07 --supports--> v09\n- v02 --refutes--> v04\n- v09 --refutes--> v08\n- v05 --supports--> v06\n- v0c --supports--> v07\n- v01 --supports--> v05\n- v04 --supports--> v00\n- v00 --refutes--> v0d\n- v06 --supports--> v0e\n- v0c --supports--> v0f\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c47e4f7a06c98af14736b5c26419d50aa9e7ec73082f00ce05377fb7cae02113","schema":"livegraph.boolean-propagation.v1","spec_hash":"25c2b1bf46f50c4650c75bf4e270b50e14b6fda74a6b5e1346508b493975c909","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":10,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-9ad1cc5dbd883ae6cf09","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v07\n- v09 --supports--> v05\n- v08 --refutes--> v01\n- v03 --supports--> v0d\n- v05 --supports--> v00\n- v02 --supports--> v0c\n- v0a --refutes--> v0b\n- v0f --refutes--> v03\n- v02 --refutes--> v0a\n- v00 --supports--> v0f\n- v0d --refutes--> v02\n- v0a --refutes--> v04\n- v01 --refutes--> v06\n- v01 --supports--> v09\n- v05 --supports--> v0e\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"02ec2a63a3d8942594da9e1d666b314cac2006b4b9fccc1b6c77721d9b17d680","schema":"livegraph.boolean-propagation.v1","spec_hash":"9ad1cc5dbd883ae6cf0989e8dc11750169e7b81e49b37464654d297dfe1363f5","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-49c0c82bc7cafc21c1dc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v0a\n- v00 --refutes--> v0d\n- v07 --refutes--> v06\n- v00 --supports--> v04\n- v05 --supports--> v0f\n- v09 --supports--> v02\n- v09 --supports--> v01\n- v03 --refutes--> v0e\n- v07 --refutes--> v0b\n- v0f --supports--> v07\n- v0c --supports--> v05\n- v09 --refutes--> v08\n- v0b --refutes--> v03\n- v02 --refutes--> v0c\n- v0a --refutes--> v00\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7d8b613f8f85f64b147e45d78837ff37ff074245ff98a9ea6ec22f5185dc7067","schema":"livegraph.boolean-propagation.v1","spec_hash":"49c0c82bc7cafc21c1dc069d96996817177ce2313311353025a66f34ef7e13c9","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-09e0db69302aa843be9f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v0e\n- v0f --refutes--> v09\n- v0e --refutes--> v07\n- v06 --supports--> v0a\n- v0c --supports--> v00\n- v05 --supports--> v06\n- v0e --refutes--> v08\n- v06 --supports--> v01\n- v00 --supports--> v0d\n- v06 --refutes--> v04\n- v05 --refutes--> v02\n- v08 --refutes--> v0c\n- v03 --supports--> v0b\n- v0f --supports--> v03\n- v0d --supports--> v0f\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6b43f55f464ce49feb0d4b70e46f5569fc090f0ac74c2721aa04e6b20b1915bf","schema":"livegraph.boolean-propagation.v1","spec_hash":"09e0db69302aa843be9fc3161628669ed7477376f2fd6e332a3ff4f0323cbb7a","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-d252158c6bf900702c21","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v0e\n- v0f --supports--> v03\n- v02 --supports--> v0b\n- v00 --supports--> v05\n- v03 --refutes--> v01\n- v04 --supports--> v0d\n- v0c --supports--> v0f\n- v0d --refutes--> v02\n- v03 --supports--> v08\n- v0a --supports--> v07\n- v03 --refutes--> v06\n- v05 --supports--> v0a\n- v05 --refutes--> v09\n- v02 --supports--> v0c\n- v07 --refutes--> v04\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"639dfbd6f3c0fdfb41f6950447d7850a0627864724668d04e7b7fc599551abdc","schema":"livegraph.boolean-propagation.v1","spec_hash":"d252158c6bf900702c2188fcaed89e7192e742017c2700d6c6c272ceee74585a","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-14d9ccc99aa0e7452cea","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v0b\n- v0f --supports--> v00\n- v07 --supports--> v02\n- v0f --refutes--> v05\n- v0f --supports--> v03\n- v05 --supports--> v0d\n- v0b --refutes--> v07\n- v08 --refutes--> v0f\n- v0a --refutes--> v09\n- v01 --supports--> v06\n- v0f --supports--> v04\n- v01 --refutes--> v0a\n- v0d --refutes--> v0c\n- v09 --supports--> v0e\n- v02 --refutes--> v08\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d298459cb20287960da7e72e85c2d5739267f44198e656908e8fde4fbd15d807","schema":"livegraph.boolean-propagation.v1","spec_hash":"14d9ccc99aa0e7452ceab4d14e2f6946551d78419af24a5f809c2edaa1b9fc8a","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-3b393429b3974ee7c087","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v0e\n- v0d --refutes--> v05\n- v06 --refutes--> v08\n- v0c --supports--> v06\n- v00 --supports--> v02\n- v0f --refutes--> v0a\n- v0a --refutes--> v03\n- v0a --refutes--> v07\n- v0b --supports--> v00\n- v05 --supports--> v0b\n- v0a --refutes--> v01\n- v07 --refutes--> v0c\n- v08 --refutes--> v0d\n- v0d --supports--> v04\n- v0f --refutes--> v09\n\n[VERIFIED ROOT]\nv0f = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ffaad45835e6a0d5fb004ce7453be6b08016676b20bd74890e6cc2857b3d07de","schema":"livegraph.boolean-propagation.v1","spec_hash":"3b393429b3974ee7c08742bd3875c21eea7ff7a18132d3f9174a49a4ecc73bba","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-07b0de88c92f7ccafdd5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v05\n- v0b --refutes--> v08\n- v0a --refutes--> v04\n- v03 --supports--> v00\n- v07 --supports--> v0b\n- v0f --supports--> v06\n- v04 --refutes--> v01\n- v08 --supports--> v02\n- v0f --refutes--> v03\n- v0c --supports--> v0d\n- v0c --supports--> v0f\n- v03 --supports--> v0e\n- v02 --refutes--> v0c\n- v0c --refutes--> v09\n- v04 --supports--> v07\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"beec5ce7bed0c81f4852bfbf3b1a6bc5bce64147878a59617fcc8c2072570464","schema":"livegraph.boolean-propagation.v1","spec_hash":"07b0de88c92f7ccafdd50783a68ad6e7eaecaaf0d769cf5c17e825ce7f43414c","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-8e54ffb8b7fce00456f9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v0d\n- v0b --refutes--> v0a\n- v0a --refutes--> v08\n- v00 --refutes--> v0c\n- v09 --supports--> v0f\n- v0f --refutes--> v05\n- v0e --supports--> v0b\n- v01 --supports--> v07\n- v0c --supports--> v04\n- v03 --supports--> v01\n- v09 --supports--> v02\n- v0c --supports--> v0e\n- v0c --supports--> v06\n- v01 --refutes--> v00\n- v05 --supports--> v03\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6d893757a55005f0df2a3bbd81dc9b7e3fdbfa5a7f5f60cd2090b990d643f201","schema":"livegraph.boolean-propagation.v1","spec_hash":"8e54ffb8b7fce00456f9b146e002b6ec6482e39c7382c196f39bb48a99508a91","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-89c4b92d6197f4b5041f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v05\n- v01 --supports--> v09\n- v0a --supports--> v0b\n- v0c --refutes--> v08\n- v06 --supports--> v00\n- v00 --supports--> v01\n- v0c --refutes--> v02\n- v07 --supports--> v04\n- v03 --refutes--> v0c\n- v03 --supports--> v0f\n- v02 --refutes--> v0e\n- v0b --refutes--> v07\n- v0c --supports--> v0d\n- v01 --supports--> v0a\n- v0e --supports--> v06\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5fb53fc7dc475dbd61b92b0840b0458902451fca764449cc0c7e115f1a6db129","schema":"livegraph.boolean-propagation.v1","spec_hash":"89c4b92d6197f4b5041fceb2fe68f06b289b273de3a06000334e6f33926d5e61","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-b7afe8b2a9f4ee0be2d0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v01\n- v0e --supports--> v08\n- v0e --refutes--> v03\n- v01 --refutes--> v04\n- v0d --supports--> v06\n- v06 --supports--> v09\n- v09 --supports--> v0a\n- v06 --refutes--> v05\n- v0c --refutes--> v0f\n- v09 --refutes--> v07\n- v09 --supports--> v0b\n- v07 --supports--> v0c\n- v0c --supports--> v02\n- v02 --refutes--> v00\n- v04 --supports--> v0e\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"22dd5f8ed76a0776f3f0feb9d829cff40c56e007da876a0937f62a9d45e921b8","schema":"livegraph.boolean-propagation.v1","spec_hash":"b7afe8b2a9f4ee0be2d054240241afd373dccceb20a9f0584730880bc4136a79","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-af03eb62f5ab9716d207","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v05\n- v0c --refutes--> v00\n- v0f --supports--> v0e\n- v06 --supports--> v01\n- v09 --supports--> v06\n- v09 --supports--> v07\n- v0e --supports--> v0b\n- v0c --supports--> v09\n- v08 --supports--> v04\n- v09 --supports--> v02\n- v0c --supports--> v0d\n- v0b --supports--> v0c\n- v03 --refutes--> v0f\n- v0b --refutes--> v0a\n- v05 --refutes--> v08\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6811c83a0d66eaa4f03980920c9ff11961e8ca072d27d65fafb8ff098e35ad80","schema":"livegraph.boolean-propagation.v1","spec_hash":"af03eb62f5ab9716d207b192b221d37818e3d3eac80f383d20757569d24730f2","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-8bf3c92c67628e06532f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v06\n- v0a --refutes--> v03\n- v0b --supports--> v05\n- v02 --supports--> v09\n- v0c --supports--> v0a\n- v04 --supports--> v0f\n- v0d --refutes--> v0e\n- v0f --refutes--> v07\n- v04 --supports--> v01\n- v02 --supports--> v0d\n- v00 --refutes--> v0b\n- v0e --supports--> v08\n- v0f --supports--> v02\n- v03 --supports--> v04\n- v0e --refutes--> v00\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d64fce6e853f680eac284ea4bc608245e66a7d46819d21367975e467ec9fed1c","schema":"livegraph.boolean-propagation.v1","spec_hash":"8bf3c92c67628e06532f83c11e1f5beeea8e6ee0a6f382ae8d56a76530d9a164","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-5f2a462bd05d0319d24d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v0f\n- v0e --supports--> v09\n- v0f --refutes--> v00\n- v03 --refutes--> v04\n- v0a --refutes--> v0c\n- v08 --supports--> v0e\n- v07 --supports--> v03\n- v07 --supports--> v06\n- v05 --supports--> v0d\n- v0f --supports--> v02\n- v09 --supports--> v05\n- v0d --refutes--> v07\n- v0b --supports--> v01\n- v03 --refutes--> v0a\n- v0a --supports--> v0b\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"096b1dcd5b5548fa2b4cc59b313b90e9ec1bda5e7df18449c402c89f92700766","schema":"livegraph.boolean-propagation.v1","spec_hash":"5f2a462bd05d0319d24da8652b3d287dabfb41a40e23d73c05946d4be63539da","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-64fcd4fb53f6d1ab31af","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v07\n- v0e --refutes--> v00\n- v00 --refutes--> v0f\n- v06 --refutes--> v0b\n- v05 --refutes--> v0d\n- v0a --supports--> v05\n- v0b --refutes--> v03\n- v09 --supports--> v0e\n- v0b --refutes--> v01\n- v05 --supports--> v02\n- v06 --supports--> v04\n- v0c --supports--> v0a\n- v0f --supports--> v06\n- v06 --refutes--> v08\n- v0d --refutes--> v09\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e552aecdb150298c029cfc9afa51b9ff46f0dec62e51143f2203f784f1a8529c","schema":"livegraph.boolean-propagation.v1","spec_hash":"64fcd4fb53f6d1ab31af9aaabc97b87c1a4d9c6dbb506831dc7add6ba8ca709b","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-bc8f7a64569a3f0d540f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v04\n- v0b --refutes--> v09\n- v04 --supports--> v0c\n- v00 --refutes--> v0e\n- v02 --refutes--> v0b\n- v05 --supports--> v0d\n- v0f --refutes--> v01\n- v0d --supports--> v02\n- v02 --supports--> v07\n- v0c --supports--> v0f\n- v0c --refutes--> v08\n- v0c --supports--> v03\n- v09 --supports--> v0a\n- v0f --supports--> v06\n- v0f --refutes--> v00\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1c5499b2eff264eb9e0e05a32e6074689a4cd8574b9b81748cdc5b29d721c018","schema":"livegraph.boolean-propagation.v1","spec_hash":"bc8f7a64569a3f0d540f3531012d9a0febbdfa5ac86b98eeb3e79b49710ca7e2","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-b304294807b0b510abc2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v00\n- v05 --supports--> v0d\n- v06 --refutes--> v05\n- v05 --refutes--> v0b\n- v09 --refutes--> v03\n- v01 --supports--> v02\n- v0a --refutes--> v07\n- v03 --refutes--> v0a\n- v05 --refutes--> v0f\n- v04 --refutes--> v0e\n- v07 --supports--> v06\n- v02 --refutes--> v08\n- v0b --supports--> v0c\n- v04 --supports--> v09\n- v02 --supports--> v04\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d366be99f322a10ab577c75753b5d6c9d7726ae98f8abcf6b82b0a7697940d5c","schema":"livegraph.boolean-propagation.v1","spec_hash":"b304294807b0b510abc2e5d3aa099c69a6fa5ec54f87b2b8d68cf1f69d9fad01","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":10,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-e67f5b0406d316aebc20","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v0c\n- v0f --supports--> v05\n- v0b --refutes--> v01\n- v05 --supports--> v02\n- v0c --supports--> v04\n- v08 --refutes--> v0e\n- v01 --supports--> v0f\n- v0c --supports--> v0a\n- v0c --supports--> v00\n- v0a --supports--> v07\n- v03 --refutes--> v06\n- v07 --refutes--> v03\n- v0e --refutes--> v0d\n- v03 --refutes--> v08\n- v05 --supports--> v09\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9c982ea8b2ca78248f477aa1dac133f7b9d364eeaa1167ea418a2c39b5dae18a","schema":"livegraph.boolean-propagation.v1","spec_hash":"e67f5b0406d316aebc206b67194b77dd5285fb0263c3d768fde68d81b6446ae8","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-409d651e96dc2c260640","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v05\n- v06 --supports--> v0d\n- v06 --supports--> v0c\n- v05 --refutes--> v09\n- v06 --refutes--> v01\n- v00 --supports--> v0e\n- v07 --supports--> v02\n- v0e --supports--> v0a\n- v0b --refutes--> v04\n- v02 --supports--> v03\n- v0e --supports--> v06\n- v03 --supports--> v0f\n- v09 --refutes--> v08\n- v0f --refutes--> v00\n- v09 --supports--> v07\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2aeeb3be57a010b4d11901c9c870b9cdff26c089ac121bdd41314571aabc26e7","schema":"livegraph.boolean-propagation.v1","spec_hash":"409d651e96dc2c2606409b2f0e0570694de31f9f891b46b11c1d8298263a4320","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-7ddbd85e6ec9562b7dc7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v01\n- v0b --refutes--> v02\n- v01 --refutes--> v07\n- v09 --refutes--> v00\n- v0e --refutes--> v0b\n- v0d --refutes--> v05\n- v05 --refutes--> v04\n- v0a --supports--> v0c\n- v0b --supports--> v08\n- v0b --supports--> v0f\n- v0e --refutes--> v06\n- v0a --refutes--> v0d\n- v09 --supports--> v03\n- v0f --supports--> v09\n- v00 --refutes--> v0a\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a14e6431a12ff22efbf63a08c02bfd38f2a76a472df99a7570c1d7199ca4c556","schema":"livegraph.boolean-propagation.v1","spec_hash":"7ddbd85e6ec9562b7dc736d99280014ec77f2bb60a0b778c203d9a4bd439f983","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-17d10d35e608434ec2ec","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v0b\n- v0c --supports--> v0d\n- v0a --refutes--> v00\n- v04 --refutes--> v02\n- v0d --refutes--> v03\n- v0f --supports--> v01\n- v0d --supports--> v05\n- v0b --supports--> v0f\n- v08 --refutes--> v04\n- v0a --supports--> v09\n- v0e --supports--> v06\n- v0e --supports--> v0a\n- v03 --refutes--> v08\n- v0c --refutes--> v07\n- v01 --supports--> v0e\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b75e5c920c50a086dc4cc8f8b69671e5cc26cc3363245b0352c665e94c8555ce","schema":"livegraph.boolean-propagation.v1","spec_hash":"17d10d35e608434ec2ec3f8aaebcc82ff304b736e532b5764e8a6d06740f0a23","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-3ee0815a1648eb6e8b98","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v01\n- v00 --supports--> v0b\n- v03 --refutes--> v00\n- v03 --supports--> v0a\n- v05 --supports--> v02\n- v0d --refutes--> v08\n- v0d --refutes--> v09\n- v06 --refutes--> v0d\n- v00 --supports--> v07\n- v0e --supports--> v0c\n- v0d --supports--> v03\n- v01 --refutes--> v06\n- v02 --supports--> v04\n- v0b --refutes--> v05\n- v02 --supports--> v0f\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"152d9ba606029efbaa55779857ccced3589511168cbfce039ce52897ccc0a081","schema":"livegraph.boolean-propagation.v1","spec_hash":"3ee0815a1648eb6e8b98cbfd492a1de6e90c4ba005cad3619bf4c51b0ea88a9c","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-194ffce393f4cdd595b2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v0b\n- v00 --supports--> v0d\n- v04 --supports--> v0a\n- v0a --supports--> v06\n- v0e --refutes--> v04\n- v07 --supports--> v0c\n- v01 --refutes--> v05\n- v05 --supports--> v02\n- v04 --supports--> v0f\n- v06 --supports--> v00\n- v05 --supports--> v09\n- v04 --supports--> v08\n- v0a --supports--> v03\n- v0b --supports--> v01\n- v02 --refutes--> v07\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bdb2134df26bcd9c65640c9705b8f3d540ace181d80ff83ee305eb17b8f9ddb9","schema":"livegraph.boolean-propagation.v1","spec_hash":"194ffce393f4cdd595b271dbe639dc9974dfce7aa5b4755005fdd2eae6c3f7d1","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-38b45b32c868c206cc28","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v03\n- v05 --refutes--> v00\n- v0f --supports--> v0a\n- v00 --refutes--> v08\n- v02 --refutes--> v09\n- v04 --refutes--> v06\n- v0b --supports--> v04\n- v04 --refutes--> v0e\n- v08 --refutes--> v0d\n- v0e --refutes--> v0f\n- v08 --supports--> v07\n- v0f --supports--> v01\n- v01 --refutes--> v05\n- v09 --supports--> v0b\n- v0e --refutes--> v0c\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b5484899031e807504e9848a9e0bb41df5d0fbfba75e32ff635e3411b5e9a777","schema":"livegraph.boolean-propagation.v1","spec_hash":"38b45b32c868c206cc28c6dea1cf8c134049e1a7f908e2fede3df426a71f3771","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-8596b8651f3fbcc80a24","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v0b\n- v01 --supports--> v0f\n- v07 --refutes--> v03\n- v05 --refutes--> v02\n- v06 --supports--> v08\n- v00 --supports--> v0e\n- v07 --supports--> v0c\n- v07 --refutes--> v06\n- v02 --refutes--> v04\n- v0a --refutes--> v0d\n- v04 --supports--> v01\n- v06 --refutes--> v05\n- v0d --refutes--> v07\n- v0e --supports--> v09\n- v01 --supports--> v00\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"72b11c546154077357bef8f99923a23756528bf5c89b5055e146675400f34fb7","schema":"livegraph.boolean-propagation.v1","spec_hash":"8596b8651f3fbcc80a2475f509fd0df00f23480c2b9b87204644818e71487802","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-e66a1d4150329cc30038","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v00\n- v08 --supports--> v09\n- v06 --supports--> v08\n- v0d --refutes--> v01\n- v0c --supports--> v0f\n- v0e --refutes--> v0b\n- v05 --refutes--> v06\n- v08 --refutes--> v03\n- v01 --refutes--> v04\n- v0a --refutes--> v0d\n- v0c --supports--> v02\n- v03 --refutes--> v0e\n- v02 --supports--> v05\n- v01 --refutes--> v0c\n- v01 --supports--> v07\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"640b646fb84347a26a57bc006d810589b16cb1b7fa5a34833fd44b673aae0a28","schema":"livegraph.boolean-propagation.v1","spec_hash":"e66a1d4150329cc300384356141f619b5439cf926a8ce8d58044e88dbe20cc3b","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-4e555b670945a95f4a74","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v04\n- v09 --refutes--> v0a\n- v02 --supports--> v07\n- v0c --refutes--> v05\n- v05 --supports--> v00\n- v0d --refutes--> v02\n- v0a --supports--> v0e\n- v0e --refutes--> v06\n- v07 --supports--> v03\n- v06 --supports--> v0c\n- v06 --supports--> v08\n- v07 --refutes--> v0f\n- v00 --supports--> v0d\n- v0e --refutes--> v0b\n- v0c --refutes--> v01\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bdab4ccb0b9ccab76b7f48e417712c8fb9f30f27d61868722bbce03a41b80745","schema":"livegraph.boolean-propagation.v1","spec_hash":"4e555b670945a95f4a748599a2e5abccd4c37856ea5e43e78f4f92edc44b0bdc","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-a863e7cb0b2acf311c9f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v0a\n- v09 --supports--> v05\n- v0f --refutes--> v06\n- v00 --supports--> v0f\n- v08 --supports--> v0d\n- v06 --supports--> v02\n- v00 --supports--> v0e\n- v0a --supports--> v04\n- v0c --refutes--> v0b\n- v0f --supports--> v07\n- v06 --supports--> v0c\n- v01 --refutes--> v03\n- v0a --supports--> v09\n- v09 --supports--> v01\n- v03 --supports--> v00\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7a846d444c481c0044d8b6a2dc9e460bc60be042b72c6412344dad0f886d558e","schema":"livegraph.boolean-propagation.v1","spec_hash":"a863e7cb0b2acf311c9f10f19a10addf1af71251bd8121337073ae414b04978a","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-50fecf8e29206f4e011c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v09\n- v05 --supports--> v08\n- v0e --refutes--> v07\n- v0b --supports--> v03\n- v05 --supports--> v00\n- v02 --supports--> v0d\n- v04 --refutes--> v06\n- v07 --supports--> v04\n- v0f --refutes--> v0c\n- v05 --refutes--> v0e\n- v0f --refutes--> v01\n- v0a --refutes--> v02\n- v03 --supports--> v05\n- v0f --supports--> v0a\n- v06 --refutes--> v0f\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"48364b32402cad1c186fe2becc8e2432f712f1909b1323b4ff18856eebcb971c","schema":"livegraph.boolean-propagation.v1","spec_hash":"50fecf8e29206f4e011ce920149e3b4f41e80f2209fef915682181b3df7dcf14","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-28355ecc9958a42d6a47","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v00\n- v03 --refutes--> v0e\n- v05 --refutes--> v0d\n- v01 --supports--> v0a\n- v09 --refutes--> v08\n- v0f --refutes--> v03\n- v0a --supports--> v02\n- v0b --supports--> v06\n- v02 --supports--> v05\n- v0c --refutes--> v0f\n- v03 --refutes--> v09\n- v0a --refutes--> v07\n- v0d --supports--> v0c\n- v0b --refutes--> v01\n- v01 --refutes--> v04\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f0b5664375b6d923f760ca0ac6293a38ee49291a89dd46ee72b2421b2f27bab8","schema":"livegraph.boolean-propagation.v1","spec_hash":"28355ecc9958a42d6a47e295b3fb49c4d618fb25734d121a931cfb55ea66388a","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-28b9cd34cbb0f3751b42","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v00\n- v08 --refutes--> v05\n- v0d --refutes--> v0b\n- v08 --refutes--> v0f\n- v01 --supports--> v0d\n- v0d --supports--> v03\n- v0e --refutes--> v01\n- v03 --supports--> v09\n- v04 --supports--> v08\n- v0f --supports--> v0a\n- v03 --supports--> v07\n- v02 --refutes--> v06\n- v0c --refutes--> v0e\n- v06 --supports--> v04\n- v0f --supports--> v0c\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"00c80b8ef368fd8ecdbbac5c3bf96d379ea371867ec0d390f195cf3b3b436bd8","schema":"livegraph.boolean-propagation.v1","spec_hash":"28b9cd34cbb0f3751b42dee0c9c1dd588c01b09622a684dc79ba18bf243b9e1a","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-01bbe376d758e776de19","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v0d\n- v0c --supports--> v02\n- v06 --supports--> v0a\n- v0e --refutes--> v07\n- v07 --refutes--> v09\n- v07 --supports--> v0f\n- v00 --supports--> v08\n- v0c --supports--> v01\n- v05 --supports--> v00\n- v01 --supports--> v05\n- v06 --refutes--> v03\n- v06 --supports--> v0b\n- v0d --refutes--> v06\n- v03 --refutes--> v0c\n- v03 --supports--> v04\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"31633b43863709901775697ae4d0ba473798350acfb53c3b8cbc4320715cf805","schema":"livegraph.boolean-propagation.v1","spec_hash":"01bbe376d758e776de1996f83a4ee3e53842781a7ef6f1dc6fb35fb44294f686","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-6180bb0320662031f940","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v04\n- v0b --refutes--> v06\n- v08 --refutes--> v0a\n- v09 --supports--> v0e\n- v07 --supports--> v03\n- v04 --supports--> v0c\n- v04 --refutes--> v0d\n- v08 --supports--> v01\n- v07 --supports--> v08\n- v0f --supports--> v00\n- v08 --refutes--> v05\n- v0e --supports--> v07\n- v00 --refutes--> v0b\n- v0b --supports--> v02\n- v0d --supports--> v0f\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c3ad5a01d629146f45363c336c449e95b64f7637e54eb8ec4d76d07f70f33ab6","schema":"livegraph.boolean-propagation.v1","spec_hash":"6180bb0320662031f940fd484ead98339acc0e0b25f32d34603c29e9f618e47a","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-426666c1ff3037030561","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v02\n- v04 --supports--> v06\n- v0d --refutes--> v0f\n- v0e --supports--> v09\n- v03 --refutes--> v04\n- v0a --supports--> v01\n- v0a --supports--> v08\n- v09 --supports--> v0d\n- v04 --refutes--> v00\n- v0b --refutes--> v0e\n- v07 --refutes--> v0a\n- v06 --supports--> v0c\n- v06 --supports--> v0b\n- v07 --supports--> v05\n- v0f --supports--> v07\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f79764c5df1d066522a01e34997f27ee1053ee7889668863aa6c688a6782eeaf","schema":"livegraph.boolean-propagation.v1","spec_hash":"426666c1ff30370305611e49d1a4b8222511024ac04c8eb92d63e24c1afe4734","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-2b789ebba6b76f007998","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v0c\n- v03 --supports--> v08\n- v0b --supports--> v0f\n- v08 --refutes--> v02\n- v06 --supports--> v07\n- v08 --supports--> v04\n- v01 --supports--> v03\n- v00 --refutes--> v05\n- v03 --refutes--> v0d\n- v07 --supports--> v00\n- v0f --supports--> v06\n- v04 --refutes--> v0e\n- v07 --refutes--> v09\n- v03 --refutes--> v0a\n- v04 --refutes--> v0b\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"01756ad1f96996d920613807c003b9bfc9c32d334792e3a20467f360742416a6","schema":"livegraph.boolean-propagation.v1","spec_hash":"2b789ebba6b76f007998f96c42fd1524983503e3088068ad490ff9fc6b04f2ea","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-1239f30763df6b1daac5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v0e\n- v05 --supports--> v09\n- v0b --supports--> v0a\n- v09 --supports--> v02\n- v07 --refutes--> v04\n- v0d --supports--> v07\n- v08 --supports--> v05\n- v0f --refutes--> v01\n- v02 --supports--> v0b\n- v0a --supports--> v0d\n- v03 --refutes--> v0f\n- v0d --refutes--> v00\n- v0b --refutes--> v06\n- v03 --supports--> v0c\n- v0f --refutes--> v08\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"30e26482c7a4e68675ba8aa49c049a4d8ca129a415647e586d292eac39da2270","schema":"livegraph.boolean-propagation.v1","spec_hash":"1239f30763df6b1daac52121a0320af15d19b35988ca70953f03ca3eb075624a","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-9b8a5d7e4cc212c2d52a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v0e\n- v07 --supports--> v04\n- v0f --refutes--> v06\n- v05 --refutes--> v0d\n- v0c --refutes--> v0f\n- v0d --supports--> v01\n- v07 --supports--> v0b\n- v01 --supports--> v02\n- v02 --refutes--> v08\n- v0d --supports--> v09\n- v05 --refutes--> v00\n- v03 --supports--> v0c\n- v08 --supports--> v07\n- v06 --supports--> v05\n- v01 --supports--> v0a\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"44ffc566358594eb066ddb008ffa206cd5a09b9194d53bb464432a95c0ef8c86","schema":"livegraph.boolean-propagation.v1","spec_hash":"9b8a5d7e4cc212c2d52ae02b0faea6903a7ff48c9bce4215607776b81e8cd166","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-2f376fcba87a9524974c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v00\n- v0a --supports--> v0f\n- v08 --supports--> v07\n- v02 --refutes--> v0c\n- v0c --refutes--> v05\n- v06 --supports--> v09\n- v07 --refutes--> v06\n- v0c --supports--> v0a\n- v04 --refutes--> v0e\n- v0c --refutes--> v03\n- v09 --refutes--> v04\n- v0b --refutes--> v01\n- v0a --supports--> v08\n- v0c --supports--> v0d\n- v01 --refutes--> v02\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"590cb09a2812149b60227f491d6caca536c6faed6355377be1d976d0dc1ea15d","schema":"livegraph.boolean-propagation.v1","spec_hash":"2f376fcba87a9524974ce06c137340404b02b94d731df6f3507f78b3a5706a20","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":10,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-e939d6acdc6926913df0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v05\n- v0e --supports--> v0a\n- v06 --refutes--> v09\n- v05 --refutes--> v00\n- v01 --supports--> v08\n- v01 --refutes--> v10\n- v06 --refutes--> v02\n- v07 --refutes--> v01\n- v00 --refutes--> v0f\n- v10 --refutes--> v03\n- v03 --supports--> v0d\n- v0d --refutes--> v0e\n- v0d --supports--> v0b\n- v01 --refutes--> v04\n- v0f --refutes--> v06\n- v0d --refutes--> v0c\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4d776ea1b0171505a34d801e869ad419fb6a73615af67c502f37308598992d46","schema":"livegraph.boolean-propagation.v1","spec_hash":"e939d6acdc6926913df08594416e3083b61d476c48bc77648d037f15f8d2cbd9","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-e7df027bb462e2245d02","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v10\n- v0b --supports--> v0d\n- v04 --supports--> v09\n- v0a --supports--> v05\n- v02 --refutes--> v06\n- v0e --supports--> v02\n- v05 --refutes--> v01\n- v0b --refutes--> v0c\n- v0d --supports--> v07\n- v09 --refutes--> v08\n- v05 --supports--> v04\n- v03 --refutes--> v0a\n- v03 --supports--> v0f\n- v0d --refutes--> v03\n- v08 --refutes--> v0e\n- v0d --supports--> v00\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b448dd2dcdf297c7f4a9145ccb455d49251baf69359839d460d62438c50ee46c","schema":"livegraph.boolean-propagation.v1","spec_hash":"e7df027bb462e2245d02004aabf1778231bdd768b76de5029a936c4766262ef0","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":11,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-07485b3101d39eb1f50a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v0f\n- v0c --refutes--> v0d\n- v0b --refutes--> v02\n- v00 --refutes--> v07\n- v08 --refutes--> v03\n- v05 --supports--> v04\n- v04 --supports--> v0a\n- v07 --refutes--> v0c\n- v0b --supports--> v06\n- v10 --refutes--> v05\n- v0e --refutes--> v10\n- v02 --supports--> v08\n- v0e --supports--> v01\n- v0d --refutes--> v0e\n- v04 --refutes--> v0b\n- v00 --refutes--> v09\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"41aded6b7e9eba4f501aa3dc0af881fc17d4c4f66e2e5b5b8a1d68065f5f7e9f","schema":"livegraph.boolean-propagation.v1","spec_hash":"07485b3101d39eb1f50ad192cbc211472c6c81abae0227fa7e331d8694140ab4","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-7d5e9fff7930411824c4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v0a\n- v0a --supports--> v0d\n- v06 --supports--> v05\n- v0e --refutes--> v03\n- v09 --supports--> v0c\n- v0d --supports--> v01\n- v09 --refutes--> v04\n- v01 --supports--> v00\n- v0f --supports--> v09\n- v06 --refutes--> v10\n- v0f --supports--> v08\n- v10 --supports--> v0e\n- v0a --supports--> v07\n- v04 --supports--> v0b\n- v0a --supports--> v02\n- v00 --supports--> v06\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dbef96242cc713fcf82c9e38e4cd08261d5d06c172799ae13f28e2d10f29872a","schema":"livegraph.boolean-propagation.v1","spec_hash":"7d5e9fff7930411824c4e1e13ac5609cc31c95104f6370cf87c081245d0411bf","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-d3a8902e3f5d955309a6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v0a\n- v08 --refutes--> v00\n- v09 --supports--> v02\n- v07 --supports--> v01\n- v09 --refutes--> v05\n- v0c --refutes--> v07\n- v08 --refutes--> v06\n- v07 --supports--> v09\n- v10 --refutes--> v0e\n- v0b --refutes--> v0f\n- v05 --refutes--> v08\n- v03 --refutes--> v0b\n- v0e --refutes--> v04\n- v0b --refutes--> v0c\n- v0d --supports--> v03\n- v04 --refutes--> v0d\n\n[VERIFIED ROOT]\nv10 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"67618b928746bd32d5e4b3c832fc8794ee45eef94e464323766185377559c4b3","schema":"livegraph.boolean-propagation.v1","spec_hash":"d3a8902e3f5d955309a6d9b6bc220b26fff3a05ef611510f1a725948073b0446","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-da5413693c480fe059c5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v09\n- v09 --refutes--> v05\n- v02 --supports--> v0e\n- v01 --refutes--> v00\n- v08 --refutes--> v0b\n- v09 --refutes--> v0d\n- v0b --refutes--> v10\n- v0e --refutes--> v0a\n- v10 --supports--> v0f\n- v0d --refutes--> v0c\n- v0a --refutes--> v07\n- v00 --supports--> v06\n- v0d --refutes--> v02\n- v0a --refutes--> v03\n- v06 --supports--> v04\n- v0a --supports--> v08\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0216a42490d4ab5917decc571c4395a0fbfccfa2d5b27b41e99fb9c0316a8988","schema":"livegraph.boolean-propagation.v1","spec_hash":"da5413693c480fe059c541b8d2a60d72720c0da1364a8ff3b5c3b4ebaa4d9a97","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-b2fbc075b057885ef816","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v05\n- v09 --refutes--> v0b\n- v0d --refutes--> v09\n- v07 --refutes--> v0d\n- v00 --supports--> v04\n- v02 --refutes--> v08\n- v0d --refutes--> v03\n- v10 --supports--> v01\n- v01 --supports--> v02\n- v06 --supports--> v0e\n- v0b --supports--> v06\n- v00 --refutes--> v07\n- v06 --refutes--> v0c\n- v0e --refutes--> v10\n- v06 --refutes--> v0a\n- v0b --refutes--> v0f\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"980d3c31d3f0f63bcc6f7e64de35820c8d419a00b356c306a9b3e404a077a282","schema":"livegraph.boolean-propagation.v1","spec_hash":"b2fbc075b057885ef816fe26da3eebe8daa03a62d2cc31b7d1132a6e714a785d","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-34cf70d599a0043e25bf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v0d\n- v10 --supports--> v04\n- v02 --supports--> v0b\n- v04 --supports--> v02\n- v06 --refutes--> v0f\n- v0d --supports--> v09\n- v0b --refutes--> v06\n- v04 --supports--> v08\n- v06 --supports--> v05\n- v0a --supports--> v07\n- v01 --supports--> v03\n- v09 --supports--> v10\n- v02 --supports--> v0e\n- v0f --supports--> v0a\n- v0a --refutes--> v0c\n- v0a --refutes--> v00\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e5c8739e8b41b08675f3c3e824b352a5f9e2d99726bb4eb5f5789d014c0a1a1e","schema":"livegraph.boolean-propagation.v1","spec_hash":"34cf70d599a0043e25bfb14e3893aaa7021eabf0cf68a7f8ca631a9fa59750d8","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-627f8eeaad73d42016d6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v07\n- v00 --refutes--> v08\n- v02 --supports--> v0e\n- v10 --supports--> v09\n- v03 --refutes--> v00\n- v04 --supports--> v0c\n- v01 --refutes--> v0b\n- v09 --supports--> v02\n- v04 --refutes--> v0a\n- v0e --refutes--> v05\n- v08 --supports--> v0d\n- v08 --supports--> v06\n- v0e --refutes--> v01\n- v0c --refutes--> v03\n- v0d --refutes--> v0f\n- v0d --refutes--> v10\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0eb269a46d50ffd79c2d23cf628b0952ea684373e33292a8bd5c78564d63f851","schema":"livegraph.boolean-propagation.v1","spec_hash":"627f8eeaad73d42016d6c37f9bbd558120c7d009450097125227255cc2e73f01","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-4c31253ca651acd8dea1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v03\n- v07 --supports--> v06\n- v0a --refutes--> v09\n- v07 --supports--> v08\n- v09 --supports--> v02\n- v02 --refutes--> v0f\n- v05 --refutes--> v10\n- v02 --refutes--> v0e\n- v04 --supports--> v0a\n- v0d --supports--> v0b\n- v03 --supports--> v01\n- v03 --refutes--> v07\n- v0e --refutes--> v0d\n- v09 --refutes--> v0c\n- v06 --refutes--> v00\n- v05 --supports--> v04\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a6314a6e49b6cb18e41db5a70820674ec6e294fae53b24ab2f637351f4ecc321","schema":"livegraph.boolean-propagation.v1","spec_hash":"4c31253ca651acd8dea1d986a942998cce5c3c4a32b80b7a2e7176b7539db610","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"}],\"hop\":11,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-0bf4a2083da3e9d50b6f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v0d\n- v09 --supports--> v0c\n- v05 --supports--> v04\n- v0b --refutes--> v06\n- v06 --supports--> v05\n- v04 --supports--> v0e\n- v00 --supports--> v08\n- v0d --supports--> v01\n- v0d --refutes--> v0b\n- v04 --supports--> v0a\n- v0c --refutes--> v10\n- v0c --refutes--> v03\n- v00 --refutes--> v07\n- v07 --supports--> v0f\n- v0e --supports--> v09\n- v07 --refutes--> v02\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7a806c9f411e4be5b1b2f18cfffea123ecab8da9fbfad158b6302a91374903ce","schema":"livegraph.boolean-propagation.v1","spec_hash":"0bf4a2083da3e9d50b6fd6323d16a1324383d3c0d63f54aa34e0eb0e427f6528","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-7ba95bb8f93b7e808388","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v04\n- v07 --supports--> v03\n- v0e --refutes--> v02\n- v0e --supports--> v09\n- v09 --refutes--> v0c\n- v0d --supports--> v05\n- v08 --supports--> v06\n- v04 --supports--> v0e\n- v09 --refutes--> v0a\n- v03 --supports--> v0b\n- v03 --refutes--> v08\n- v05 --refutes--> v07\n- v0f --supports--> v10\n- v08 --refutes--> v01\n- v08 --refutes--> v00\n- v10 --supports--> v0d\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1c410befcdc0568a8054397bf1068a283a2db3dc3aa43d3b031cf437d3071d88","schema":"livegraph.boolean-propagation.v1","spec_hash":"7ba95bb8f93b7e8083887e017becd563f16a7ca053a577fb631bba2c6fcb004b","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-3c66e3b0a3e97ec7f58f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v05\n- v01 --supports--> v07\n- v07 --refutes--> v08\n- v0e --supports--> v0b\n- v06 --supports--> v0f\n- v0f --refutes--> v0c\n- v10 --refutes--> v0d\n- v0f --refutes--> v10\n- v0d --supports--> v02\n- v0d --supports--> v0a\n- v09 --refutes--> v0e\n- v08 --supports--> v06\n- v0e --supports--> v04\n- v05 --refutes--> v09\n- v01 --refutes--> v00\n- v01 --refutes--> v03\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"80688ea4c56b2bb19c48a2c65bcdc87112a80039fc635cb25a529d58bf7dfb21","schema":"livegraph.boolean-propagation.v1","spec_hash":"3c66e3b0a3e97ec7f58fa1e276ce9a73ab6d363d6f2bf26f1f0ce36c4feb34bf","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-30a1f32e8e2be64364cb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v09\n- v0c --supports--> v0a\n- v02 --refutes--> v0d\n- v09 --refutes--> v07\n- v08 --refutes--> v0e\n- v0d --refutes--> v0f\n- v05 --refutes--> v0c\n- v07 --refutes--> v08\n- v0f --refutes--> v06\n- v0c --refutes--> v02\n- v0d --refutes--> v10\n- v00 --refutes--> v05\n- v00 --supports--> v04\n- v0e --refutes--> v01\n- v00 --supports--> v03\n- v0d --supports--> v0b\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8c5ce314977f303a1e768e567cd46c0356cc4eec0d951102130edea7015f62da","schema":"livegraph.boolean-propagation.v1","spec_hash":"30a1f32e8e2be64364cb0671ed5eb34eac6675d42319b7af4dd4f9ee0c6eed69","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-e556d12a208dae10862f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v0a\n- v04 --supports--> v0c\n- v0e --supports--> v05\n- v0f --refutes--> v06\n- v10 --supports--> v07\n- v0a --supports--> v00\n- v02 --supports--> v0e\n- v0d --supports--> v09\n- v05 --refutes--> v10\n- v00 --refutes--> v04\n- v10 --refutes--> v01\n- v0d --refutes--> v02\n- v00 --supports--> v0b\n- v05 --refutes--> v08\n- v04 --supports--> v0d\n- v07 --refutes--> v0f\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"df2b39829cfbbd3570e0e9a1e3f8178c48b4d851997b82bdc1ec4a4948d6b28a","schema":"livegraph.boolean-propagation.v1","spec_hash":"e556d12a208dae10862fc7858e2c0063ed49419af58f150ee9b95094ede58c61","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-012ed75ba7ab8bb51350","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v0d\n- v0d --supports--> v01\n- v07 --supports--> v03\n- v02 --refutes--> v00\n- v00 --refutes--> v10\n- v01 --supports--> v07\n- v09 --refutes--> v04\n- v03 --supports--> v0a\n- v0a --supports--> v02\n- v09 --refutes--> v05\n- v03 --refutes--> v0b\n- v03 --refutes--> v0f\n- v02 --supports--> v0c\n- v0e --supports--> v09\n- v08 --supports--> v0e\n- v0e --supports--> v06\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a1e85bdb51dc1901574ff6067098c3b0b89146a3437e6fd547f4b32e96bb14d6","schema":"livegraph.boolean-propagation.v1","spec_hash":"012ed75ba7ab8bb51350a72926c254aaec099b47dec545d86ffd161e5a69e86c","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":11,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-bfbd1ea42ebf32d0db70","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v0a\n- v0f --refutes--> v03\n- v0e --refutes--> v06\n- v0d --supports--> v08\n- v02 --supports--> v07\n- v0f --supports--> v09\n- v10 --supports--> v0e\n- v01 --refutes--> v0b\n- v01 --supports--> v05\n- v02 --refutes--> v0f\n- v06 --refutes--> v0d\n- v04 --supports--> v10\n- v05 --supports--> v0c\n- v09 --refutes--> v01\n- v0d --supports--> v00\n- v08 --refutes--> v02\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7cc42598d9fb2ed3d488c5314eba8c9ebb5b5b43c60d79cb34fb18b90af38d89","schema":"livegraph.boolean-propagation.v1","spec_hash":"bfbd1ea42ebf32d0db7081ae5b3388aecfab317b56076e462f1b6d949f3ea87e","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-06c7ac8884e6b23d8ac7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v03\n- v0c --supports--> v10\n- v10 --refutes--> v0a\n- v03 --refutes--> v07\n- v04 --supports--> v0d\n- v04 --refutes--> v0b\n- v0a --supports--> v06\n- v09 --supports--> v08\n- v06 --supports--> v02\n- v10 --refutes--> v00\n- v08 --refutes--> v0c\n- v0b --supports--> v0f\n- v0f --refutes--> v05\n- v02 --supports--> v0e\n- v03 --supports--> v09\n- v02 --supports--> v01\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3e55ff63f3f29b689d81970887c7fbdf9f707845da0a480d4f39f0986c35f7d2","schema":"livegraph.boolean-propagation.v1","spec_hash":"06c7ac8884e6b23d8ac76d52f64af34aa0bec4f685d565070d37b1fdd80df61e","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-d2e7a8a5d5455c52c978","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v00\n- v10 --supports--> v0e\n- v07 --supports--> v0d\n- v0f --refutes--> v0c\n- v0d --refutes--> v10\n- v04 --supports--> v05\n- v0a --supports--> v07\n- v05 --supports--> v0f\n- v0f --supports--> v09\n- v04 --refutes--> v08\n- v03 --supports--> v0a\n- v01 --refutes--> v04\n- v0c --supports--> v0b\n- v0a --supports--> v06\n- v0e --supports--> v01\n- v0a --refutes--> v02\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7a740a68919ccef3743041862564ebd07d9b95316983ea800fd01806b533c0d3","schema":"livegraph.boolean-propagation.v1","spec_hash":"d2e7a8a5d5455c52c97872cc48a9d93f464efd3d7443c01fe9f099eb415b14e3","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-cbb3564f48793da86c4e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v08\n- v0a --refutes--> v0c\n- v00 --refutes--> v0b\n- v0b --refutes--> v10\n- v10 --supports--> v0a\n- v02 --refutes--> v01\n- v00 --supports--> v09\n- v0b --refutes--> v06\n- v07 --refutes--> v0e\n- v10 --refutes--> v04\n- v03 --supports--> v05\n- v0c --supports--> v03\n- v0e --refutes--> v0d\n- v05 --refutes--> v07\n- v0b --supports--> v0f\n- v01 --supports--> v00\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7d62130d10b9285fcac08b46e84ef1dc5a461d33e90dc0f8959ceeb767f74bdc","schema":"livegraph.boolean-propagation.v1","spec_hash":"cbb3564f48793da86c4edfe0ce44c74e981e34009cb39100edd90838aafae6ed","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-5aa319b539a5276a1289","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v0a\n- v06 --supports--> v02\n- v0d --refutes--> v09\n- v0c --supports--> v04\n- v00 --refutes--> v08\n- v01 --refutes--> v03\n- v06 --refutes--> v0e\n- v04 --supports--> v00\n- v07 --refutes--> v01\n- v0a --refutes--> v07\n- v01 --refutes--> v06\n- v06 --supports--> v0f\n- v00 --refutes--> v10\n- v09 --supports--> v05\n- v02 --refutes--> v0c\n- v09 --supports--> v0b\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cafdc2531a54c4030a5951ce0676ee494278aa2470bd64bbd1d5aa69bc037ea9","schema":"livegraph.boolean-propagation.v1","spec_hash":"5aa319b539a5276a12893b7fbd0e52e8f3cea8528569630b5f8bc21c53aa1939","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":11,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-1ced723c601118888db6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v08\n- v0e --supports--> v02\n- v0a --supports--> v09\n- v0d --supports--> v01\n- v06 --supports--> v10\n- v0f --refutes--> v0d\n- v03 --refutes--> v06\n- v0f --refutes--> v00\n- v01 --supports--> v0b\n- v02 --supports--> v0a\n- v08 --supports--> v04\n- v01 --refutes--> v0e\n- v0e --supports--> v05\n- v04 --supports--> v07\n- v02 --refutes--> v0c\n- v07 --supports--> v0f\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"95cf714b1a4d1d01b2289b74b832ca1bd9c17e200a678ee62ee584cd4deb9e6e","schema":"livegraph.boolean-propagation.v1","spec_hash":"1ced723c601118888db6cb1a912a503a82e414a4efab9ca015dbd714dc8d523a","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-b9938c7d083ff9d52f82","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v10\n- v0f --supports--> v08\n- v10 --refutes--> v0d\n- v06 --supports--> v0f\n- v0f --refutes--> v02\n- v0d --refutes--> v0a\n- v01 --refutes--> v0b\n- v0f --refutes--> v04\n- v0e --refutes--> v0c\n- v03 --refutes--> v07\n- v09 --supports--> v01\n- v01 --supports--> v06\n- v03 --supports--> v09\n- v0e --refutes--> v00\n- v08 --supports--> v0e\n- v0a --refutes--> v05\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b802f33e55ba81912743aeaefaee32a5f0dd0aa89593dea98d7271520758e9ab","schema":"livegraph.boolean-propagation.v1","spec_hash":"b9938c7d083ff9d52f82c8b2f5eac6c4e333240dfce1bdf41a0d3b5776c1bb3a","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-f8301f28492bed253dc9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v09\n- v0a --supports--> v00\n- v0d --refutes--> v06\n- v08 --refutes--> v03\n- v09 --supports--> v0e\n- v04 --supports--> v0c\n- v0e --supports--> v04\n- v0d --refutes--> v05\n- v04 --supports--> v08\n- v0e --supports--> v07\n- v02 --supports--> v10\n- v03 --refutes--> v0a\n- v02 --refutes--> v0b\n- v01 --refutes--> v02\n- v05 --supports--> v01\n- v02 --refutes--> v0f\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"12780e685f80e4a320ccdff7e3c747b70b1ae2d5a95b0efe4d27f726aaa6385e","schema":"livegraph.boolean-propagation.v1","spec_hash":"f8301f28492bed253dc93a1ad2a7a54190034725d670bda4f7fed2ad0eb5f784","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":11,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-ee72ddf215014ffbf26d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v05\n- v01 --supports--> v03\n- v0a --supports--> v0c\n- v06 --refutes--> v0b\n- v08 --supports--> v10\n- v0b --supports--> v04\n- v08 --supports--> v02\n- v0d --supports--> v0a\n- v03 --supports--> v08\n- v08 --supports--> v0f\n- v01 --refutes--> v00\n- v04 --refutes--> v07\n- v09 --supports--> v06\n- v04 --refutes--> v0d\n- v05 --refutes--> v0e\n- v0a --supports--> v01\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a708ebf01266524ee204d9852a4fd6007fa8881630f8e145263addd450817495","schema":"livegraph.boolean-propagation.v1","spec_hash":"ee72ddf215014ffbf26db2c60417e68f2c9615be46dee89694c92fabfddd8137","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":11,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-cc22144c999f511967da","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v01\n- v0f --supports--> v02\n- v0a --refutes--> v0d\n- v01 --refutes--> v0b\n- v02 --supports--> v0c\n- v01 --refutes--> v05\n- v08 --supports--> v00\n- v03 --supports--> v10\n- v10 --supports--> v0a\n- v02 --refutes--> v0e\n- v04 --refutes--> v08\n- v02 --refutes--> v07\n- v0d --supports--> v06\n- v08 --supports--> v03\n- v02 --supports--> v09\n- v06 --refutes--> v0f\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"10155e2ba9141a298d5623ca2524092d9685cf2d960887106da3fe90d9c23ade","schema":"livegraph.boolean-propagation.v1","spec_hash":"cc22144c999f511967dae491e80d79ef9b97516843c59dc0e9b264814bd51ffc","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-0eeba0708d448603dc31","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v02\n- v0f --supports--> v06\n- v00 --refutes--> v07\n- v0d --refutes--> v08\n- v05 --refutes--> v09\n- v06 --refutes--> v04\n- v08 --refutes--> v01\n- v10 --supports--> v0f\n- v03 --refutes--> v0b\n- v05 --refutes--> v0e\n- v08 --refutes--> v0c\n- v03 --refutes--> v05\n- v09 --refutes--> v00\n- v05 --supports--> v0a\n- v07 --supports--> v0d\n- v0c --refutes--> v10\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a2f0792ee9b28ab9401de30c086b0e7f04af1b023245109340f4cfd1241bb4ce","schema":"livegraph.boolean-propagation.v1","spec_hash":"0eeba0708d448603dc31eaa45187b265e4a88420d102696827e52acda2b7cec5","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-d6d2a2a2941a123aa24f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v07\n- v07 --supports--> v04\n- v0b --refutes--> v00\n- v0b --supports--> v0d\n- v0a --refutes--> v0b\n- v0e --supports--> v06\n- v03 --supports--> v0a\n- v09 --refutes--> v0f\n- v10 --refutes--> v0e\n- v09 --refutes--> v01\n- v06 --supports--> v0c\n- v09 --refutes--> v05\n- v07 --refutes--> v09\n- v08 --refutes--> v10\n- v08 --refutes--> v02\n- v05 --supports--> v03\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"59416b457585bc6b7b1a18f033e16b054752081bf482de2a5b991c3a3716385f","schema":"livegraph.boolean-propagation.v1","spec_hash":"d6d2a2a2941a123aa24f7a0e378054f4f164e1362f43cf3c3796baaf01640f66","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-dd6e4dd0e3708c5cfd7d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v03\n- v03 --refutes--> v02\n- v09 --refutes--> v01\n- v07 --refutes--> v0e\n- v02 --supports--> v09\n- v0a --supports--> v0c\n- v0a --refutes--> v00\n- v08 --supports--> v0a\n- v0e --refutes--> v0d\n- v0e --supports--> v0f\n- v0b --refutes--> v06\n- v06 --refutes--> v08\n- v01 --supports--> v05\n- v01 --refutes--> v07\n- v0e --refutes--> v04\n- v0f --supports--> v10\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"56a8d8630ba9236f4f6fd53fc7b0baee074ffc0d246e3415218b2d5ce5c42379","schema":"livegraph.boolean-propagation.v1","spec_hash":"dd6e4dd0e3708c5cfd7d061e20d706085686657262f61b2892a6301dca31df1c","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-14608dd1b05c16e2b1ae","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v02\n- v05 --refutes--> v06\n- v09 --supports--> v0d\n- v03 --supports--> v0a\n- v07 --supports--> v01\n- v0c --refutes--> v0f\n- v0d --supports--> v05\n- v10 --supports--> v09\n- v02 --supports--> v10\n- v07 --supports--> v03\n- v03 --refutes--> v08\n- v06 --refutes--> v0c\n- v03 --supports--> v0e\n- v04 --refutes--> v07\n- v06 --supports--> v0b\n- v03 --supports--> v00\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"21101c4cbc1a6357a4ea486474a8d0b5498c352fb9c284e5d36e60a93fdd7298","schema":"livegraph.boolean-propagation.v1","spec_hash":"14608dd1b05c16e2b1aea8cae7993161d9eba12bd89b66527eea3da229a13d77","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-ceeffb598683c91ded20","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v04\n- v09 --refutes--> v01\n- v03 --refutes--> v00\n- v07 --supports--> v06\n- v04 --refutes--> v05\n- v02 --supports--> v0d\n- v0c --supports--> v07\n- v0f --supports--> v10\n- v04 --supports--> v08\n- v05 --supports--> v02\n- v01 --refutes--> v0b\n- v02 --supports--> v0e\n- v0d --supports--> v0f\n- v0f --refutes--> v0c\n- v00 --refutes--> v09\n- v02 --supports--> v0a\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7785746c251803c229ad6b95057d1bd3fae064bc9dfa65a37e40c264add403f6","schema":"livegraph.boolean-propagation.v1","spec_hash":"ceeffb598683c91ded201355c9abc90285e2486115f06c81f382de8cead8fef8","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-22ae4615b5d455e6f99e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v06\n- v01 --refutes--> v0c\n- v0e --refutes--> v0f\n- v04 --refutes--> v0a\n- v06 --supports--> v07\n- v0c --refutes--> v03\n- v0a --supports--> v02\n- v07 --supports--> v0e\n- v0b --refutes--> v09\n- v0f --supports--> v01\n- v05 --supports--> v0b\n- v06 --supports--> v00\n- v04 --supports--> v10\n- v0a --supports--> v05\n- v0e --refutes--> v0d\n- v08 --supports--> v04\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cad5e1b0f6d40783f82955217b29f2428a8c98b3179a246276cd04065f68b60f","schema":"livegraph.boolean-propagation.v1","spec_hash":"22ae4615b5d455e6f99edc944cb3d77926fbb6cace69c27bb700dca30608512b","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-12ea25315d2042a43b2f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v06\n- v0a --supports--> v0d\n- v0a --supports--> v05\n- v0a --refutes--> v04\n- v05 --supports--> v08\n- v0e --refutes--> v0b\n- v01 --supports--> v09\n- v10 --refutes--> v03\n- v08 --refutes--> v01\n- v07 --refutes--> v0e\n- v0c --supports--> v0a\n- v0e --supports--> v0f\n- v0f --refutes--> v0c\n- v00 --supports--> v10\n- v0e --refutes--> v02\n- v03 --refutes--> v07\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f12e7c6c1a8525360c55e3194042b844f329860a6af0d52fa5b030632ae68710","schema":"livegraph.boolean-propagation.v1","spec_hash":"12ea25315d2042a43b2fd488de4821d97661fb3178e55a9625a157d41905e2cc","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-b9012208dba425ff22d1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v09\n- v07 --refutes--> v08\n- v05 --refutes--> v0b\n- v0c --supports--> v01\n- v06 --supports--> v0c\n- v08 --refutes--> v10\n- v06 --supports--> v03\n- v06 --supports--> v0f\n- v00 --supports--> v02\n- v02 --supports--> v0e\n- v02 --refutes--> v06\n- v10 --supports--> v05\n- v06 --refutes--> v04\n- v0a --supports--> v07\n- v0b --refutes--> v00\n- v01 --refutes--> v0d\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cb2e53cf8fe1c623bdbbdad94bf40ed60e354a1aaef22ed539c1900c1466c0ec","schema":"livegraph.boolean-propagation.v1","spec_hash":"b9012208dba425ff22d11c84e6b407642fce9132b30cdc54ab4d6d0f33a3c78c","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-077e04eee834b3b4a9d1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v01\n- v08 --supports--> v06\n- v05 --supports--> v03\n- v04 --refutes--> v08\n- v07 --refutes--> v00\n- v07 --supports--> v10\n- v00 --refutes--> v05\n- v07 --supports--> v0d\n- v02 --supports--> v0e\n- v0a --refutes--> v0f\n- v0a --refutes--> v07\n- v09 --refutes--> v0a\n- v0e --refutes--> v0c\n- v0e --refutes--> v0b\n- v0e --refutes--> v04\n- v06 --supports--> v09\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e2fdc12f9550ee8b779650303ee22eebfeecc3bc228382351a14ed4a6be9a59b","schema":"livegraph.boolean-propagation.v1","spec_hash":"077e04eee834b3b4a9d13d9225bbb7b4544750ac198771c462d0c63686c21260","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"}],\"hop\":11,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-1a4a5ace655315211f4d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v08\n- v0c --refutes--> v0e\n- v00 --refutes--> v07\n- v08 --refutes--> v03\n- v08 --refutes--> v01\n- v0a --refutes--> v0b\n- v0c --supports--> v09\n- v0f --supports--> v05\n- v10 --refutes--> v0a\n- v00 --supports--> v02\n- v04 --supports--> v00\n- v00 --supports--> v0d\n- v01 --supports--> v04\n- v05 --supports--> v10\n- v02 --supports--> v0c\n- v0e --refutes--> v06\n\n[VERIFIED ROOT]\nv0f = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1485149cb0049b3dd10cd61bb6489815f43b8a41e2fb6a1229c5ee7f24a3c908","schema":"livegraph.boolean-propagation.v1","spec_hash":"1a4a5ace655315211f4d599f732057a849a929e9d639be50401b947638e16ce6","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-83d757d2d35a74e8d540","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v0c\n- v03 --supports--> v0e\n- v09 --refutes--> v01\n- v09 --refutes--> v00\n- v0f --supports--> v06\n- v0e --supports--> v08\n- v08 --supports--> v0f\n- v08 --refutes--> v02\n- v06 --refutes--> v0d\n- v06 --supports--> v05\n- v0c --supports--> v04\n- v07 --refutes--> v10\n- v0a --refutes--> v0b\n- v09 --supports--> v07\n- v04 --refutes--> v0a\n- v0b --refutes--> v03\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a254ab6bc6a40c04f468d9a502519fbe0704b0946dc9a6f0d420e5b4a1f8111d","schema":"livegraph.boolean-propagation.v1","spec_hash":"83d757d2d35a74e8d5400e4492e95ec40b7a48a4b9f9db2084c6a66cd77cd83e","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-a25d53105c72f05545a9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v0f\n- v10 --refutes--> v0a\n- v05 --supports--> v01\n- v01 --supports--> v10\n- v00 --supports--> v0e\n- v07 --supports--> v04\n- v04 --supports--> v0d\n- v00 --supports--> v08\n- v0c --refutes--> v0b\n- v04 --supports--> v09\n- v0b --refutes--> v05\n- v0a --refutes--> v07\n- v09 --supports--> v03\n- v10 --refutes--> v02\n- v08 --refutes--> v0c\n- v09 --supports--> v06\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"461ed8d9738b2e35fc963a934eee3847b34e023a8f912b20176dd4df715ce5ac","schema":"livegraph.boolean-propagation.v1","spec_hash":"a25d53105c72f05545a9f56a9f2714e404f7bf209047cb2d873c581af9430244","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-08abc90099b82f218dd5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v02\n- v00 --supports--> v07\n- v03 --refutes--> v0c\n- v08 --refutes--> v10\n- v06 --supports--> v03\n- v0f --supports--> v05\n- v00 --refutes--> v0d\n- v0a --refutes--> v01\n- v01 --refutes--> v06\n- v00 --supports--> v09\n- v09 --supports--> v0b\n- v05 --refutes--> v08\n- v0b --supports--> v0a\n- v0c --supports--> v0f\n- v0a --supports--> v04\n- v00 --refutes--> v0e\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b71849f58a624656826cbe12f2328f916b8bf72138532429ff10fea12193129f","schema":"livegraph.boolean-propagation.v1","spec_hash":"08abc90099b82f218dd5655e73be94f202b02af638d697f3f97b4db238050a94","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-2da46ecddc0d821e8b1c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --supports--> v00\n- v06 --supports--> v03\n- v00 --supports--> v05\n- v06 --refutes--> v08\n- v06 --refutes--> v0c\n- v00 --supports--> v0d\n- v09 --refutes--> v06\n- v0b --refutes--> v0f\n- v02 --refutes--> v04\n- v00 --refutes--> v01\n- v07 --refutes--> v0a\n- v04 --refutes--> v07\n- v03 --supports--> v02\n- v0a --refutes--> v0b\n- v0f --refutes--> v10\n- v00 --supports--> v0e\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"014e84694ca093b752f5a1d1f18266c8e0dacf4d8f7194d92a9ee0a3c4700ca8","schema":"livegraph.boolean-propagation.v1","spec_hash":"2da46ecddc0d821e8b1cec2b0ca854a348752b1ba4e4092bce4780cfc4dd6446","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-c637e3959b99561eb7d2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v0e\n- v0e --refutes--> v0f\n- v01 --refutes--> v0a\n- v01 --refutes--> v08\n- v0c --supports--> v0d\n- v01 --refutes--> v0b\n- v0a --refutes--> v05\n- v04 --supports--> v09\n- v02 --refutes--> v07\n- v0a --refutes--> v0c\n- v03 --refutes--> v04\n- v0d --supports--> v00\n- v06 --supports--> v01\n- v0f --supports--> v10\n- v0d --refutes--> v03\n- v10 --supports--> v06\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5e3a7edbfa741410007a6e51a8ca9c7ed543c4ca0deecccf68430b70b462c8a8","schema":"livegraph.boolean-propagation.v1","spec_hash":"c637e3959b99561eb7d259af80c368329e7536cc1e596158370e00b2112675a3","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-154c319b17cfb30dac8b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --supports--> v0c\n- v04 --refutes--> v06\n- v09 --refutes--> v07\n- v0f --supports--> v03\n- v07 --supports--> v04\n- v0a --supports--> v01\n- v08 --supports--> v0d\n- v02 --supports--> v0a\n- v01 --refutes--> v0e\n- v0f --refutes--> v09\n- v06 --supports--> v10\n- v06 --refutes--> v00\n- v04 --supports--> v0b\n- v0f --supports--> v05\n- v0d --refutes--> v02\n- v01 --refutes--> v0f\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dbc4c98c439c0e2058b29e9090f57b4d0af9500ff2b61496e3884f506d5806de","schema":"livegraph.boolean-propagation.v1","spec_hash":"154c319b17cfb30dac8ba83b2e8e994ec81f39ade1bc431d47a489a46aa3ebea","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-0316f12db546ce500630","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v06\n- v00 --refutes--> v0e\n- v0a --supports--> v10\n- v08 --refutes--> v00\n- v07 --supports--> v0f\n- v09 --refutes--> v0a\n- v01 --refutes--> v05\n- v0f --supports--> v04\n- v05 --supports--> v07\n- v01 --supports--> v03\n- v00 --supports--> v0b\n- v0b --supports--> v0d\n- v0a --refutes--> v01\n- v0c --supports--> v09\n- v0d --supports--> v0c\n- v08 --supports--> v02\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a49ee43d97b5eda5d977681dd9a88bbc273b06fd79354670d5934b17a330d02d","schema":"livegraph.boolean-propagation.v1","spec_hash":"0316f12db546ce500630441dd2345806c826640927d570c68103b561c5042fbe","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-9f076b2cfc5c7b8e0d54","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v0d\n- v0b --refutes--> v07\n- v0c --supports--> v09\n- v06 --supports--> v0a\n- v05 --supports--> v02\n- v02 --supports--> v03\n- v10 --refutes--> v0c\n- v0c --refutes--> v04\n- v0b --supports--> v01\n- v03 --refutes--> v0b\n- v0e --refutes--> v06\n- v01 --supports--> v00\n- v09 --supports--> v08\n- v00 --refutes--> v10\n- v09 --supports--> v0e\n- v10 --supports--> v0f\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6392f6a5489ccd001623a204c5f815f3edb9cc49deaaa4ea71d74948ee9f76e5","schema":"livegraph.boolean-propagation.v1","spec_hash":"9f076b2cfc5c7b8e0d540a709a4349f25f51fcbeda2b22a90f28a34df3a2a6c3","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-6580add879d54fc92a98","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v0f\n- v0e --supports--> v02\n- v08 --supports--> v07\n- v0c --refutes--> v03\n- v0f --refutes--> v05\n- v0e --refutes--> v00\n- v05 --refutes--> v0a\n- v0b --refutes--> v06\n- v0a --refutes--> v04\n- v0e --supports--> v0c\n- v06 --refutes--> v09\n- v01 --supports--> v0b\n- v03 --refutes--> v08\n- v0a --supports--> v0e\n- v0f --refutes--> v10\n- v05 --refutes--> v0d\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"086984bf4f563603d09819690825636821f761b7d3d84d44757e3ea7b8b096c6","schema":"livegraph.boolean-propagation.v1","spec_hash":"6580add879d54fc92a9897ed88055c50910fa685dad9eb6b7018f7dffc24c598","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-17de145e777a6a8ea1c6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v07\n- v00 --refutes--> v0b\n- v10 --supports--> v05\n- v09 --supports--> v02\n- v0b --refutes--> v0e\n- v09 --supports--> v06\n- v02 --refutes--> v00\n- v0a --refutes--> v0f\n- v00 --refutes--> v01\n- v10 --refutes--> v0d\n- v05 --refutes--> v09\n- v08 --supports--> v0a\n- v04 --supports--> v10\n- v0c --refutes--> v08\n- v09 --refutes--> v03\n- v0e --supports--> v0c\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"86084e4f06675a599ddc6c213e493379dc95a02eee9538f3c76b984030d59fc3","schema":"livegraph.boolean-propagation.v1","spec_hash":"17de145e777a6a8ea1c66f78af96584200848ce55b6596d2a233206e046cec90","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-1656b30f5abcea97a541","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v04\n- v01 --supports--> v0b\n- v08 --refutes--> v09\n- v08 --supports--> v00\n- v07 --supports--> v0d\n- v0f --refutes--> v01\n- v04 --supports--> v02\n- v05 --refutes--> v0c\n- v0f --refutes--> v10\n- v06 --supports--> v0f\n- v00 --supports--> v06\n- v0c --supports--> v03\n- v04 --refutes--> v0a\n- v0d --supports--> v08\n- v06 --refutes--> v0e\n- v0c --refutes--> v07\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"80454dc17b9dd3612e7959226a85fb9ad7f9dad6826e8365ae98f7d5afcf7f7a","schema":"livegraph.boolean-propagation.v1","spec_hash":"1656b30f5abcea97a541bebf38f12e7440f4cd4943ed9f36bb2e587073bc559a","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-31c8c95de702d61b0550","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v04\n- v06 --supports--> v0a\n- v03 --refutes--> v10\n- v0e --supports--> v06\n- v01 --refutes--> v03\n- v0b --supports--> v00\n- v10 --refutes--> v0f\n- v00 --refutes--> v0d\n- v06 --supports--> v0c\n- v0e --refutes--> v02\n- v0d --supports--> v09\n- v0f --supports--> v0b\n- v07 --refutes--> v01\n- v0a --refutes--> v05\n- v03 --supports--> v08\n- v0d --refutes--> v0e\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8c69b49ed66c86d28521e2be92dbfe9d24a90977cfe889f72de64bc016b6d9f1","schema":"livegraph.boolean-propagation.v1","spec_hash":"31c8c95de702d61b05502d4ca4f2ac36c80973e0058d0280f3181dfe8884656c","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-bd92dbeb61de5dd73ac5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v10\n- v0b --supports--> v08\n- v0e --supports--> v09\n- v0c --supports--> v06\n- v0b --supports--> v0f\n- v02 --refutes--> v01\n- v09 --refutes--> v03\n- v02 --refutes--> v07\n- v03 --refutes--> v0c\n- v0d --refutes--> v0a\n- v0f --refutes--> v0e\n- v09 --refutes--> v05\n- v0a --refutes--> v0b\n- v06 --refutes--> v04\n- v06 --supports--> v02\n- v03 --refutes--> v00\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5064fb006220953a82f2cf763740c0689ea432aec929ceec3babc26d4e3c5e10","schema":"livegraph.boolean-propagation.v1","spec_hash":"bd92dbeb61de5dd73ac5780ebac2025cd998f4bf79b54cec0efac4c622231843","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-aa719607a30bf206d481","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v07\n- v0d --supports--> v05\n- v05 --supports--> v08\n- v09 --supports--> v03\n- v0c --refutes--> v0a\n- v0e --supports--> v0f\n- v03 --supports--> v01\n- v10 --supports--> v0d\n- v03 --refutes--> v02\n- v07 --refutes--> v04\n- v01 --refutes--> v00\n- v05 --supports--> v06\n- v0f --refutes--> v10\n- v0c --supports--> v0e\n- v00 --supports--> v0c\n- v00 --refutes--> v0b\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"af182171f18a66abed0fd9152df5fc0cec5409e381a1af32b2ddd4e2cb404598","schema":"livegraph.boolean-propagation.v1","spec_hash":"aa719607a30bf206d48165544560005980d66b79887c8a9c8c044de7acc8dc6a","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-ca40d1ac9725847fb7fc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v00\n- v06 --refutes--> v0c\n- v05 --refutes--> v09\n- v09 --refutes--> v0e\n- v0d --refutes--> v03\n- v08 --supports--> v04\n- v04 --refutes--> v05\n- v0e --supports--> v0d\n- v03 --supports--> v01\n- v09 --supports--> v10\n- v01 --supports--> v0a\n- v0c --refutes--> v0b\n- v08 --supports--> v07\n- v0b --supports--> v08\n- v0c --refutes--> v0f\n- v0b --refutes--> v02\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"36c83e3b42dd8e9e04753fa3e5345821f7b7fe4cf255d0a677ce9c673746f0aa","schema":"livegraph.boolean-propagation.v1","spec_hash":"ca40d1ac9725847fb7fc606c4a3b90bc9ed2b6b6681ab3486f568e03041347f6","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":11,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-00b1807c0121eb17a8a4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v0f\n- v04 --refutes--> v03\n- v0b --refutes--> v01\n- v0e --refutes--> v0b\n- v0b --supports--> v0d\n- v0f --refutes--> v00\n- v0c --supports--> v0a\n- v0f --supports--> v02\n- v03 --refutes--> v0c\n- v03 --supports--> v09\n- v0d --supports--> v04\n- v00 --refutes--> v0e\n- v0a --supports--> v08\n- v07 --supports--> v06\n- v0c --refutes--> v10\n- v07 --supports--> v05\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"66e6234654f38ca9e3e102ea2fc1be637142808aa44ef6f48f047736faee2d23","schema":"livegraph.boolean-propagation.v1","spec_hash":"00b1807c0121eb17a8a45ecaacb6a7ad5ef27a4c65847aaf9f5c6f727a7b18b9","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-f1c293bfda7945a212fb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --supports--> v09\n- v0e --refutes--> v0c\n- v02 --supports--> v05\n- v0c --refutes--> v0a\n- v0f --supports--> v01\n- v0d --refutes--> v0e\n- v03 --supports--> v0f\n- v02 --refutes--> v08\n- v06 --refutes--> v07\n- v02 --supports--> v04\n- v01 --supports--> v00\n- v00 --supports--> v0d\n- v07 --refutes--> v03\n- v02 --supports--> v10\n- v0d --supports--> v0b\n- v10 --refutes--> v06\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2b1ed44252fb7ee3e2991911895c21969944a7488a6d9ab1dd94ce4b7a0f6cf6","schema":"livegraph.boolean-propagation.v1","spec_hash":"f1c293bfda7945a212fb02b2bf01377b723ce5df00dd45b7cd68a30fe2dd754d","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-4c72af5dbb69deca98a3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v0e\n- v0b --refutes--> v05\n- v08 --supports--> v07\n- v00 --supports--> v0f\n- v10 --supports--> v00\n- v0e --refutes--> v10\n- v04 --refutes--> v0c\n- v07 --supports--> v03\n- v04 --supports--> v0d\n- v01 --supports--> v04\n- v00 --supports--> v06\n- v07 --supports--> v0b\n- v09 --supports--> v01\n- v05 --supports--> v09\n- v04 --refutes--> v02\n- v08 --refutes--> v0a\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b582b34e9cd31b773eca72388ef55eadb48f5ab597fa860900c686147de9bab2","schema":"livegraph.boolean-propagation.v1","spec_hash":"4c72af5dbb69deca98a320f97f23c2da1ae10ae6b948e7ecee43b36ac6b7cce7","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-3b02243cad5e2ef479e7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v03\n- v01 --refutes--> v0d\n- v02 --supports--> v0c\n- v06 --supports--> v10\n- v0e --supports--> v08\n- v07 --refutes--> v02\n- v0c --refutes--> v0b\n- v05 --supports--> v0a\n- v00 --supports--> v0e\n- v0c --supports--> v0f\n- v08 --refutes--> v05\n- v01 --refutes--> v07\n- v0c --supports--> v06\n- v04 --supports--> v01\n- v06 --supports--> v00\n- v0a --supports--> v09\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"01af277fe1e50e23fa0307c9b4b976cd1f8b622acb659ce8760cc4e22258ec3f","schema":"livegraph.boolean-propagation.v1","spec_hash":"3b02243cad5e2ef479e72c5d9707d0c8897e35b644823f9781e3ee8f2630aab8","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":11,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-1996da1e5fd1f1a9ca91","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v09\n- v07 --supports--> v02\n- v05 --supports--> v06\n- v05 --refutes--> v01\n- v0e --supports--> v0c\n- v0d --supports--> v07\n- v0e --refutes--> v08\n- v02 --supports--> v04\n- v04 --supports--> v0f\n- v10 --refutes--> v0e\n- v09 --refutes--> v0d\n- v07 --refutes--> v03\n- v05 --supports--> v0b\n- v0c --refutes--> v05\n- v0f --refutes--> v00\n- v09 --refutes--> v0a\n\n[VERIFIED ROOT]\nv10 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ad309336b18576d2f354dcd108324d0683ab06c806fd93c319ac4ff4b8d62d2d","schema":"livegraph.boolean-propagation.v1","spec_hash":"1996da1e5fd1f1a9ca91c72ac3a3038b82a9a9e79dfeb108c9b57ce76c1f3e91","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-c876cb61476b6a60979e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v10\n- v07 --supports--> v05\n- v05 --supports--> v09\n- v09 --refutes--> v03\n- v0f --refutes--> v0d\n- v07 --supports--> v0c\n- v01 --refutes--> v04\n- v00 --refutes--> v0b\n- v0e --supports--> v06\n- v09 --refutes--> v0e\n- v08 --supports--> v07\n- v0d --refutes--> v0a\n- v10 --refutes--> v08\n- v10 --refutes--> v02\n- v0b --refutes--> v0f\n- v0d --supports--> v01\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"28ec1dd64d12bb43e1617d0039f5197d053e90132c522e2fa75de69c7b01fedd","schema":"livegraph.boolean-propagation.v1","spec_hash":"c876cb61476b6a60979e727a4a3af6a1cc28d5a35c1ad331117846cd5716c75e","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-252347304a4ec70ec654","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v09\n- v0c --supports--> v02\n- v10 --supports--> v0e\n- v05 --supports--> v07\n- v0e --supports--> v01\n- v0d --supports--> v0f\n- v03 --supports--> v0b\n- v04 --supports--> v08\n- v10 --refutes--> v06\n- v08 --refutes--> v10\n- v0e --refutes--> v0d\n- v02 --supports--> v03\n- v09 --refutes--> v0c\n- v02 --refutes--> v00\n- v0e --supports--> v0a\n- v0b --supports--> v04\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"96a03f2e02603a213a5dd81c81740bcfbaa73b0530c8a91f666f00561374dc45","schema":"livegraph.boolean-propagation.v1","spec_hash":"252347304a4ec70ec654ed45730885b7f1f3fba67aa0582377052694692f7994","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-e1a80450014e116c9b32","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v0b\n- v02 --refutes--> v01\n- v08 --refutes--> v09\n- v04 --supports--> v0e\n- v09 --supports--> v0a\n- v0a --supports--> v0c\n- v02 --refutes--> v03\n- v04 --refutes--> v00\n- v06 --refutes--> v05\n- v0f --supports--> v04\n- v0a --supports--> v02\n- v10 --supports--> v06\n- v00 --refutes--> v10\n- v02 --refutes--> v07\n- v05 --refutes--> v08\n- v0d --supports--> v0f\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"066ac14144b06df088e03bbe6a5ef71f90ab732dab68047a2ca6489a80bb4c6d","schema":"livegraph.boolean-propagation.v1","spec_hash":"e1a80450014e116c9b3253bb2018ee5e41461eb05486bb489573688a3751c328","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-befb3c590d61f0ed0f2f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v0c\n- v04 --refutes--> v0e\n- v05 --supports--> v03\n- v05 --refutes--> v00\n- v06 --refutes--> v08\n- v0a --refutes--> v10\n- v04 --supports--> v02\n- v0e --supports--> v0b\n- v0d --refutes--> v01\n- v01 --supports--> v04\n- v06 --refutes--> v0d\n- v03 --refutes--> v06\n- v05 --supports--> v07\n- v10 --refutes--> v0f\n- v0f --supports--> v05\n- v0b --supports--> v09\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ff7192aaf1bb3ff00cc9f086589edb0a0e94273c098cd6f2b60e6cf11b75e7cb","schema":"livegraph.boolean-propagation.v1","spec_hash":"befb3c590d61f0ed0f2f5b96951b8921235dd01d536d777c1aa3969b58fcbb9c","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-c2cf8085d9e84b79abbc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v01\n- v0d --supports--> v05\n- v04 --refutes--> v03\n- v0a --refutes--> v0d\n- v07 --supports--> v0e\n- v0f --refutes--> v10\n- v0e --supports--> v0f\n- v0d --refutes--> v0c\n- v02 --refutes--> v0a\n- v0f --supports--> v06\n- v03 --refutes--> v00\n- v09 --supports--> v07\n- v03 --supports--> v0b\n- v01 --supports--> v09\n- v0f --supports--> v02\n- v08 --supports--> v04\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9589b5ad4c445c5ed3fb8fdb8adf542c8ab634dcaff2cbcc2c63ec12a8be16da","schema":"livegraph.boolean-propagation.v1","spec_hash":"c2cf8085d9e84b79abbc060d29bfa5350d3432b2e9a100a3b80e11df0eb7bcaa","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-675163ed9ae991adc207","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v06\n- v03 --refutes--> v01\n- v05 --supports--> v0a\n- v06 --supports--> v0c\n- v05 --refutes--> v00\n- v07 --supports--> v04\n- v0b --refutes--> v07\n- v0f --refutes--> v03\n- v10 --supports--> v0f\n- v04 --supports--> v08\n- v06 --refutes--> v0d\n- v0e --refutes--> v0b\n- v03 --supports--> v0e\n- v09 --supports--> v02\n- v02 --supports--> v05\n- v06 --supports--> v09\n\n[VERIFIED ROOT]\nv10 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"de54b7e883e80566b8f028b96222662a42a25efe724b834d290b1c7916f07bb1","schema":"livegraph.boolean-propagation.v1","spec_hash":"675163ed9ae991adc207aa490c69721f709f361fa235c91fac3fb3f3a349c9a6","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-9e9cb3c05c0cb0a6307f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v04\n- v0b --refutes--> v07\n- v0a --supports--> v05\n- v03 --refutes--> v02\n- v06 --refutes--> v0b\n- v0f --supports--> v0d\n- v06 --supports--> v0e\n- v0a --supports--> v01\n- v08 --supports--> v00\n- v0c --refutes--> v08\n- v02 --refutes--> v09\n- v09 --supports--> v06\n- v10 --supports--> v0f\n- v0b --supports--> v0a\n- v08 --refutes--> v10\n- v01 --supports--> v0c\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0a0c3aa5ecd34ffa80afeef31248059b9a076cd6d058f0886253de8847a1d146","schema":"livegraph.boolean-propagation.v1","spec_hash":"9e9cb3c05c0cb0a6307f980dc6304c0c4c67afd7ae282ca7cf7957a43d8a706e","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-806b95e058e1fb683d7f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v07\n- v0a --refutes--> v10\n- v00 --supports--> v01\n- v0b --refutes--> v0d\n- v0e --supports--> v00\n- v0d --refutes--> v09\n- v0d --refutes--> v04\n- v0a --refutes--> v0c\n- v0c --refutes--> v0f\n- v08 --supports--> v06\n- v06 --supports--> v0a\n- v0f --refutes--> v0e\n- v04 --supports--> v08\n- v01 --supports--> v02\n- v0f --supports--> v03\n- v01 --refutes--> v05\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ca6a426bb180a5c5ee5b1dbd4e82035893df78af473e51cf7c076ea77911efd6","schema":"livegraph.boolean-propagation.v1","spec_hash":"806b95e058e1fb683d7f911056589a1a9503cca39da7e92d7fd8db084002b311","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-2a7341cd7222d5434800","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v07\n- v09 --refutes--> v04\n- v10 --supports--> v0d\n- v00 --supports--> v01\n- v07 --refutes--> v00\n- v0a --refutes--> v06\n- v10 --refutes--> v0c\n- v06 --supports--> v0e\n- v04 --supports--> v0a\n- v01 --supports--> v02\n- v07 --supports--> v08\n- v0e --refutes--> v0b\n- v01 --refutes--> v03\n- v01 --refutes--> v09\n- v0e --refutes--> v05\n- v05 --supports--> v10\n\n[VERIFIED ROOT]\nv0f = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e41070f313e7f08e42a38a35313e906a59966d6210025ad806f786458bf9428d","schema":"livegraph.boolean-propagation.v1","spec_hash":"2a7341cd7222d54348007d4aa761fc70e2a1870f804c196c24887601399c4b6b","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-d5747a9d76c69bebaf0b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v0b\n- v0e --refutes--> v01\n- v0e --supports--> v06\n- v04 --refutes--> v09\n- v04 --supports--> v00\n- v03 --supports--> v10\n- v0f --supports--> v0d\n- v08 --supports--> v0e\n- v0a --refutes--> v08\n- v0c --refutes--> v03\n- v10 --refutes--> v0a\n- v08 --refutes--> v02\n- v00 --refutes--> v0c\n- v0d --supports--> v04\n- v06 --refutes--> v07\n- v00 --refutes--> v05\n\n[VERIFIED ROOT]\nv0f = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0b5d21afacca267839acb5b007f0c02a6b1a64b56c7cd700db88b7d67a225533","schema":"livegraph.boolean-propagation.v1","spec_hash":"d5747a9d76c69bebaf0b94257e72de28e3ca9194426a1d0e29685ff3a1d44c03","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-dfe38c6c9df5d06f846c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v0e\n- v03 --refutes--> v10\n- v0c --supports--> v00\n- v0d --supports--> v06\n- v0c --refutes--> v0a\n- v04 --refutes--> v07\n- v01 --refutes--> v0d\n- v0c --refutes--> v08\n- v09 --supports--> v0c\n- v0d --refutes--> v03\n- v0e --supports--> v0b\n- v10 --supports--> v0f\n- v04 --refutes--> v09\n- v0e --supports--> v02\n- v08 --refutes--> v01\n- v02 --refutes--> v05\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a2999dbbe15560b6bdaecbd67b8cb1b1929fbc2a02a46cf7ea684bfb91ca08ad","schema":"livegraph.boolean-propagation.v1","spec_hash":"dfe38c6c9df5d06f846c423c03c731a9d7ec7a9a9764291b97194014c90612a1","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-1eed3885d538ed0db3c3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v07\n- v00 --refutes--> v09\n- v0a --refutes--> v0e\n- v0d --refutes--> v06\n- v10 --refutes--> v0c\n- v0a --supports--> v0f\n- v0d --supports--> v04\n- v09 --supports--> v01\n- v09 --supports--> v0b\n- v07 --supports--> v10\n- v08 --refutes--> v03\n- v08 --refutes--> v05\n- v04 --supports--> v08\n- v0e --supports--> v0d\n- v00 --refutes--> v02\n- v03 --refutes--> v00\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"aaf96bfdf8557e2b4dd36710ce24f9b234148fa9e028f360fc8df09d36b03e91","schema":"livegraph.boolean-propagation.v1","spec_hash":"1eed3885d538ed0db3c3db770994a57223c2e06515e170d8af4ab791536a4bc8","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-e3ed15f78832056c7701","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v0f\n- v0b --supports--> v0c\n- v0f --refutes--> v07\n- v0b --supports--> v04\n- v0e --refutes--> v06\n- v00 --refutes--> v09\n- v07 --supports--> v0b\n- v01 --supports--> v05\n- v05 --supports--> v02\n- v06 --supports--> v08\n- v05 --refutes--> v03\n- v06 --supports--> v01\n- v00 --supports--> v10\n- v05 --supports--> v0a\n- v0d --refutes--> v0e\n- v0c --supports--> v0d\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bffbe5a83aa032b1d6c8baec1f733f0afecabf34466aa08a1b6a845a426b0366","schema":"livegraph.boolean-propagation.v1","spec_hash":"e3ed15f78832056c7701e643021f776ff11f5beffb1464359d10f85c4bfdfc1f","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-56c86694715b55fabf8a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v0c\n- v0c --supports--> v0a\n- v02 --refutes--> v0f\n- v05 --refutes--> v04\n- v01 --refutes--> v00\n- v0c --supports--> v03\n- v0f --supports--> v09\n- v04 --refutes--> v0b\n- v10 --refutes--> v05\n- v04 --supports--> v0e\n- v04 --refutes--> v08\n- v00 --supports--> v06\n- v0b --supports--> v02\n- v03 --refutes--> v10\n- v05 --supports--> v07\n- v01 --supports--> v0d\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b81a8d96860915fceb318d0e9e8a03e89442c10bbc7f38d894f24e4814b78d1f","schema":"livegraph.boolean-propagation.v1","spec_hash":"56c86694715b55fabf8a213fa6932c909583ce2064df35b89097c4a6a80d2986","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-586a87e036fa6c856870","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v02\n- v09 --supports--> v0a\n- v10 --supports--> v06\n- v01 --refutes--> v0e\n- v05 --supports--> v10\n- v0f --refutes--> v08\n- v04 --supports--> v00\n- v08 --refutes--> v04\n- v07 --refutes--> v0f\n- v06 --refutes--> v01\n- v0f --refutes--> v03\n- v00 --refutes--> v05\n- v02 --supports--> v0d\n- v00 --supports--> v0c\n- v09 --supports--> v0b\n- v01 --supports--> v09\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7b16e6fa442e3ff54e9576b748932aa39cbf9dd5cd6b59ec58a5047ad5ca32f3","schema":"livegraph.boolean-propagation.v1","spec_hash":"586a87e036fa6c856870b76b65f2333c774d4ade21c31c8074d05f2622d4ccee","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-f20b1d32d49d3b4d8486","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v0e\n- v0e --refutes--> v05\n- v0b --supports--> v0c\n- v08 --refutes--> v07\n- v06 --refutes--> v0b\n- v0d --refutes--> v09\n- v0a --supports--> v06\n- v0c --refutes--> v02\n- v07 --refutes--> v0a\n- v09 --supports--> v03\n- v00 --refutes--> v08\n- v0e --refutes--> v0f\n- v02 --refutes--> v10\n- v0e --supports--> v04\n- v07 --refutes--> v01\n- v02 --refutes--> v0d\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a15506fd36340b0a4864845c4cb39fba08b5a1fbea1d27e8da7a0f9d3298f188","schema":"livegraph.boolean-propagation.v1","spec_hash":"f20b1d32d49d3b4d848646bbb15869e6dc677eb73227c57fb3239fae0cebf0a2","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":11,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-4b057d82aee6faae8f2a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v04\n- v07 --supports--> v05\n- v00 --refutes--> v0b\n- v06 --refutes--> v0d\n- v0e --supports--> v0c\n- v0a --refutes--> v02\n- v0b --supports--> v01\n- v0d --refutes--> v00\n- v04 --supports--> v0f\n- v0f --refutes--> v0e\n- v05 --refutes--> v0a\n- v0a --supports--> v03\n- v0b --refutes--> v07\n- v0a --refutes--> v08\n- v07 --supports--> v10\n- v06 --refutes--> v09\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3829b8376c98e33a6e616e5076e25cafbdea9070646c5c756423c5e4a51df82d","schema":"livegraph.boolean-propagation.v1","spec_hash":"4b057d82aee6faae8f2ab0ff6c1ae3400ef65a539c551c0696d870f74f5cd9ed","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-8b8fd1c36e87474d8226","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v0d\n- v0e --refutes--> v0a\n- v0d --supports--> v00\n- v03 --refutes--> v01\n- v10 --supports--> v0b\n- v05 --supports--> v06\n- v0d --refutes--> v07\n- v0a --supports--> v08\n- v08 --refutes--> v09\n- v04 --supports--> v0e\n- v09 --supports--> v02\n- v10 --refutes--> v04\n- v10 --supports--> v0c\n- v00 --refutes--> v05\n- v05 --supports--> v0f\n- v01 --refutes--> v10\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a7eb6faccac142b35e5776598c88eb1381235dcb7ad9f2be15a6ecfcb55665db","schema":"livegraph.boolean-propagation.v1","spec_hash":"8b8fd1c36e87474d8226689feb981577f30316253a1501f933cc5cbd5f14ac9b","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-d3a67f32312d9659ed48","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v0e\n- v0b --supports--> v10\n- v01 --supports--> v09\n- v05 --supports--> v06\n- v0e --supports--> v0b\n- v0f --supports--> v0a\n- v10 --supports--> v01\n- v09 --supports--> v03\n- v02 --refutes--> v0d\n- v03 --refutes--> v04\n- v02 --supports--> v08\n- v0f --supports--> v00\n- v0e --refutes--> v07\n- v03 --supports--> v05\n- v06 --supports--> v0f\n- v0d --supports--> v0c\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0eda3177101fb4f6ffc6759da022277a80a8253d8eba42ee69d1dbde9dbdbb1f","schema":"livegraph.boolean-propagation.v1","spec_hash":"d3a67f32312d9659ed48aa897d56915ecdbc950fe9dcfa608a99af7a1e6a278f","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-55666b386233f02c4954","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v00\n- v0d --refutes--> v10\n- v0a --supports--> v06\n- v05 --refutes--> v02\n- v10 --refutes--> v0b\n- v08 --supports--> v0d\n- v02 --refutes--> v07\n- v0b --supports--> v0c\n- v0c --refutes--> v0f\n- v09 --refutes--> v05\n- v03 --supports--> v0e\n- v02 --supports--> v0a\n- v0d --refutes--> v04\n- v03 --refutes--> v08\n- v06 --refutes--> v03\n- v0c --supports--> v01\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d6c4904d4bec9e0f7890d8523baedcd5a71b716585386e69f3729c9c53989d2c","schema":"livegraph.boolean-propagation.v1","spec_hash":"55666b386233f02c4954415ae78b92382231f54c03411bc3fe79a9001835c5cd","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-0cbecda4c90ada4d9efa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v0b\n- v01 --supports--> v05\n- v04 --refutes--> v0e\n- v0c --supports--> v07\n- v00 --supports--> v0a\n- v08 --refutes--> v03\n- v06 --supports--> v0c\n- v10 --supports--> v09\n- v02 --supports--> v04\n- v09 --refutes--> v02\n- v04 --supports--> v0f\n- v0a --refutes--> v0d\n- v0a --refutes--> v01\n- v0f --refutes--> v00\n- v01 --supports--> v06\n- v08 --supports--> v10\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"aa54ea298ead7af18ddb45c9b10ff6de95d3378bec83a70b2395e8ac0b94f207","schema":"livegraph.boolean-propagation.v1","spec_hash":"0cbecda4c90ada4d9efa49446160e3d6e9505be6ab6bbd7fc9e3a238c37980d3","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-747a8c872c1a9500008b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v02\n- v0c --refutes--> v07\n- v0a --supports--> v06\n- v00 --refutes--> v03\n- v0a --refutes--> v01\n- v0d --refutes--> v05\n- v0d --refutes--> v08\n- v0d --refutes--> v0e\n- v06 --supports--> v0b\n- v05 --refutes--> v00\n- v06 --refutes--> v10\n- v07 --refutes--> v09\n- v03 --refutes--> v04\n- v09 --refutes--> v0f\n- v0b --refutes--> v0d\n- v04 --refutes--> v0c\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8a4953c50fd3dd0a9e1dc5d1458b46873c6063fac523fdddd8789c810dd19a1d","schema":"livegraph.boolean-propagation.v1","spec_hash":"747a8c872c1a9500008bb035a4516a84c1173e1fc38ba4f4ddf5735a6c75c2ff","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":11,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-89292443675aa448bd07","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v0c\n- v0c --refutes--> v03\n- v0b --supports--> v05\n- v06 --refutes--> v02\n- v02 --refutes--> v0f\n- v07 --supports--> v10\n- v03 --supports--> v0b\n- v03 --refutes--> v00\n- v0a --supports--> v08\n- v06 --refutes--> v04\n- v0b --refutes--> v09\n- v0d --supports--> v06\n- v0f --refutes--> v0e\n- v0e --refutes--> v0a\n- v08 --supports--> v01\n- v0b --supports--> v07\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"36f290fd48d3d706fb747f4ee043480a126df433e343c1989de33b3c3d1a5422","schema":"livegraph.boolean-propagation.v1","spec_hash":"89292443675aa448bd0762fa0fa674ded6752d20fcb7f1c8b97aa9fe2825a756","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-af6920e908ae43aeb48f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --supports--> v05\n- v0d --refutes--> v07\n- v08 --refutes--> v0d\n- v06 --supports--> v01\n- v07 --refutes--> v03\n- v03 --refutes--> v00\n- v0a --refutes--> v04\n- v0f --refutes--> v0e\n- v0d --refutes--> v0c\n- v06 --refutes--> v02\n- v0f --supports--> v0a\n- v01 --refutes--> v0b\n- v04 --supports--> v06\n- v01 --refutes--> v08\n- v01 --refutes--> v09\n- v00 --supports--> v10\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5c0d2ed905aa058ba68c54bf552710dcec210e64e0e6af75b054aa57f649c953","schema":"livegraph.boolean-propagation.v1","spec_hash":"af6920e908ae43aeb48f8d382da6bee59ff3dd180b7a48f918dccdb9c69e5211","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-6c4d4f6f692117f0e4da","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v09\n- v0c --supports--> v0b\n- v05 --refutes--> v01\n- v10 --supports--> v07\n- v01 --refutes--> v0d\n- v06 --refutes--> v04\n- v03 --supports--> v0c\n- v0a --refutes--> v0e\n- v0a --refutes--> v00\n- v08 --supports--> v05\n- v0b --refutes--> v08\n- v0a --supports--> v0f\n- v0f --refutes--> v10\n- v02 --refutes--> v06\n- v01 --refutes--> v0a\n- v10 --refutes--> v02\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b5fcb7d888fc23e8a91833769f09f457f3dca1f9c68c73dfbeca3d8471ccc70c","schema":"livegraph.boolean-propagation.v1","spec_hash":"6c4d4f6f692117f0e4dade52e1109e993a978df8cac4bd07fee5db8e29adda06","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-2c4833acf1bb6447fb97","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v04\n- v01 --supports--> v09\n- v0a --refutes--> v08\n- v03 --supports--> v02\n- v09 --supports--> v07\n- v06 --refutes--> v0b\n- v04 --refutes--> v0d\n- v06 --refutes--> v01\n- v04 --refutes--> v0f\n- v05 --supports--> v10\n- v03 --refutes--> v0e\n- v0e --supports--> v06\n- v0d --supports--> v00\n- v08 --supports--> v05\n- v10 --supports--> v03\n- v0d --supports--> v0c\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4cb503b7707b6d5bb2af98a0a78885176f17b40f6396a622684cd0ce82326eed","schema":"livegraph.boolean-propagation.v1","spec_hash":"2c4833acf1bb6447fb9754d6fefae38d91cf5c45577ec1f2796a53761ed0d29e","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-b53416a9a7af772743b0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v0b\n- v0c --refutes--> v06\n- v04 --supports--> v00\n- v0e --refutes--> v0d\n- v0a --supports--> v0c\n- v05 --refutes--> v0a\n- v06 --supports--> v07\n- v07 --refutes--> v08\n- v0c --refutes--> v10\n- v07 --refutes--> v09\n- v00 --refutes--> v03\n- v09 --refutes--> v04\n- v06 --refutes--> v0f\n- v0a --refutes--> v01\n- v0b --refutes--> v0e\n- v09 --refutes--> v02\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3b8dd3286991626a74cb625b2791edbd86bb584f66d339791145068d0956f328","schema":"livegraph.boolean-propagation.v1","spec_hash":"b53416a9a7af772743b0646e318434300d5129122091ef40fce8da65586fd098","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"}],\"hop\":11,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-0c0f5507085b2d0e50a8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v08\n- v09 --supports--> v02\n- v0a --supports--> v01\n- v00 --refutes--> v05\n- v03 --supports--> v0c\n- v0e --refutes--> v0b\n- v0e --supports--> v09\n- v04 --refutes--> v0a\n- v10 --supports--> v00\n- v09 --supports--> v0d\n- v02 --supports--> v03\n- v07 --supports--> v10\n- v06 --refutes--> v0e\n- v05 --supports--> v04\n- v0a --refutes--> v06\n- v07 --refutes--> v0f\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8036e18e62eecc6348c4243d561f16722b6544a65b303956ab6ab5353a843d28","schema":"livegraph.boolean-propagation.v1","spec_hash":"0c0f5507085b2d0e50a8129f4d0b1c10fe27711b616c40f9ba71d4f38fda93bb","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-b13df53503f694441bea","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v0b\n- v03 --refutes--> v02\n- v01 --refutes--> v04\n- v03 --refutes--> v0d\n- v0f --supports--> v08\n- v0a --supports--> v07\n- v0d --supports--> v01\n- v0a --refutes--> v10\n- v05 --refutes--> v00\n- v01 --refutes--> v0e\n- v0f --refutes--> v03\n- v00 --supports--> v0f\n- v0b --supports--> v06\n- v10 --refutes--> v09\n- v06 --supports--> v0a\n- v10 --refutes--> v0c\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"41e1e4d53edb7c26c4216a6155e37860930ad91ecbc6f1f90d624064c62c7454","schema":"livegraph.boolean-propagation.v1","spec_hash":"b13df53503f694441beaba022537848c12743f0828cbf01861b5cc7210a80c15","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-826fcaceae0f9ed0f2f3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v0d\n- v09 --refutes--> v08\n- v06 --supports--> v05\n- v08 --refutes--> v06\n- v0b --supports--> v07\n- v04 --refutes--> v0c\n- v04 --refutes--> v03\n- v04 --supports--> v0e\n- v0f --refutes--> v0b\n- v0b --supports--> v09\n- v03 --supports--> v02\n- v01 --refutes--> v10\n- v10 --refutes--> v04\n- v01 --refutes--> v0a\n- v06 --supports--> v00\n- v02 --supports--> v0f\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"68dfde0208edcba72fa78591e9a412697a8bb417f0f57136d62de9dd8ced8ddc","schema":"livegraph.boolean-propagation.v1","spec_hash":"826fcaceae0f9ed0f2f32e844cf6e3807f5e09573ef2667df9692a5ba7fe09ff","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-a3d4671bf024095d07b1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v0d\n- v02 --supports--> v05\n- v08 --supports--> v0e\n- v0b --refutes--> v06\n- v01 --refutes--> v10\n- v10 --refutes--> v0c\n- v05 --refutes--> v0f\n- v03 --supports--> v07\n- v0d --refutes--> v0b\n- v05 --refutes--> v03\n- v0d --supports--> v09\n- v0b --refutes--> v0a\n- v07 --refutes--> v04\n- v04 --refutes--> v01\n- v0b --supports--> v02\n- v10 --supports--> v08\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6b4707b575eb2c53f36046e949261f098797ea20e5f8ee1641db10fd15308408","schema":"livegraph.boolean-propagation.v1","spec_hash":"a3d4671bf024095d07b1e7394c8015444ad8f1bdf6e585bcf08c1761333de8c0","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-37c56b9eb8018f09403d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v0d\n- v08 --refutes--> v05\n- v0b --supports--> v08\n- v06 --supports--> v01\n- v07 --refutes--> v09\n- v0f --supports--> v04\n- v10 --refutes--> v0a\n- v00 --supports--> v0b\n- v06 --refutes--> v00\n- v04 --supports--> v0c\n- v08 --supports--> v03\n- v06 --supports--> v0e\n- v04 --supports--> v02\n- v05 --supports--> v10\n- v09 --supports--> v06\n- v0a --supports--> v0f\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dd9b47b7e6230c666eaeed28c479586f953223e8b7486e3a9f5646161a9de5a7","schema":"livegraph.boolean-propagation.v1","spec_hash":"37c56b9eb8018f09403ddd2e2bbf9f94213a3587568c160c24ec64ee83c9e094","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-0e1aafa686c43ee4d8c5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v06\n- v09 --supports--> v03\n- v10 --refutes--> v0b\n- v0d --supports--> v02\n- v03 --supports--> v08\n- v10 --supports--> v0a\n- v02 --supports--> v0c\n- v04 --supports--> v10\n- v03 --supports--> v05\n- v06 --refutes--> v0f\n- v01 --refutes--> v04\n- v07 --supports--> v0d\n- v02 --refutes--> v01\n- v0b --refutes--> v09\n- v04 --supports--> v0e\n- v04 --refutes--> v00\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"057dd2a3af28cfccce3e0509a16d19e2eb451d56eb897ae9731b85978302cabf","schema":"livegraph.boolean-propagation.v1","spec_hash":"0e1aafa686c43ee4d8c5e9237c5cf8c88a8656a0c5f8a480a32021aae36263f3","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-7b5e607417b0ef4692ba","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v0c\n- v00 --refutes--> v03\n- v07 --refutes--> v08\n- v0f --supports--> v0e\n- v03 --supports--> v02\n- v01 --supports--> v04\n- v04 --refutes--> v0d\n- v03 --supports--> v0b\n- v10 --supports--> v00\n- v03 --supports--> v09\n- v0c --supports--> v06\n- v03 --supports--> v05\n- v09 --supports--> v0a\n- v07 --refutes--> v10\n- v0d --refutes--> v0f\n- v0e --supports--> v07\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5059ae52cd5742ac56268d58a6ef54539cf6552af25f790ed1136bc86f131219","schema":"livegraph.boolean-propagation.v1","spec_hash":"7b5e607417b0ef4692badc32faa41076fd049251af600486e2a0bc0523013b8c","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-cef5618ce69af3ddd993","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v0c\n- v07 --refutes--> v04\n- v10 --refutes--> v03\n- v05 --refutes--> v0a\n- v08 --refutes--> v0b\n- v10 --refutes--> v01\n- v05 --supports--> v09\n- v0c --refutes--> v05\n- v0d --refutes--> v10\n- v10 --refutes--> v08\n- v00 --refutes--> v0e\n- v0f --refutes--> v06\n- v06 --refutes--> v00\n- v0c --refutes--> v02\n- v09 --refutes--> v0f\n- v08 --refutes--> v07\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0b9c687e1721c21f9a9ff235388aad180deb9b0745056eb6d88985423e10d993","schema":"livegraph.boolean-propagation.v1","spec_hash":"cef5618ce69af3ddd993910dbf547566a463ebd93de6d0e05fb2b7f09746c9f8","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-3cd8d4735bd0eeb27775","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v09\n- v0c --refutes--> v06\n- v09 --refutes--> v03\n- v01 --supports--> v02\n- v01 --refutes--> v0d\n- v01 --refutes--> v04\n- v02 --supports--> v0c\n- v09 --refutes--> v0e\n- v00 --refutes--> v08\n- v08 --refutes--> v0b\n- v03 --supports--> v01\n- v06 --supports--> v07\n- v08 --supports--> v0a\n- v05 --refutes--> v10\n- v06 --supports--> v00\n- v0a --supports--> v05\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1cbea3b4694fd74de6846f5755e9c19c2546bafbbb14eddff67fc3872188cd70","schema":"livegraph.boolean-propagation.v1","spec_hash":"3cd8d4735bd0eeb277757241f8c3a9f47ad036c5104fa69e9077b7cb3cf29edd","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-3453e2f87faf4387d6a5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v0e\n- v09 --supports--> v10\n- v01 --supports--> v04\n- v10 --supports--> v0f\n- v0e --supports--> v08\n- v02 --supports--> v0a\n- v00 --refutes--> v05\n- v08 --refutes--> v02\n- v0b --refutes--> v09\n- v0a --refutes--> v07\n- v00 --supports--> v0c\n- v05 --refutes--> v01\n- v08 --supports--> v06\n- v0a --supports--> v03\n- v04 --supports--> v0b\n- v04 --supports--> v0d\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"02eb7eae49ea3126d428cfaf132943f9299d33b384699ecd1469d2b96170747b","schema":"livegraph.boolean-propagation.v1","spec_hash":"3453e2f87faf4387d6a58db8e2d97c69f0bf5c302f0692fe349c58a7e34761ea","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-775ac99b886f68da0079","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v0b\n- v07 --supports--> v09\n- v0e --refutes--> v01\n- v01 --supports--> v06\n- v05 --supports--> v08\n- v0f --supports--> v0e\n- v01 --supports--> v03\n- v04 --refutes--> v07\n- v08 --supports--> v00\n- v01 --supports--> v05\n- v0b --refutes--> v0d\n- v0c --refutes--> v0a\n- v0a --supports--> v0f\n- v00 --supports--> v02\n- v07 --supports--> v10\n- v07 --refutes--> v0c\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fccae6d7a43439d8b38ca49fc8d29740dc1ed3b8817944b922abd01baf8430f0","schema":"livegraph.boolean-propagation.v1","spec_hash":"775ac99b886f68da0079ffa8adb6fe4a96861597d80baccf4f32e75f09081eb7","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-f9975209c99e38515500","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v03\n- v03 --refutes--> v09\n- v03 --supports--> v08\n- v0a --supports--> v05\n- v0b --refutes--> v06\n- v01 --refutes--> v07\n- v08 --supports--> v0c\n- v04 --supports--> v10\n- v01 --refutes--> v0a\n- v0b --refutes--> v0f\n- v05 --supports--> v02\n- v05 --supports--> v0d\n- v10 --refutes--> v01\n- v02 --refutes--> v0e\n- v0e --supports--> v0b\n- v10 --refutes--> v00\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1dd74dbfb3acde62dc705a26c5195a07f0f7f9fd5ff35970418438f01309d31b","schema":"livegraph.boolean-propagation.v1","spec_hash":"f9975209c99e38515500be632355cafb25cbe84beae4342d26d426b9c716f2c6","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-4a82352c58fbb71dc71b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v04\n- v08 --supports--> v0d\n- v0c --refutes--> v10\n- v0e --supports--> v05\n- v0c --supports--> v07\n- v0a --supports--> v01\n- v08 --refutes--> v00\n- v09 --supports--> v0e\n- v00 --supports--> v02\n- v05 --refutes--> v0a\n- v0c --supports--> v0b\n- v03 --refutes--> v0f\n- v00 --supports--> v03\n- v04 --supports--> v06\n- v10 --refutes--> v08\n- v06 --refutes--> v0c\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9ddb81b56b0afbecdb5827306ce7e318856ba0fbc6ab2a0743d7acb0cebe476a","schema":"livegraph.boolean-propagation.v1","spec_hash":"4a82352c58fbb71dc71bce3d734607c854b0b2942096fb9f65bc2fb195af713d","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-9034784f0c6d2a4eb5f5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v00\n- v09 --supports--> v0d\n- v0a --refutes--> v01\n- v07 --refutes--> v03\n- v0b --supports--> v09\n- v05 --refutes--> v0c\n- v03 --refutes--> v08\n- v09 --refutes--> v02\n- v05 --refutes--> v10\n- v07 --supports--> v0f\n- v0e --supports--> v0b\n- v0a --refutes--> v06\n- v10 --refutes--> v0e\n- v04 --supports--> v05\n- v03 --supports--> v04\n- v00 --refutes--> v07\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8b84be39d8fc6180d424ecce6202924fb68fd4d4f39bbe885a858e769f86d4dc","schema":"livegraph.boolean-propagation.v1","spec_hash":"9034784f0c6d2a4eb5f55bdf5129fd7d66b120d5d4398296758841ae48a46d45","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-9598a2871405d15b0c1b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v02\n- v0d --supports--> v0a\n- v0a --refutes--> v0f\n- v0c --supports--> v07\n- v00 --refutes--> v0e\n- v06 --supports--> v0b\n- v0b --supports--> v04\n- v05 --refutes--> v08\n- v0b --refutes--> v10\n- v02 --refutes--> v00\n- v08 --refutes--> v09\n- v10 --supports--> v05\n- v03 --refutes--> v0d\n- v0c --supports--> v06\n- v0b --refutes--> v01\n- v08 --supports--> v03\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"19eacd425da4e0d0ab8621b8616bb4d349e5ce99e78330c096c9a487887c0e3b","schema":"livegraph.boolean-propagation.v1","spec_hash":"9598a2871405d15b0c1b520453c24a8e014029f18be8e207ad5b4caca9d0d221","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-f15ce98d0d8f1d1f8e5f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v05\n- v02 --supports--> v04\n- v06 --supports--> v08\n- v0a --refutes--> v0c\n- v0f --refutes--> v0d\n- v09 --supports--> v0a\n- v00 --refutes--> v01\n- v00 --supports--> v0b\n- v01 --supports--> v0e\n- v0d --supports--> v09\n- v06 --refutes--> v02\n- v01 --refutes--> v07\n- v0c --supports--> v00\n- v10 --refutes--> v0f\n- v04 --refutes--> v10\n- v02 --supports--> v03\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c9af1c87dafafcd66f4171d098f4e7ecba2bcaf734410e7a6e82c8adad619c7c","schema":"livegraph.boolean-propagation.v1","spec_hash":"f15ce98d0d8f1d1f8e5f94d0df69366de8159540610b473abb6ac5b60edcaed5","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-2e10738e4419ffe90816","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v06\n- v01 --supports--> v0f\n- v10 --supports--> v03\n- v0f --supports--> v10\n- v03 --refutes--> v02\n- v10 --supports--> v05\n- v0d --refutes--> v0c\n- v03 --supports--> v00\n- v08 --refutes--> v0b\n- v08 --refutes--> v09\n- v0a --refutes--> v08\n- v0c --refutes--> v0e\n- v09 --refutes--> v07\n- v00 --refutes--> v0d\n- v0a --supports--> v04\n- v0e --supports--> v0a\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3b851cfb95aeb64e32ed2e6b0a427bbf7fc2a1e284a2cbff338e6d4ca35dff33","schema":"livegraph.boolean-propagation.v1","spec_hash":"2e10738e4419ffe908163be9d8ccc46bd989ecd799fa9c9b56ee3fcb3b00e861","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-3ca891032c8f61d7fe59","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v10\n- v0a --supports--> v0f\n- v0e --refutes--> v0a\n- v10 --supports--> v06\n- v07 --supports--> v0e\n- v08 --refutes--> v09\n- v00 --supports--> v04\n- v0b --supports--> v02\n- v0c --refutes--> v03\n- v04 --refutes--> v0b\n- v04 --refutes--> v05\n- v0c --supports--> v07\n- v09 --supports--> v00\n- v0a --refutes--> v08\n- v10 --supports--> v0d\n- v0d --refutes--> v01\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4b3d3bab25bb2245579008018990cc68bc7474ad7d574cf43fee782ea338a17d","schema":"livegraph.boolean-propagation.v1","spec_hash":"3ca891032c8f61d7fe59bcaf7378d2537863062e01324ba15bd6e05d49c91b05","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-bd68a9073e2321c81ad4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v0f\n- v06 --refutes--> v0b\n- v07 --refutes--> v02\n- v06 --supports--> v08\n- v08 --refutes--> v0c\n- v02 --refutes--> v10\n- v0d --refutes--> v07\n- v08 --refutes--> v01\n- v10 --supports--> v03\n- v09 --supports--> v0d\n- v00 --supports--> v09\n- v04 --refutes--> v06\n- v02 --refutes--> v0a\n- v01 --supports--> v00\n- v0e --supports--> v04\n- v10 --refutes--> v05\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"af3a01d4a48fb03951c8b8f85f6df2fb857eae8013495c66584f2828a106eb54","schema":"livegraph.boolean-propagation.v1","spec_hash":"bd68a9073e2321c81ad4a90a4db842f29afefefeb8efb19208daf4515eaf5eba","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-a09658cb44f79744a211","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v0e\n- v06 --supports--> v0f\n- v0b --refutes--> v04\n- v03 --supports--> v09\n- v0e --supports--> v06\n- v05 --supports--> v01\n- v06 --supports--> v08\n- v10 --refutes--> v0a\n- v0d --refutes--> v0b\n- v01 --supports--> v10\n- v10 --refutes--> v02\n- v09 --supports--> v0c\n- v0b --supports--> v00\n- v06 --supports--> v05\n- v0c --refutes--> v0d\n- v0b --refutes--> v07\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b1d672d7370997b6246e11aa996e1892258fc7e2391c6d57d0cfd84fd3e3a7ce","schema":"livegraph.boolean-propagation.v1","spec_hash":"a09658cb44f79744a2115efa4432365e160a22ee1c882a90201965640c702a06","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-e660a4a4368df0248220","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v07\n- v0b --supports--> v02\n- v06 --refutes--> v0d\n- v07 --supports--> v05\n- v09 --supports--> v04\n- v0f --refutes--> v09\n- v0c --supports--> v08\n- v10 --supports--> v0e\n- v06 --supports--> v0c\n- v09 --refutes--> v03\n- v03 --supports--> v00\n- v00 --refutes--> v06\n- v0e --supports--> v0f\n- v07 --refutes--> v0b\n- v0c --supports--> v0a\n- v09 --supports--> v01\n\n[VERIFIED ROOT]\nv10 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f32b367f28eb0cfce20392a539fb7495b430f299c32c6c885693fb9481bfebfa","schema":"livegraph.boolean-propagation.v1","spec_hash":"e660a4a4368df0248220720bce3c6182b3b8ce72d6513b2e867cc6638224c2e2","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-0ca42309c6c8a062fd88","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v05\n- v01 --supports--> v0e\n- v08 --supports--> v0a\n- v00 --refutes--> v0f\n- v06 --supports--> v0c\n- v01 --refutes--> v0b\n- v07 --refutes--> v01\n- v0c --supports--> v0d\n- v0d --supports--> v07\n- v0f --supports--> v03\n- v0f --supports--> v09\n- v10 --supports--> v00\n- v10 --refutes--> v02\n- v05 --supports--> v06\n- v01 --refutes--> v10\n- v0c --supports--> v04\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3ae634e78dec4b6e5a508b3d620108f4744a48630ff1cee9cda4344c4c2f09f7","schema":"livegraph.boolean-propagation.v1","spec_hash":"0ca42309c6c8a062fd88bbe663c23180ac3a77a082b02b91ab6048cc11ca8206","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-062845219ce2d52060ac","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v09\n- v0a --supports--> v0c\n- v03 --refutes--> v05\n- v02 --supports--> v06\n- v08 --refutes--> v01\n- v01 --refutes--> v10\n- v08 --refutes--> v0f\n- v02 --refutes--> v04\n- v0c --supports--> v03\n- v05 --refutes--> v0d\n- v0e --supports--> v0b\n- v02 --refutes--> v08\n- v03 --supports--> v07\n- v0b --refutes--> v02\n- v0b --refutes--> v00\n- v09 --refutes--> v0e\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"53675b688584cc7a14f8c0c8c1c7f51187daf4d64dca867d3fd97a0a5b0158ea","schema":"livegraph.boolean-propagation.v1","spec_hash":"062845219ce2d52060ac6faadcb6ebe6243e4f0489d38bea86f347e357819852","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":11,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-aecfe43fd35365199a94","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v00\n- v04 --supports--> v0a\n- v0e --refutes--> v0c\n- v06 --supports--> v0e\n- v10 --refutes--> v08\n- v08 --refutes--> v0b\n- v0a --supports--> v02\n- v0a --refutes--> v0d\n- v09 --refutes--> v01\n- v05 --refutes--> v09\n- v01 --refutes--> v07\n- v08 --refutes--> v0f\n- v0e --refutes--> v10\n- v0f --supports--> v04\n- v0a --refutes--> v03\n- v07 --refutes--> v06\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a851585ab41721a2cca1c45617a02219dbcdcf57bad939c17a9aa9afe07bb619","schema":"livegraph.boolean-propagation.v1","spec_hash":"aecfe43fd35365199a94faa0fc936e253b1291b526e030ce4ddf70b9cbfbaaba","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-ec553b0359fb4808e5c0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v09\n- v0c --refutes--> v07\n- v0a --supports--> v00\n- v00 --refutes--> v0b\n- v03 --supports--> v0e\n- v05 --refutes--> v0d\n- v02 --supports--> v05\n- v05 --supports--> v01\n- v0b --refutes--> v03\n- v0c --refutes--> v10\n- v0f --refutes--> v06\n- v0f --supports--> v0c\n- v07 --supports--> v0a\n- v04 --supports--> v0f\n- v03 --refutes--> v08\n- v0e --supports--> v02\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"df50f584dd3485f732ecf4f8badd2dcf59f5f0f9f4b78392599dfbaaab462330","schema":"livegraph.boolean-propagation.v1","spec_hash":"ec553b0359fb4808e5c07b337df382b6e646f86cc4b449945c5a95dffa6078ba","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-6ee9bffed221540f377e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v03\n- v03 --refutes--> v06\n- v0c --refutes--> v10\n- v08 --supports--> v04\n- v06 --supports--> v02\n- v06 --supports--> v0b\n- v10 --supports--> v0f\n- v04 --refutes--> v05\n- v04 --refutes--> v07\n- v07 --refutes--> v0d\n- v0d --supports--> v0c\n- v0e --refutes--> v09\n- v10 --refutes--> v00\n- v02 --refutes--> v01\n- v10 --refutes--> v0a\n- v02 --supports--> v08\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"07487f9ef4160b6f5d38d579f8b38a0b5b55dd4a047f776fa9600ddbc5f096da","schema":"livegraph.boolean-propagation.v1","spec_hash":"6ee9bffed221540f377e30220437305568f93084b8e0863f9a4ad042ae09dce1","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-85ba20680f93ebde3e1c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v02\n- v09 --supports--> v08\n- v0c --refutes--> v05\n- v0c --refutes--> v07\n- v02 --supports--> v10\n- v08 --refutes--> v0a\n- v02 --refutes--> v0b\n- v02 --refutes--> v04\n- v01 --supports--> v03\n- v00 --supports--> v0e\n- v08 --refutes--> v06\n- v10 --supports--> v00\n- v0d --refutes--> v0c\n- v0e --refutes--> v01\n- v07 --supports--> v09\n- v0a --supports--> v0f\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ea30d427dd7098990e55243dfe8ee0d3d2f5060e874e04535c942d248c4a9117","schema":"livegraph.boolean-propagation.v1","spec_hash":"85ba20680f93ebde3e1c88ced33cb119f0bf3bccfca5ce5780b86e0239a34bff","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-c03b35b1df8cc77a7cbb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v01\n- v0a --refutes--> v07\n- v01 --supports--> v10\n- v06 --refutes--> v0f\n- v04 --refutes--> v02\n- v02 --supports--> v0b\n- v01 --supports--> v04\n- v09 --refutes--> v0c\n- v08 --refutes--> v06\n- v09 --supports--> v08\n- v07 --refutes--> v0e\n- v0a --refutes--> v03\n- v02 --refutes--> v0a\n- v0d --supports--> v05\n- v00 --refutes--> v0d\n- v0e --refutes--> v09\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"19f2be7fcbd5babf759f3caaba39656f22a813300a3fee286c8015f95ec6a3b1","schema":"livegraph.boolean-propagation.v1","spec_hash":"c03b35b1df8cc77a7cbb448fe351ebfce9f4870f7e2a9cc200e2a18249d968b9","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"}],\"hop\":11,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-5beaacdef4cb10edabc0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v02\n- v03 --refutes--> v06\n- v05 --refutes--> v0a\n- v01 --supports--> v00\n- v00 --refutes--> v07\n- v06 --refutes--> v05\n- v02 --refutes--> v0c\n- v0f --refutes--> v08\n- v0e --refutes--> v0f\n- v0a --supports--> v09\n- v09 --supports--> v10\n- v09 --refutes--> v0d\n- v05 --supports--> v0b\n- v0c --supports--> v0e\n- v01 --refutes--> v04\n- v0f --supports--> v01\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5a1fc089124c368bae7cc9d46e802ccefc1a3e71a447e23b6d5a0a0488576276","schema":"livegraph.boolean-propagation.v1","spec_hash":"5beaacdef4cb10edabc09bc2607883841b058924308df281dea7bbf5c842e50f","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-08531a61fdfd8992be60","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v0b\n- v10 --supports--> v05\n- v0f --refutes--> v0a\n- v04 --supports--> v10\n- v03 --supports--> v01\n- v0f --refutes--> v02\n- v00 --supports--> v0d\n- v0e --supports--> v04\n- v09 --refutes--> v07\n- v09 --supports--> v03\n- v0c --supports--> v08\n- v08 --supports--> v0e\n- v00 --refutes--> v0f\n- v03 --refutes--> v06\n- v01 --supports--> v00\n- v05 --refutes--> v09\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"59c4578e8f98456a7324e04f636a5c993d4619b31678b555d368d86015eea99e","schema":"livegraph.boolean-propagation.v1","spec_hash":"08531a61fdfd8992be600d51ccfe5a237336f21a10e52dba1a3949f67b706e6f","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-b8ed8db3ab8a0127e552","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v0f\n- v0f --refutes--> v05\n- v04 --refutes--> v0b\n- v0a --refutes--> v00\n- v0e --refutes--> v03\n- v10 --supports--> v0e\n- v05 --refutes--> v07\n- v07 --refutes--> v0a\n- v03 --supports--> v01\n- v00 --supports--> v04\n- v04 --supports--> v09\n- v0c --refutes--> v10\n- v03 --refutes--> v0d\n- v0f --refutes--> v02\n- v0c --supports--> v08\n- v10 --refutes--> v06\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d50022afb29cf76c8d745c46c55cba6f2a2b0a8489eaa717810a23230f794f6e","schema":"livegraph.boolean-propagation.v1","spec_hash":"b8ed8db3ab8a0127e55216ba6299c71dcf8e14455c668caa74fea436dd207899","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-ae1cbc47d935512e81e8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v0f\n- v02 --refutes--> v01\n- v0b --refutes--> v10\n- v10 --refutes--> v0c\n- v07 --supports--> v0d\n- v04 --supports--> v06\n- v08 --refutes--> v0e\n- v0b --refutes--> v00\n- v05 --refutes--> v03\n- v0c --refutes--> v09\n- v10 --refutes--> v0a\n- v0e --refutes--> v07\n- v01 --refutes--> v05\n- v0c --supports--> v04\n- v03 --refutes--> v08\n- v04 --refutes--> v02\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7719fdadc8fcb72d9a97a0222c4bad032f91981be764c0c0d129c9c0b4c167f2","schema":"livegraph.boolean-propagation.v1","spec_hash":"ae1cbc47d935512e81e89f5af17dd63c95854abf3a95d43c0228dcd1f74cf751","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-3409fed8ec3901ea96ba","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v06\n- v07 --supports--> v0b\n- v06 --supports--> v0e\n- v0c --refutes--> v05\n- v03 --supports--> v04\n- v08 --refutes--> v10\n- v07 --refutes--> v03\n- v0d --supports--> v09\n- v10 --supports--> v02\n- v0c --supports--> v01\n- v0e --supports--> v08\n- v02 --refutes--> v07\n- v09 --refutes--> v00\n- v08 --supports--> v0a\n- v08 --supports--> v0f\n- v01 --refutes--> v0d\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8fc67ea5cae96a4f15a40d79f4a4e424e4dd0ee2fdd20f56b146a8495a212aa4","schema":"livegraph.boolean-propagation.v1","spec_hash":"3409fed8ec3901ea96bab13f502906d2fd8bff5d287a99516ffd09a26d98158d","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-59eaa7c8e90d6c8cb724","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v05\n- v0c --refutes--> v0b\n- v05 --refutes--> v0f\n- v03 --supports--> v01\n- v0d --refutes--> v10\n- v05 --refutes--> v09\n- v02 --supports--> v04\n- v10 --supports--> v07\n- v02 --refutes--> v08\n- v0c --supports--> v06\n- v0e --refutes--> v02\n- v00 --refutes--> v0d\n- v0c --supports--> v0a\n- v08 --refutes--> v0c\n- v03 --supports--> v0e\n- v06 --refutes--> v00\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fb4f8c5972fd8017564025c423acab93904fdf096db29a06b3f4299bc1f8625a","schema":"livegraph.boolean-propagation.v1","spec_hash":"59eaa7c8e90d6c8cb724f4412bec56ca98b994519d1490a8e3b63f6d04d232f5","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-5fd03e1447059492b3db","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v0b\n- v09 --refutes--> v00\n- v06 --refutes--> v0a\n- v08 --refutes--> v05\n- v07 --refutes--> v0d\n- v10 --supports--> v04\n- v04 --supports--> v01\n- v02 --refutes--> v09\n- v10 --refutes--> v03\n- v0d --supports--> v10\n- v05 --refutes--> v02\n- v00 --supports--> v07\n- v07 --refutes--> v0e\n- v04 --supports--> v0c\n- v0f --refutes--> v08\n- v06 --supports--> v0f\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"021457e43342c486a53dad1d66d4f8486ed0c5a759ee45141ff5599eb1a66312","schema":"livegraph.boolean-propagation.v1","spec_hash":"5fd03e1447059492b3db289eaac087fb1c0d59f3dcb5cb43dd80f585acb0c9a6","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-bfbf5216f91f02571952","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v04\n- v0f --refutes--> v03\n- v00 --refutes--> v01\n- v07 --refutes--> v0c\n- v09 --refutes--> v0d\n- v04 --refutes--> v02\n- v02 --refutes--> v08\n- v07 --supports--> v0a\n- v00 --supports--> v0b\n- v00 --refutes--> v0e\n- v08 --refutes--> v00\n- v0c --supports--> v09\n- v0f --refutes--> v05\n- v03 --supports--> v07\n- v01 --supports--> v0f\n- v00 --supports--> v10\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6a461c8a56b5d0266925d0bb332bcf0b1124cb72c1430fedb4159c06616b5192","schema":"livegraph.boolean-propagation.v1","spec_hash":"bfbf5216f91f025719525744524da39e5eeb2991c8357b503ab0cefba229c7bc","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-59ba2292555144266d7b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v0d\n- v0e --refutes--> v07\n- v10 --supports--> v01\n- v01 --supports--> v0a\n- v02 --refutes--> v10\n- v0a --supports--> v08\n- v0c --supports--> v09\n- v05 --supports--> v0e\n- v08 --refutes--> v05\n- v01 --supports--> v0f\n- v0c --refutes--> v06\n- v05 --refutes--> v03\n- v07 --supports--> v0c\n- v00 --refutes--> v04\n- v09 --refutes--> v00\n- v0a --supports--> v0b\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"01b45b2132e416025677b76aebc8f7fb211f99bb5627c5c6bd6120aefa36f04b","schema":"livegraph.boolean-propagation.v1","spec_hash":"59ba2292555144266d7bcb8c5221c457a40a83bd7317cc8eb1a927f13c4ed7f6","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-da65f43037c7b590a737","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --supports--> v04\n- v0b --supports--> v0d\n- v05 --supports--> v0f\n- v03 --supports--> v07\n- v0e --refutes--> v03\n- v02 --refutes--> v00\n- v10 --supports--> v0c\n- v0b --refutes--> v10\n- v04 --supports--> v05\n- v09 --supports--> v08\n- v10 --refutes--> v06\n- v0e --refutes--> v0a\n- v0f --supports--> v0e\n- v07 --supports--> v09\n- v08 --supports--> v02\n- v0f --supports--> v01\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b211dac0ad0b53f2e95ebf4487a784c956864e37e9760bb94b05ad986faa58bb","schema":"livegraph.boolean-propagation.v1","spec_hash":"da65f43037c7b590a737fab01fa6fdfc8cd76dd7f9f746eb96c443871cf65cdd","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-717fa0e08999b9b1633f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v0e\n- v01 --refutes--> v03\n- v10 --supports--> v08\n- v01 --supports--> v04\n- v06 --supports--> v05\n- v02 --supports--> v0a\n- v05 --refutes--> v0c\n- v07 --supports--> v0f\n- v0f --refutes--> v01\n- v00 --supports--> v02\n- v01 --refutes--> v0d\n- v05 --refutes--> v00\n- v02 --supports--> v09\n- v0e --supports--> v10\n- v0b --refutes--> v07\n- v10 --refutes--> v0b\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"573a6de14d67622b6b532d2fcf0e9f191010933f3d89f58063a1e98a952427b3","schema":"livegraph.boolean-propagation.v1","spec_hash":"717fa0e08999b9b1633fba2101088b02acd3f18eae33cd79abe7fb9f019e81c6","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-c68486f01ed1fc3285c1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v0b\n- v00 --supports--> v03\n- v0a --supports--> v0e\n- v0d --supports--> v05\n- v0f --refutes--> v09\n- v09 --refutes--> v0c\n- v06 --refutes--> v01\n- v05 --refutes--> v08\n- v01 --supports--> v04\n- v04 --refutes--> v0d\n- v03 --refutes--> v06\n- v0a --supports--> v10\n- v0e --refutes--> v00\n- v00 --supports--> v07\n- v05 --supports--> v0f\n- v00 --supports--> v02\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1a19546d72f60d7bbbf5167567ca14fd55ca69cd73dd3bbf7aa0ea3e96da6dbc","schema":"livegraph.boolean-propagation.v1","spec_hash":"c68486f01ed1fc3285c11961f66b291ad0a2b31aeada8d573458e81f0f4f6c17","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-53834e0a52430400bbcf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v0f\n- v0f --refutes--> v04\n- v0d --refutes--> v07\n- v0f --refutes--> v03\n- v02 --supports--> v0e\n- v10 --refutes--> v0b\n- v07 --supports--> v01\n- v0f --supports--> v0d\n- v0e --supports--> v0a\n- v0b --refutes--> v0c\n- v00 --refutes--> v05\n- v0c --supports--> v06\n- v07 --refutes--> v02\n- v05 --supports--> v09\n- v00 --supports--> v08\n- v0a --refutes--> v00\n\n[VERIFIED ROOT]\nv10 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6fc9e7c737eaef8d62f84c61a3024d28136a69ef0b41b3c7481d0908b8259173","schema":"livegraph.boolean-propagation.v1","spec_hash":"53834e0a52430400bbcff68076ab1b0e21f8c96227a9628fb348ef597b303ee7","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-ae29863e4b4b97451270","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v01\n- v0c --refutes--> v08\n- v07 --supports--> v03\n- v09 --refutes--> v0d\n- v07 --supports--> v02\n- v00 --refutes--> v07\n- v09 --supports--> v10\n- v0a --supports--> v0e\n- v04 --refutes--> v0c\n- v0d --refutes--> v00\n- v0c --supports--> v0f\n- v06 --supports--> v09\n- v0f --supports--> v0b\n- v07 --refutes--> v05\n- v03 --supports--> v04\n- v0b --refutes--> v0a\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9ac55b9db257dc6be76e9ce2eaf9582464a24eec774e4f41e88034e87a0c0aaf","schema":"livegraph.boolean-propagation.v1","spec_hash":"ae29863e4b4b9745127094fddda8ad321d5438dd60773bf1827e89790586365e","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-897f30fb1117190d3526","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v02\n- v10 --refutes--> v0c\n- v0f --refutes--> v06\n- v03 --refutes--> v0f\n- v05 --supports--> v03\n- v0e --refutes--> v0a\n- v04 --refutes--> v00\n- v09 --supports--> v04\n- v0c --supports--> v01\n- v0e --supports--> v10\n- v08 --refutes--> v0b\n- v06 --supports--> v0e\n- v0e --refutes--> v07\n- v00 --refutes--> v05\n- v0c --refutes--> v08\n- v09 --refutes--> v0d\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7e1d3b2c68a03f257fb13097e535db0642d6dea77d13e9abbdaca08d979405d3","schema":"livegraph.boolean-propagation.v1","spec_hash":"897f30fb1117190d352620eba3595e49d979cd50b0b4c13f00bda3d0f668df2f","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-a9d961412002ee033171","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v10\n- v05 --refutes--> v07\n- v0b --refutes--> v0e\n- v02 --supports--> v0c\n- v02 --refutes--> v06\n- v10 --supports--> v04\n- v0e --refutes--> v08\n- v07 --supports--> v0a\n- v01 --refutes--> v05\n- v01 --supports--> v0d\n- v01 --refutes--> v00\n- v04 --refutes--> v03\n- v03 --refutes--> v0b\n- v0a --supports--> v02\n- v03 --supports--> v0f\n- v01 --supports--> v09\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9ced4154b9676a44e248c23b9246d9b386d020a6a0a0bf866182475351c224e8","schema":"livegraph.boolean-propagation.v1","spec_hash":"a9d961412002ee0331710337d374014a8bc784cf22a4a3c67d30300ec35b6b8f","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-b4ee8f374f0b83a89d09","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v10\n- v10 --refutes--> v0a\n- v07 --refutes--> v05\n- v01 --supports--> v04\n- v07 --supports--> v0b\n- v10 --refutes--> v03\n- v10 --refutes--> v0f\n- v01 --refutes--> v0c\n- v0d --supports--> v00\n- v00 --refutes--> v02\n- v09 --refutes--> v01\n- v03 --supports--> v0e\n- v00 --refutes--> v08\n- v0e --refutes--> v0d\n- v05 --supports--> v09\n- v08 --supports--> v07\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c66663e788f4e5a386199a106e3a9f936a8ea583c457d556afc477540677f0da","schema":"livegraph.boolean-propagation.v1","spec_hash":"b4ee8f374f0b83a89d09ea3afdd7f6213cc086ceb83507693bf666fb0dd42b2b","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-b8551a5656e2eb36607d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v0c\n- v04 --supports--> v01\n- v0d --supports--> v06\n- v04 --supports--> v0b\n- v09 --refutes--> v0f\n- v06 --refutes--> v04\n- v03 --supports--> v0a\n- v0b --refutes--> v07\n- v0b --supports--> v10\n- v03 --refutes--> v0d\n- v0c --refutes--> v08\n- v0d --refutes--> v02\n- v0f --supports--> v0e\n- v08 --refutes--> v05\n- v05 --supports--> v03\n- v0f --supports--> v00\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"32243f11e7bd3e81d213cbaee3db62931478ea51a0bd1509c183adac21d53e17","schema":"livegraph.boolean-propagation.v1","spec_hash":"b8551a5656e2eb36607d07bb94e66df4c6afaec3c8347c8261ede364e88cfc1f","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-aa46b8bb041d03c22838","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v0c\n- v00 --supports--> v10\n- v07 --refutes--> v09\n- v02 --refutes--> v0e\n- v0f --supports--> v03\n- v0e --supports--> v0d\n- v0a --supports--> v0b\n- v00 --supports--> v01\n- v0c --refutes--> v00\n- v00 --supports--> v04\n- v08 --refutes--> v06\n- v05 --supports--> v07\n- v0a --supports--> v05\n- v07 --refutes--> v02\n- v03 --refutes--> v08\n- v01 --refutes--> v0f\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9d3d3014e3f488b7af4c7c59c3bfc7a4fb504d23f57b582521f4ae0ba95f9ff4","schema":"livegraph.boolean-propagation.v1","spec_hash":"aa46b8bb041d03c22838d60aaca97c4a90a0d5827c4af3fdb19a1f80ab9d8045","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-b9f1e41270aaac3bfb8e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v0f\n- v0d --refutes--> v10\n- v0d --supports--> v01\n- v0a --refutes--> v04\n- v10 --supports--> v08\n- v0f --supports--> v07\n- v02 --supports--> v06\n- v05 --supports--> v0d\n- v0a --supports--> v02\n- v0f --refutes--> v0e\n- v0e --supports--> v0b\n- v0b --supports--> v0c\n- v06 --supports--> v05\n- v10 --refutes--> v00\n- v10 --refutes--> v09\n- v0c --supports--> v03\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"61a34103f5e9fef3cfd432c87a26b62f62a36552e2870e8ab80bb412b0276613","schema":"livegraph.boolean-propagation.v1","spec_hash":"b9f1e41270aaac3bfb8e127a0fc18cfa34e0c93e55c5411aecd2d6ccd8693b99","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-09aee780bca81ef01f28","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v09\n- v00 --refutes--> v07\n- v05 --supports--> v04\n- v03 --refutes--> v06\n- v07 --refutes--> v0f\n- v06 --supports--> v0e\n- v07 --supports--> v08\n- v00 --supports--> v02\n- v0d --refutes--> v00\n- v0c --refutes--> v10\n- v10 --supports--> v03\n- v09 --refutes--> v0c\n- v0e --supports--> v0d\n- v0f --supports--> v01\n- v0e --refutes--> v0b\n- v10 --refutes--> v0a\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ebf409fa7735e6ea0208998ab840dc75a60f079ccdd91c04a2e374c8bab5b10d","schema":"livegraph.boolean-propagation.v1","spec_hash":"09aee780bca81ef01f28344eff83032f15eebbbf8ebadde59f8e0ba3f8d18315","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-62f0c87e7a64c2400142","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v0a\n- v10 --refutes--> v00\n- v04 --supports--> v0d\n- v09 --refutes--> v10\n- v0d --supports--> v09\n- v0b --refutes--> v01\n- v0d --refutes--> v05\n- v10 --refutes--> v06\n- v01 --supports--> v0f\n- v0f --refutes--> v02\n- v07 --supports--> v0c\n- v0c --refutes--> v0b\n- v08 --supports--> v04\n- v10 --refutes--> v03\n- v06 --refutes--> v07\n- v08 --supports--> v0e\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"18b6a589285400b0bf0ddc4be3efa932cc63b1805d104e37eb97dd9c43068563","schema":"livegraph.boolean-propagation.v1","spec_hash":"62f0c87e7a64c24001427cf43790918c92d2edd32533b3744b9d28f8f70276f9","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-09be27d2f89829b6859c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v10\n- v07 --refutes--> v00\n- v00 --refutes--> v05\n- v02 --refutes--> v06\n- v0f --supports--> v0c\n- v06 --refutes--> v0a\n- v04 --refutes--> v0b\n- v0f --supports--> v08\n- v00 --supports--> v09\n- v06 --supports--> v07\n- v0c --supports--> v0d\n- v09 --supports--> v04\n- v01 --refutes--> v02\n- v0e --refutes--> v03\n- v0b --supports--> v0e\n- v0c --supports--> v01\n\n[VERIFIED ROOT]\nv0f = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"97bc42c02e9986d6d0617a4caece4eddcc9d3f1dfdf7f96e60ed111ad6d38b19","schema":"livegraph.boolean-propagation.v1","spec_hash":"09be27d2f89829b6859cd5aa9d9c1cf530bace1d32ecb9893b01e8299a6f14ba","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-75fa93e55640618b5899","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v05\n- v0f --refutes--> v07\n- v09 --supports--> v04\n- v0b --supports--> v10\n- v06 --refutes--> v00\n- v02 --refutes--> v06\n- v07 --supports--> v01\n- v0b --supports--> v02\n- v08 --refutes--> v09\n- v0d --refutes--> v0a\n- v06 --refutes--> v0f\n- v04 --supports--> v0b\n- v01 --refutes--> v0d\n- v07 --refutes--> v0c\n- v0b --refutes--> v0e\n- v03 --refutes--> v08\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5b55032085ee850b7af4c5bd5dfa75d15a8b684516c393f3cfb7e065cfa08d2a","schema":"livegraph.boolean-propagation.v1","spec_hash":"75fa93e55640618b58994abf064d66de29983bdb8a052e08f766e8edbd8c8d90","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-d70b59ab6604e44f77f9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v05\n- v0d --refutes--> v04\n- v0c --supports--> v06\n- v01 --refutes--> v0f\n- v0b --supports--> v0d\n- v0d --refutes--> v0e\n- v06 --supports--> v10\n- v04 --refutes--> v01\n- v07 --supports--> v09\n- v02 --refutes--> v03\n- v10 --refutes--> v07\n- v00 --supports--> v02\n- v0f --supports--> v0c\n- v07 --refutes--> v00\n- v00 --supports--> v08\n- v02 --refutes--> v0a\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1065871f25c93afc7f7f34d86d9a040114fd1961ea8064171c46890982bf0299","schema":"livegraph.boolean-propagation.v1","spec_hash":"d70b59ab6604e44f77f976cb2e3e8be7611edcfa52dfe58a337386fcead69712","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-b0efb3160eaf081f9f54","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v0f\n- v0a --supports--> v05\n- v08 --supports--> v07\n- v0e --supports--> v0a\n- v0b --supports--> v10\n- v06 --refutes--> v02\n- v0d --refutes--> v0c\n- v0e --refutes--> v00\n- v02 --refutes--> v01\n- v0b --supports--> v03\n- v0c --supports--> v06\n- v09 --supports--> v04\n- v02 --supports--> v0e\n- v05 --refutes--> v09\n- v09 --supports--> v0b\n- v10 --refutes--> v08\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f1a5d579c85af9fe3d4e7c66da20a1bb81d3bbf8d73977e9816f8ed624a6d311","schema":"livegraph.boolean-propagation.v1","spec_hash":"b0efb3160eaf081f9f54312212def313722eb35c9df82477f6e1ac66ace6e99a","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-d22cf0906ec594c99939","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v01\n- v06 --supports--> v07\n- v09 --refutes--> v10\n- v04 --supports--> v0d\n- v03 --supports--> v02\n- v0a --refutes--> v0b\n- v06 --supports--> v08\n- v0b --supports--> v00\n- v0b --supports--> v0c\n- v05 --supports--> v04\n- v0a --supports--> v0f\n- v10 --refutes--> v06\n- v02 --supports--> v0e\n- v0c --supports--> v09\n- v07 --supports--> v05\n- v04 --supports--> v03\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"88c4a2c49a20caed369835169e99c1e51dcf3da5aa5080183883d5fdbff00cc6","schema":"livegraph.boolean-propagation.v1","spec_hash":"d22cf0906ec594c999396fbdd9356d19e3d8b32ea3005415539d1af840620fc0","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-876ecb1c51932f6c0012","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v0f\n- v01 --supports--> v03\n- v05 --supports--> v0c\n- v00 --refutes--> v0d\n- v02 --refutes--> v0b\n- v0f --supports--> v04\n- v06 --supports--> v01\n- v0f --refutes--> v09\n- v0d --refutes--> v0a\n- v0c --supports--> v06\n- v04 --supports--> v05\n- v00 --supports--> v08\n- v10 --supports--> v02\n- v02 --refutes--> v07\n- v0c --supports--> v0e\n- v03 --supports--> v10\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5e8054d00190797755ed4f1a9dd7e181cf0ce02ee95746e61f2478d94caa02bf","schema":"livegraph.boolean-propagation.v1","spec_hash":"876ecb1c51932f6c001247764dc2975101b30298f2eaad954e073b8c8b84e102","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-005e57f513b12f0cd5fa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v10\n- v04 --refutes--> v06\n- v0c --refutes--> v02\n- v08 --refutes--> v01\n- v08 --supports--> v0a\n- v08 --refutes--> v03\n- v0d --refutes--> v07\n- v0f --refutes--> v0e\n- v07 --refutes--> v09\n- v05 --refutes--> v08\n- v0b --supports--> v00\n- v0c --refutes--> v0b\n- v00 --refutes--> v0f\n- v06 --refutes--> v0d\n- v07 --refutes--> v0c\n- v03 --refutes--> v04\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1bc245e9891a09bc56871ff1ae5d5224b5f0a6ba2ba4da67cfe54186924db507","schema":"livegraph.boolean-propagation.v1","spec_hash":"005e57f513b12f0cd5fa22e15ecc03b1c8629fd2b0d0de772bf371594c96692d","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-167584c1ca42ca28bdbe","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v05\n- v10 --refutes--> v00\n- v03 --refutes--> v0f\n- v0b --refutes--> v04\n- v0e --supports--> v01\n- v02 --refutes--> v0e\n- v0f --supports--> v07\n- v00 --supports--> v03\n- v0c --refutes--> v09\n- v10 --refutes--> v0d\n- v07 --supports--> v0c\n- v04 --refutes--> v0a\n- v0a --refutes--> v02\n- v0c --supports--> v0b\n- v0e --supports--> v06\n- v10 --supports--> v08\n\n[VERIFIED ROOT]\nv10 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"04abe93aa0107a822d1b802c80ffc3cab1b675aa0146f99e6a69e561a0d93a04","schema":"livegraph.boolean-propagation.v1","spec_hash":"167584c1ca42ca28bdbec74e7e58471ad0e9fb0118fdef661d2b6252fc662d21","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-d1902c98c1a500783f3f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v0a\n- v0d --refutes--> v07\n- v02 --supports--> v04\n- v07 --supports--> v09\n- v0f --refutes--> v0c\n- v0a --refutes--> v0e\n- v08 --refutes--> v06\n- v0f --supports--> v01\n- v03 --supports--> v08\n- v08 --supports--> v0d\n- v0c --refutes--> v0b\n- v0a --supports--> v03\n- v09 --refutes--> v0f\n- v00 --refutes--> v10\n- v00 --refutes--> v05\n- v0b --refutes--> v00\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fe24d9ea49c072c377bea9682c62bbe5f51020be44ab2d57a3081c899a310c2d","schema":"livegraph.boolean-propagation.v1","spec_hash":"d1902c98c1a500783f3f94cac60fed05002461858b7afb98bee0512578d1357d","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-3e415ba6f9e36cad6cdf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v00\n- v0f --refutes--> v0d\n- v10 --supports--> v09\n- v10 --refutes--> v02\n- v05 --refutes--> v03\n- v05 --refutes--> v0b\n- v00 --refutes--> v04\n- v04 --supports--> v0f\n- v08 --refutes--> v01\n- v0c --refutes--> v05\n- v0d --supports--> v10\n- v05 --supports--> v0a\n- v02 --refutes--> v08\n- v08 --refutes--> v0c\n- v07 --refutes--> v06\n- v07 --supports--> v0e\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d961cf6b03438d7768dc053cb342145718d25646e60b67e484bafbb3ad10d7f3","schema":"livegraph.boolean-propagation.v1","spec_hash":"3e415ba6f9e36cad6cdfef075e700e01eff0379722efc42d2b40cfcf169f6d19","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-60b36a22c07af8cbd3f6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v0e\n- v00 --refutes--> v0a\n- v0f --refutes--> v09\n- v07 --supports--> v0f\n- v10 --supports--> v06\n- v01 --supports--> v0b\n- v0a --refutes--> v10\n- v06 --supports--> v0c\n- v01 --supports--> v02\n- v05 --supports--> v01\n- v0e --refutes--> v0d\n- v10 --refutes--> v04\n- v07 --supports--> v08\n- v06 --supports--> v03\n- v0c --refutes--> v05\n- v0e --refutes--> v07\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"19c531fca3f95ae9351338864869229bc8679c76a77b80a0b3078348b1ce5f18","schema":"livegraph.boolean-propagation.v1","spec_hash":"60b36a22c07af8cbd3f6cdb383a559122af52c52185dfdeac85f0a90db83a786","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-4f4a877b33b3a4c8d031","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v0d\n- v0a --refutes--> v08\n- v05 --supports--> v06\n- v10 --supports--> v02\n- v06 --refutes--> v01\n- v00 --supports--> v07\n- v08 --refutes--> v0c\n- v07 --refutes--> v04\n- v10 --refutes--> v0f\n- v05 --supports--> v0e\n- v09 --refutes--> v00\n- v00 --refutes--> v03\n- v0c --supports--> v0b\n- v0c --refutes--> v10\n- v04 --supports--> v0a\n- v02 --refutes--> v05\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"97a23c2a9f80d61c7ce5f11d3ab0e959ab77e8dc02c1dbc7011939a9953a020c","schema":"livegraph.boolean-propagation.v1","spec_hash":"4f4a877b33b3a4c8d031a7b61d518fd64a2729d978c7582f4546a5e7006ab90b","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-3ce6a77c8f0e65d0e92e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v04\n- v06 --supports--> v09\n- v06 --refutes--> v0e\n- v02 --supports--> v0b\n- v10 --refutes--> v01\n- v0c --supports--> v03\n- v08 --refutes--> v02\n- v06 --refutes--> v0f\n- v05 --refutes--> v06\n- v03 --supports--> v00\n- v01 --refutes--> v0a\n- v0b --supports--> v0c\n- v06 --refutes--> v10\n- v0a --refutes--> v0d\n- v0c --refutes--> v07\n- v0d --refutes--> v08\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"127024bbeab1ac163c00c3dbc5ce8689fe524b59db174f6b2f1b672cc9051b03","schema":"livegraph.boolean-propagation.v1","spec_hash":"3ce6a77c8f0e65d0e92eecb639b98baef8c6d76a2594ac41652b2758529686a0","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-016d7874bce094b700eb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v02\n- v0b --refutes--> v09\n- v0f --supports--> v06\n- v0e --refutes--> v08\n- v0c --supports--> v04\n- v0d --supports--> v0c\n- v02 --supports--> v05\n- v02 --supports--> v01\n- v02 --supports--> v0d\n- v0a --supports--> v0e\n- v0c --refutes--> v07\n- v0e --supports--> v03\n- v0a --supports--> v10\n- v04 --supports--> v0f\n- v06 --supports--> v0a\n- v03 --supports--> v0b\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"42ea8419a54ccb32a03a26fb4e031f698707492e5196701b9b6590540848ffc8","schema":"livegraph.boolean-propagation.v1","spec_hash":"016d7874bce094b700ebce55835fc313718a980a19d2ef0483f6d64e78436fbc","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-5719d460118c1371c117","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v0b\n- v09 --supports--> v0c\n- v0f --refutes--> v00\n- v09 --refutes--> v0f\n- v02 --refutes--> v05\n- v0e --supports--> v02\n- v03 --supports--> v01\n- v07 --supports--> v0e\n- v05 --refutes--> v0d\n- v0b --refutes--> v10\n- v0b --refutes--> v08\n- v0a --supports--> v04\n- v01 --refutes--> v07\n- v0e --refutes--> v06\n- v0f --supports--> v0a\n- v0b --refutes--> v09\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"15bef2489f709e1512d07c0dc3dfaded963ed1bd4a139f9225b037ab491d3953","schema":"livegraph.boolean-propagation.v1","spec_hash":"5719d460118c1371c11726afaceb0c10e9ad1cb3f85e857897d47493556a0f7e","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-2a3391b636d746ce14ed","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v06\n- v06 --refutes--> v09\n- v0e --refutes--> v0f\n- v07 --supports--> v03\n- v0b --supports--> v07\n- v10 --supports--> v04\n- v01 --refutes--> v0a\n- v03 --refutes--> v0e\n- v03 --refutes--> v02\n- v04 --supports--> v01\n- v06 --supports--> v05\n- v07 --supports--> v08\n- v00 --supports--> v10\n- v04 --refutes--> v0c\n- v06 --supports--> v0b\n- v0f --refutes--> v0d\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2519fda91d0fc3843afa025bb6d7ebd63dfd70c864eecab78e0e85aac3d6473a","schema":"livegraph.boolean-propagation.v1","spec_hash":"2a3391b636d746ce14ed85e16c95d49ca247c32677eaedef55cd4c32e2b1b565","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-545fbbeadeb00b078753","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v06\n- v02 --refutes--> v07\n- v08 --refutes--> v10\n- v09 --refutes--> v01\n- v09 --supports--> v0a\n- v10 --refutes--> v0b\n- v01 --supports--> v0e\n- v09 --refutes--> v0c\n- v0d --supports--> v05\n- v06 --refutes--> v03\n- v10 --refutes--> v04\n- v0d --refutes--> v00\n- v0e --refutes--> v0d\n- v02 --supports--> v0f\n- v04 --refutes--> v02\n- v0f --supports--> v09\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b47a474a5f10f271659ab5c81cfa0f515b057ba95ef756f04961b5fcbf297231","schema":"livegraph.boolean-propagation.v1","spec_hash":"545fbbeadeb00b078753528ca80c8dff7410432d99fdf65e2a2021794c5ed86d","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-184d523437507220011d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v04\n- v01 --supports--> v0e\n- v09 --supports--> v01\n- v0a --supports--> v10\n- v0e --supports--> v0a\n- v10 --supports--> v0c\n- v03 --supports--> v09\n- v01 --refutes--> v02\n- v07 --supports--> v06\n- v0e --refutes--> v0f\n- v0c --refutes--> v07\n- v01 --refutes--> v05\n- v0b --supports--> v03\n- v01 --supports--> v0d\n- v06 --supports--> v00\n- v0c --refutes--> v08\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2b2eacfef1c90e185ebc7cc5fedafedec020814cfa31e783f3f6b4f658ee76c1","schema":"livegraph.boolean-propagation.v1","spec_hash":"184d523437507220011d710a3969ea4a00f115cad209d47d51f2428f9d40a1de","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-0f15edd46a0ad0fa9c90","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v05\n- v01 --refutes--> v02\n- v0c --refutes--> v0e\n- v08 --refutes--> v04\n- v00 --supports--> v07\n- v0e --supports--> v00\n- v0d --supports--> v01\n- v0b --supports--> v06\n- v0d --supports--> v10\n- v0e --supports--> v09\n- v0a --supports--> v08\n- v07 --refutes--> v0f\n- v02 --supports--> v0c\n- v02 --refutes--> v03\n- v04 --supports--> v0d\n- v0b --supports--> v0a\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4dbe9828014a52ab711086bd05f30048142655f10225086f696724ba0db692f5","schema":"livegraph.boolean-propagation.v1","spec_hash":"0f15edd46a0ad0fa9c9022de0e58f0c96ff75d2a50b78950dc70fd31d243be68","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":11,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-bff724b8107fb388741a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v0d\n- v07 --refutes--> v01\n- v06 --supports--> v08\n- v08 --supports--> v00\n- v02 --refutes--> v05\n- v08 --supports--> v10\n- v03 --supports--> v0a\n- v03 --supports--> v0b\n- v05 --supports--> v03\n- v0d --supports--> v0c\n- v0c --supports--> v06\n- v03 --supports--> v04\n- v08 --refutes--> v0f\n- v0b --supports--> v07\n- v06 --refutes--> v09\n- v00 --supports--> v02\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"894e9c734676ed171f4bc9798224a48e3c92fd854c6bd24d7d45ef8af268ad70","schema":"livegraph.boolean-propagation.v1","spec_hash":"bff724b8107fb388741a3294d6249140ac29175932a10b6e962c0584e84be583","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"}],\"hop\":11,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-b1f81d8d871815836a13","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v0e\n- v01 --supports--> v06\n- v09 --refutes--> v08\n- v0c --supports--> v0a\n- v0c --refutes--> v0b\n- v10 --refutes--> v0c\n- v0e --supports--> v05\n- v07 --refutes--> v03\n- v02 --supports--> v09\n- v07 --refutes--> v00\n- v00 --supports--> v0f\n- v0b --supports--> v02\n- v00 --supports--> v10\n- v05 --supports--> v01\n- v01 --refutes--> v0d\n- v04 --supports--> v07\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"aa79fa9aee563841292598d2b8c29da7166353c050de9694583e551b3cd205c4","schema":"livegraph.boolean-propagation.v1","spec_hash":"b1f81d8d871815836a139be27774d0bf22c84511a973959523b3d3d8e7b9aa01","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-5028e98f7121c823f087","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v0c\n- v01 --supports--> v00\n- v0e --supports--> v06\n- v0c --supports--> v08\n- v04 --supports--> v03\n- v06 --supports--> v02\n- v01 --supports--> v05\n- v06 --supports--> v09\n- v0d --supports--> v10\n- v09 --supports--> v0b\n- v0b --refutes--> v0f\n- v08 --refutes--> v07\n- v0b --supports--> v04\n- v0a --supports--> v0e\n- v0d --supports--> v01\n- v08 --refutes--> v0a\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8bb312d016b9111318f7678afd28d04177c1ca4668671ebf8ad0a08949abd781","schema":"livegraph.boolean-propagation.v1","spec_hash":"5028e98f7121c823f08703a7f8059362e266454325d8b288b6eae48a62e613fb","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":11,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-ae2e6b44a9c3717f35d1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v07\n- v05 --supports--> v0d\n- v05 --supports--> v10\n- v0e --refutes--> v05\n- v05 --supports--> v0f\n- v09 --refutes--> v0c\n- v01 --refutes--> v08\n- v0f --supports--> v0a\n- v08 --supports--> v06\n- v08 --supports--> v04\n- v00 --refutes--> v03\n- v0c --refutes--> v02\n- v03 --refutes--> v09\n- v05 --supports--> v0b\n- v06 --supports--> v00\n- v0a --refutes--> v01\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e2ed37861a331350ed86878b10bb2bbb2cc979399edd3b55b53c9b3a30594f18","schema":"livegraph.boolean-propagation.v1","spec_hash":"ae2e6b44a9c3717f35d1be17a15f48e1a393052e8aa7d3aad92e247ce4e20c9d","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-b59bbdea8d045dfa6417","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v0c\n- v03 --refutes--> v0a\n- v0f --supports--> v06\n- v02 --supports--> v04\n- v0f --supports--> v0b\n- v05 --refutes--> v0e\n- v09 --refutes--> v03\n- v0a --supports--> v07\n- v04 --refutes--> v08\n- v07 --supports--> v00\n- v02 --refutes--> v01\n- v07 --supports--> v10\n- v0f --refutes--> v05\n- v0e --supports--> v09\n- v0c --supports--> v0d\n- v0c --supports--> v0f\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"adbfd2f63b04a0e0bbde0f8b744c9a343602f53ce9286fcac43b8b8e903e4d3d","schema":"livegraph.boolean-propagation.v1","spec_hash":"b59bbdea8d045dfa6417fdbf09a3b1996219a491a816d0c06da2e73221667a51","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-6c4cbc9249b91295faf9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v0b\n- v0d --refutes--> v0e\n- v0d --supports--> v02\n- v01 --refutes--> v03\n- v04 --refutes--> v10\n- v0b --refutes--> v04\n- v04 --refutes--> v06\n- v08 --refutes--> v07\n- v03 --supports--> v0c\n- v0d --refutes--> v09\n- v06 --supports--> v01\n- v06 --supports--> v0a\n- v08 --refutes--> v0f\n- v0f --refutes--> v05\n- v0c --supports--> v0d\n- v00 --refutes--> v08\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9e12553a8f037fd00ff85d45d772afd915c5dcaa71ad1fa8ab87b889086e4ef9","schema":"livegraph.boolean-propagation.v1","spec_hash":"6c4cbc9249b91295faf92ba531e016ddcbc47711e12b83156375fb6d3541690c","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-0c87d3723a8ef6e1e53b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --supports--> v05\n- v07 --refutes--> v0c\n- v04 --supports--> v06\n- v05 --supports--> v0f\n- v04 --supports--> v01\n- v03 --supports--> v00\n- v06 --supports--> v0d\n- v00 --supports--> v0e\n- v0e --refutes--> v0b\n- v0d --refutes--> v07\n- v0a --supports--> v09\n- v0a --refutes--> v04\n- v0b --supports--> v0a\n- v00 --supports--> v08\n- v10 --supports--> v02\n- v07 --refutes--> v10\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f35705266df64e5e2ac21d2711d9fbabb93cf1f913b0c4e89c86f74dea0dbd6c","schema":"livegraph.boolean-propagation.v1","spec_hash":"0c87d3723a8ef6e1e53b6dd064eab1ae7ae41c8aaf9fcf606f3209d00d039802","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-5c67f3f0e1b2af5eccdb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v0f\n- v07 --supports--> v10\n- v10 --refutes--> v03\n- v00 --supports--> v09\n- v03 --refutes--> v05\n- v0f --refutes--> v0e\n- v01 --refutes--> v0d\n- v05 --refutes--> v0c\n- v00 --supports--> v0a\n- v09 --refutes--> v07\n- v10 --supports--> v08\n- v04 --supports--> v01\n- v05 --supports--> v06\n- v10 --supports--> v02\n- v0e --refutes--> v00\n- v05 --supports--> v0b\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cb78978604e7f97d7c862a258ebd8d65321a3bbd015665043a951a42e6ca565d","schema":"livegraph.boolean-propagation.v1","spec_hash":"5c67f3f0e1b2af5eccdbafd10a17a04e42c14ba264cb8f7440ae96fdfeea587a","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-e098575e9775790c6d23","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v02\n- v0d --supports--> v0a\n- v06 --supports--> v04\n- v0c --refutes--> v0f\n- v08 --refutes--> v0d\n- v05 --supports--> v0e\n- v02 --supports--> v10\n- v09 --supports--> v0c\n- v05 --supports--> v07\n- v06 --supports--> v03\n- v09 --supports--> v0b\n- v0f --refutes--> v06\n- v01 --refutes--> v09\n- v0a --refutes--> v00\n- v02 --refutes--> v01\n- v00 --supports--> v05\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"25b40255d7f16bf2d12e9a0d82993f9b8729958cd430a608aa2692db123cab6e","schema":"livegraph.boolean-propagation.v1","spec_hash":"e098575e9775790c6d23600a1e8ad00ceb2a812b701ac68b0dd2dc20fc51e35d","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-38cb2c0093c284c26587","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v0c\n- v05 --refutes--> v08\n- v03 --refutes--> v0d\n- v07 --supports--> v0b\n- v02 --supports--> v01\n- v01 --supports--> v05\n- v0e --refutes--> v02\n- v0b --supports--> v0f\n- v0a --supports--> v0e\n- v0a --supports--> v10\n- v0d --supports--> v00\n- v0e --supports--> v06\n- v08 --refutes--> v03\n- v0f --supports--> v0a\n- v0e --refutes--> v09\n- v0d --refutes--> v04\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6ac5951e8a967a91af8f5ca8db48ecd43fcf7da7ed035220a101bc9b511855ed","schema":"livegraph.boolean-propagation.v1","spec_hash":"38cb2c0093c284c26587d64c616d70167482c29121e98b8d636fa1cc4c0bfbc5","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-a5d84685e91126379228","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v03\n- v0f --refutes--> v0e\n- v03 --supports--> v0a\n- v03 --supports--> v05\n- v0a --supports--> v01\n- v0e --refutes--> v10\n- v0c --refutes--> v0f\n- v09 --supports--> v0c\n- v0b --supports--> v07\n- v07 --supports--> v02\n- v06 --supports--> v0b\n- v0c --supports--> v08\n- v07 --refutes--> v00\n- v04 --supports--> v09\n- v0e --refutes--> v06\n- v09 --refutes--> v0d\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"912fdc1e63d394f3b6fdbf3a0adbf8f0244b906e52a9a7ece44b9b32719d5191","schema":"livegraph.boolean-propagation.v1","spec_hash":"a5d84685e91126379228642c9618cbcbfbe85fae9b24d84ee9bd4540bb099188","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-4bb6a77ab8485f445ca7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v0a\n- v0e --refutes--> v0c\n- v10 --refutes--> v05\n- v03 --refutes--> v07\n- v06 --supports--> v00\n- v0a --refutes--> v0d\n- v06 --supports--> v08\n- v0c --refutes--> v10\n- v07 --supports--> v0b\n- v0d --refutes--> v0f\n- v0f --supports--> v0e\n- v08 --refutes--> v09\n- v10 --supports--> v06\n- v08 --supports--> v04\n- v0a --refutes--> v01\n- v08 --supports--> v03\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"910126260c3b85f7d1c37b08b93b09a13d9573d8221755bbd98dbea73307bf0e","schema":"livegraph.boolean-propagation.v1","spec_hash":"4bb6a77ab8485f445ca74ab39818895da841f066d7f0bc50437f463a8b5deab4","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-69a49ae555623993872b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v03\n- v0b --refutes--> v00\n- v10 --supports--> v0b\n- v0c --refutes--> v08\n- v06 --refutes--> v10\n- v0a --refutes--> v04\n- v01 --supports--> v09\n- v01 --supports--> v05\n- v07 --refutes--> v0f\n- v00 --refutes--> v07\n- v0c --refutes--> v0a\n- v01 --supports--> v0d\n- v09 --supports--> v0c\n- v04 --supports--> v06\n- v0c --supports--> v0e\n- v03 --refutes--> v02\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1d63a423e11ba3972ee95e1a9c562505d135fec5a10a4c84160b85e982663f46","schema":"livegraph.boolean-propagation.v1","spec_hash":"69a49ae555623993872bb431519425290cf29bd38c6fb8116d067e1e3ba5a6ae","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-af2305e8f518f695bcb5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v06\n- v0d --supports--> v10\n- v06 --refutes--> v04\n- v07 --supports--> v0d\n- v0f --supports--> v0a\n- v0e --refutes--> v05\n- v05 --supports--> v0c\n- v02 --supports--> v0e\n- v0c --supports--> v01\n- v0e --supports--> v08\n- v07 --supports--> v00\n- v10 --supports--> v02\n- v0f --refutes--> v07\n- v06 --refutes--> v0f\n- v01 --refutes--> v0b\n- v01 --refutes--> v03\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e1906ebb8ec1629e2bf279976e3148f591494a6533458fa909657e6239923df3","schema":"livegraph.boolean-propagation.v1","spec_hash":"af2305e8f518f695bcb5d256a7f72ab2f3840c6c3b11da8e85ec6628e2131c6a","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-b7ce694751d0a0a57a39","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v03\n- v0d --supports--> v09\n- v06 --refutes--> v0a\n- v02 --supports--> v05\n- v0d --supports--> v0b\n- v06 --refutes--> v04\n- v07 --supports--> v0f\n- v08 --supports--> v0d\n- v0b --refutes--> v07\n- v05 --supports--> v10\n- v0c --refutes--> v01\n- v03 --refutes--> v02\n- v04 --refutes--> v0c\n- v05 --supports--> v00\n- v08 --supports--> v0e\n- v10 --supports--> v08\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a66ab4edc86616ea9bbe2529fd88126b3623ed795a69b56fff142ea0971fa375","schema":"livegraph.boolean-propagation.v1","spec_hash":"b7ce694751d0a0a57a39eed198a9746d15edd2cedaac392e351629fca4e3938e","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-73e595fa8b5ff0e0c0bc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v03\n- v03 --supports--> v09\n- v09 --refutes--> v06\n- v10 --supports--> v05\n- v08 --refutes--> v02\n- v0b --refutes--> v0d\n- v05 --supports--> v0f\n- v08 --supports--> v0b\n- v00 --supports--> v08\n- v05 --refutes--> v01\n- v07 --refutes--> v04\n- v01 --supports--> v07\n- v0b --supports--> v0e\n- v0d --supports--> v0c\n- v10 --supports--> v0a\n- v06 --refutes--> v00\n\n[VERIFIED ROOT]\nv10 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d1df2dad82d491361e74957dca2aeff3f7379455323208a852f95fade58b38e6","schema":"livegraph.boolean-propagation.v1","spec_hash":"73e595fa8b5ff0e0c0bcf330d9ef4261430d65c29b8f2a64f64895746dd77f0c","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-730a6a26d27a3d74378e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v02\n- v09 --supports--> v01\n- v02 --refutes--> v00\n- v0b --supports--> v03\n- v0a --refutes--> v0d\n- v03 --supports--> v04\n- v00 --supports--> v0b\n- v01 --supports--> v0f\n- v03 --supports--> v07\n- v03 --supports--> v0e\n- v04 --supports--> v10\n- v06 --refutes--> v09\n- v08 --supports--> v06\n- v0a --supports--> v08\n- v05 --refutes--> v0a\n- v0a --refutes--> v0c\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c4bd35639dd9ff7547d278f926dd2e58665693e9fabae3cfa2bbcb672c21ad25","schema":"livegraph.boolean-propagation.v1","spec_hash":"730a6a26d27a3d74378eca9fe90778b767e20269ed9c2235c0e9c22d7c1f6ef8","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":11,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-6a60a65767659a98b934","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v00\n- v01 --refutes--> v0c\n- v0b --refutes--> v09\n- v06 --refutes--> v08\n- v08 --supports--> v0b\n- v05 --supports--> v07\n- v04 --supports--> v06\n- v07 --supports--> v01\n- v06 --supports--> v0e\n- v0d --supports--> v05\n- v01 --refutes--> v02\n- v07 --refutes--> v0a\n- v09 --supports--> v10\n- v10 --refutes--> v0d\n- v02 --refutes--> v0f\n- v08 --supports--> v03\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ad8a6dee63d529588ddc70192c1815615d3f1bd1482e5a192cd917ab8f7ab8c1","schema":"livegraph.boolean-propagation.v1","spec_hash":"6a60a65767659a98b934d05935c5949d2a1faf5e21214d916eff59eecca10fe2","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-a86eb4d8f03d05731258","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v03\n- v07 --supports--> v00\n- v0d --refutes--> v07\n- v05 --refutes--> v10\n- v06 --supports--> v05\n- v0a --refutes--> v02\n- v0c --refutes--> v06\n- v0b --refutes--> v04\n- v10 --supports--> v0d\n- v0d --refutes--> v09\n- v00 --refutes--> v0e\n- v06 --supports--> v01\n- v02 --supports--> v0c\n- v0f --supports--> v0b\n- v00 --refutes--> v0f\n- v00 --refutes--> v08\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e3d3fd51ebd5b7e65b13a891c12ab8c61d9e5bee2f198e8d687669a5f2e0bb35","schema":"livegraph.boolean-propagation.v1","spec_hash":"a86eb4d8f03d05731258affba51c99b2d2fbd79a081373496fc9ce6ffde79190","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-a71c7e033cdcc97b0224","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v06\n- v0b --supports--> v0e\n- v01 --supports--> v0f\n- v07 --refutes--> v08\n- v0d --refutes--> v02\n- v03 --supports--> v07\n- v05 --supports--> v0c\n- v0e --refutes--> v10\n- v07 --supports--> v0b\n- v01 --supports--> v09\n- v0e --supports--> v00\n- v06 --supports--> v01\n- v0b --supports--> v04\n- v07 --supports--> v0a\n- v0c --supports--> v0d\n- v0f --supports--> v03\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"31d4e2afa181f47448fef0be4d1f6922a41cbe2c5c2ebba032e8dcefe6ed3cfb","schema":"livegraph.boolean-propagation.v1","spec_hash":"a71c7e033cdcc97b0224bba30d749038e29bed1ffce8246a5b23f923bf2df98c","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-1c967e44a45e1cc6ced7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v06\n- v07 --supports--> v09\n- v01 --supports--> v0e\n- v02 --refutes--> v0f\n- v04 --refutes--> v01\n- v03 --supports--> v04\n- v02 --refutes--> v03\n- v0b --refutes--> v0a\n- v00 --refutes--> v0b\n- v0a --refutes--> v0d\n- v0d --supports--> v05\n- v01 --refutes--> v10\n- v06 --refutes--> v00\n- v0d --refutes--> v07\n- v0a --supports--> v0c\n- v04 --supports--> v08\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9243ec46812d79a12c5eb932a7e420ef98bdf60d6e41733b6cb8fde260915aff","schema":"livegraph.boolean-propagation.v1","spec_hash":"1c967e44a45e1cc6ced769e2715081b603b1f3f8c92dd1e526d0adb7c3579c4c","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"}],\"hop\":11,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-e1037892f4737ebc6224","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v05\n- v04 --supports--> v03\n- v08 --supports--> v0f\n- v0d --refutes--> v06\n- v09 --supports--> v00\n- v0c --refutes--> v0a\n- v0b --supports--> v04\n- v02 --supports--> v0c\n- v08 --refutes--> v0e\n- v10 --refutes--> v08\n- v0b --refutes--> v07\n- v0e --refutes--> v01\n- v0c --supports--> v0b\n- v03 --supports--> v09\n- v00 --refutes--> v10\n- v0e --refutes--> v0d\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f6c86f87a5891f91cd4ec72038ce94ae4e1d33900fd975ec77326fe71ee653e7","schema":"livegraph.boolean-propagation.v1","spec_hash":"e1037892f4737ebc622475427a9dab5d50e84ff926b27fce6de58dfac553fafe","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-fc8d2173d6b6db9188c5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v05\n- v08 --refutes--> v07\n- v04 --supports--> v0a\n- v0d --refutes--> v09\n- v0d --refutes--> v10\n- v0f --supports--> v04\n- v0b --refutes--> v03\n- v0f --supports--> v00\n- v0c --refutes--> v0d\n- v08 --supports--> v0c\n- v03 --supports--> v01\n- v0a --supports--> v08\n- v0e --supports--> v0f\n- v06 --supports--> v02\n- v02 --refutes--> v0e\n- v03 --refutes--> v06\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"090f58b47daba36c8364c89ab5d394f64d1cbe759994d8a7e0a650de0b665e17","schema":"livegraph.boolean-propagation.v1","spec_hash":"fc8d2173d6b6db9188c5384361a7764ba85f48d1ed2da127958517bc3a58236b","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-810c1a50be93d56de136","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v04\n- v00 --refutes--> v01\n- v10 --supports--> v0a\n- v02 --refutes--> v00\n- v0a --refutes--> v05\n- v02 --refutes--> v0b\n- v07 --refutes--> v08\n- v01 --supports--> v0f\n- v0c --refutes--> v03\n- v06 --refutes--> v10\n- v0c --refutes--> v0e\n- v0c --supports--> v06\n- v05 --supports--> v0d\n- v08 --supports--> v0c\n- v01 --refutes--> v07\n- v07 --supports--> v09\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"382643c8849858bf0a6e41353c5f8430e8de03dd05f73bf53be7091392d379f1","schema":"livegraph.boolean-propagation.v1","spec_hash":"810c1a50be93d56de1363582121a290e3622e69608776c5cf45ad4730e852483","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-9e53574e9d1568379550","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v03\n- v07 --supports--> v09\n- v01 --refutes--> v07\n- v0c --refutes--> v0a\n- v02 --refutes--> v08\n- v0d --supports--> v0c\n- v0b --refutes--> v00\n- v07 --refutes--> v04\n- v0c --refutes--> v0e\n- v0d --supports--> v05\n- v09 --refutes--> v10\n- v0e --supports--> v06\n- v0d --supports--> v0f\n- v08 --supports--> v01\n- v0b --supports--> v0d\n- v03 --supports--> v02\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fb4ce71bea827c78c2b6905dec775b446af5cf8a1e87c9a1df73fbd8c64b6d24","schema":"livegraph.boolean-propagation.v1","spec_hash":"9e53574e9d156837955071b4c6ff3be9f7093bf53530b934eba8638cf2c48911","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":11,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-a1b11fbbfaddec52f91a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v03\n- v03 --supports--> v07\n- v0d --refutes--> v04\n- v03 --supports--> v09\n- v01 --refutes--> v0b\n- v07 --refutes--> v10\n- v08 --supports--> v05\n- v0f --refutes--> v0e\n- v10 --supports--> v0c\n- v03 --supports--> v0a\n- v02 --supports--> v06\n- v0c --refutes--> v01\n- v02 --refutes--> v0f\n- v05 --refutes--> v02\n- v04 --refutes--> v08\n- v05 --refutes--> v00\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3afa994f7c3b586c8883feb2afabb16beb308602266a82bd152cb627bf5bc574","schema":"livegraph.boolean-propagation.v1","spec_hash":"a1b11fbbfaddec52f91ac2380eabdf7c5c6b57596c726e76dce4ec2c4d8df6d0","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-30745367ad74abf26e6a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v10\n- v0b --refutes--> v09\n- v04 --refutes--> v0c\n- v0c --supports--> v02\n- v06 --supports--> v03\n- v03 --refutes--> v01\n- v07 --supports--> v0a\n- v03 --supports--> v07\n- v07 --refutes--> v08\n- v0a --supports--> v0d\n- v0f --supports--> v0e\n- v09 --refutes--> v0f\n- v05 --supports--> v06\n- v05 --refutes--> v00\n- v10 --supports--> v05\n- v0c --refutes--> v0b\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"90845b5eed2bbca34a2c137aeb6b5271e482d17960f111d7e8f8324b8bf4323c","schema":"livegraph.boolean-propagation.v1","spec_hash":"30745367ad74abf26e6a41f87e7f77246879231c0e84fec34cdfaeae9e4f791b","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-9d40af41cce263114612","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v07\n- v0b --refutes--> v02\n- v0b --refutes--> v0f\n- v10 --supports--> v0b\n- v03 --refutes--> v05\n- v0d --supports--> v10\n- v05 --refutes--> v09\n- v00 --refutes--> v06\n- v00 --supports--> v0d\n- v0f --refutes--> v0a\n- v05 --supports--> v0c\n- v04 --refutes--> v00\n- v0c --refutes--> v01\n- v0c --refutes--> v04\n- v0e --supports--> v03\n- v0a --refutes--> v08\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ae8b4b6ddf2dfc6039bbe94c367aa81d74d02576a590e8a6cdffe3ce1cc75666","schema":"livegraph.boolean-propagation.v1","spec_hash":"9d40af41cce263114612589755b157353ac8903ffda40e36be6041cc0a0fe783","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-d673301dd5278eb95bb0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v03\n- v10 --refutes--> v05\n- v04 --refutes--> v10\n- v0f --supports--> v09\n- v09 --refutes--> v0b\n- v00 --supports--> v04\n- v05 --supports--> v0e\n- v08 --refutes--> v0a\n- v08 --refutes--> v00\n- v00 --refutes--> v0d\n- v0e --supports--> v0f\n- v0b --supports--> v02\n- v0b --refutes--> v07\n- v05 --supports--> v0c\n- v06 --refutes--> v08\n- v01 --refutes--> v06\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"181142d7cdeb4bdce050d83512b46b1536c22d988b22c74215b26130798c7a42","schema":"livegraph.boolean-propagation.v1","spec_hash":"d673301dd5278eb95bb0a71a6c0a9cf1f62638355bd7c00baac1c4e9207af8d6","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-c69f80e7feb90df3c9f6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v02\n- v0d --refutes--> v01\n- v03 --supports--> v0d\n- v0d --supports--> v05\n- v0e --supports--> v0b\n- v07 --refutes--> v10\n- v0e --supports--> v0c\n- v05 --refutes--> v08\n- v0a --refutes--> v00\n- v04 --supports--> v0f\n- v05 --supports--> v06\n- v02 --supports--> v04\n- v10 --refutes--> v0a\n- v0f --refutes--> v03\n- v00 --supports--> v09\n- v05 --refutes--> v07\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6cb2e99688a0265b86a90153a7777afe0e14707fd41003353ad549491005935c","schema":"livegraph.boolean-propagation.v1","spec_hash":"c69f80e7feb90df3c9f60eb62a9ae8e675ca1e5cedc264202fac80900a889d5a","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-289dec384dfeed2876f4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v02\n- v00 --supports--> v0a\n- v0e --supports--> v07\n- v0b --refutes--> v01\n- v0f --refutes--> v0b\n- v02 --refutes--> v08\n- v0c --supports--> v00\n- v10 --refutes--> v0d\n- v00 --supports--> v03\n- v0e --supports--> v04\n- v0d --refutes--> v09\n- v0d --supports--> v0f\n- v0c --refutes--> v06\n- v01 --supports--> v0e\n- v0e --supports--> v05\n- v08 --refutes--> v0c\n\n[VERIFIED ROOT]\nv10 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0d060391280e95b0f84f6c626e5463302f5a8ab78aae033f973d19192a59eed1","schema":"livegraph.boolean-propagation.v1","spec_hash":"289dec384dfeed2876f4b00f369377aa95db6f2a2fabffc0c2800f283b159967","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-f502b185721bd1c3997b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v07\n- v04 --supports--> v0a\n- v0a --supports--> v01\n- v10 --supports--> v0c\n- v0f --refutes--> v04\n- v00 --supports--> v0b\n- v03 --supports--> v00\n- v0d --supports--> v0f\n- v0f --supports--> v09\n- v0f --supports--> v05\n- v0c --supports--> v0e\n- v06 --supports--> v08\n- v0a --refutes--> v03\n- v0b --supports--> v10\n- v0b --refutes--> v02\n- v08 --refutes--> v0d\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c9a088f763487c1635124ae0cea0185d1e4f749376711ea0484bb55ee53c6989","schema":"livegraph.boolean-propagation.v1","spec_hash":"f502b185721bd1c3997bed4c35d40e25cb151fdbf29398df0ae250a44b81ba27","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-7a705ae53ca92bd505ec","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v09\n- v00 --refutes--> v02\n- v0b --supports--> v03\n- v02 --refutes--> v0f\n- v06 --refutes--> v0a\n- v08 --supports--> v06\n- v04 --refutes--> v10\n- v04 --supports--> v05\n- v03 --supports--> v0c\n- v0f --refutes--> v0e\n- v09 --refutes--> v01\n- v0b --supports--> v07\n- v06 --refutes--> v00\n- v10 --supports--> v08\n- v0e --supports--> v0d\n- v09 --supports--> v0b\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d047cbddb2fbb310d62b43f23edd30cb8c64da29a08ffb84277ed3286e946adc","schema":"livegraph.boolean-propagation.v1","spec_hash":"7a705ae53ca92bd505ece200ac98da982db79244e96784bbcd44b79a3cb0236a","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-981ac798627619f457a3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v06\n- v01 --refutes--> v03\n- v02 --refutes--> v07\n- v0f --refutes--> v0e\n- v0a --supports--> v10\n- v10 --refutes--> v00\n- v02 --refutes--> v04\n- v03 --supports--> v0a\n- v0c --refutes--> v09\n- v0f --refutes--> v08\n- v10 --refutes--> v0f\n- v07 --refutes--> v05\n- v0a --refutes--> v0d\n- v0b --refutes--> v02\n- v09 --supports--> v0b\n- v05 --supports--> v01\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f0a48db8f009913c72f09ab65d9cf77696a67ff8aaf9f0249d26994df3d12aa1","schema":"livegraph.boolean-propagation.v1","spec_hash":"981ac798627619f457a374a79ed98a7113193a52b6c5164fc6fe0f5b2bffa500","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-6448255f680f24325a21","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v01\n- v04 --refutes--> v02\n- v09 --supports--> v08\n- v01 --refutes--> v03\n- v02 --supports--> v05\n- v0f --refutes--> v10\n- v0c --refutes--> v09\n- v09 --supports--> v00\n- v0d --refutes--> v06\n- v02 --refutes--> v0e\n- v01 --refutes--> v0d\n- v10 --supports--> v0b\n- v06 --supports--> v04\n- v08 --supports--> v0a\n- v08 --refutes--> v0f\n- v02 --refutes--> v07\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e0e6704bcea50b94153c918bcebe3199cfa3265ae1d4fb9f47a73c6e7bc71fb3","schema":"livegraph.boolean-propagation.v1","spec_hash":"6448255f680f24325a2131c4ec991a3bfde299d640900711248756aed19ba5c8","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-3c7d1e4abec1e9e6365a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v0c\n- v09 --refutes--> v0e\n- v05 --supports--> v08\n- v04 --refutes--> v06\n- v0f --refutes--> v0b\n- v05 --supports--> v10\n- v0b --supports--> v01\n- v0e --supports--> v04\n- v01 --refutes--> v0a\n- v00 --supports--> v03\n- v0c --refutes--> v07\n- v01 --refutes--> v02\n- v00 --supports--> v0f\n- v0c --refutes--> v05\n- v08 --refutes--> v09\n- v0f --refutes--> v0d\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"13ffa0c0718a77bb3eba2064ec791b429ebdeae172ff9de1b00eb55eb38dea42","schema":"livegraph.boolean-propagation.v1","spec_hash":"3c7d1e4abec1e9e6365a88ce872d601d515bf751edac076ed678ccf2fea10ddd","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-77869048b77ef72227f3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v09\n- v07 --refutes--> v0b\n- v00 --supports--> v06\n- v0e --supports--> v08\n- v09 --supports--> v10\n- v01 --supports--> v07\n- v03 --supports--> v0c\n- v06 --supports--> v01\n- v09 --refutes--> v0d\n- v01 --refutes--> v02\n- v07 --supports--> v04\n- v0d --supports--> v05\n- v08 --supports--> v0f\n- v0f --refutes--> v0a\n- v0c --refutes--> v0e\n- v0d --refutes--> v00\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8dec3afb6bb7022663d675278f166f89d32b4118f6c55a19fc8acefeaddca895","schema":"livegraph.boolean-propagation.v1","spec_hash":"77869048b77ef72227f3700ff11957a98e1b4fc3acc21486e40015e4889f50a8","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-9be5584e6d199aabd858","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v0c\n- v0e --refutes--> v07\n- v03 --supports--> v00\n- v08 --refutes--> v0e\n- v0d --supports--> v0a\n- v09 --refutes--> v0d\n- v0a --refutes--> v0f\n- v03 --supports--> v06\n- v0c --supports--> v04\n- v04 --refutes--> v10\n- v02 --supports--> v08\n- v05 --supports--> v02\n- v00 --refutes--> v09\n- v04 --refutes--> v0b\n- v0a --refutes--> v05\n- v08 --refutes--> v01\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a5a8a649d757c87fe374afca0fe054f37e3d0b5c112775086520c24d1a6334ef","schema":"livegraph.boolean-propagation.v1","spec_hash":"9be5584e6d199aabd858f99d07ae5303a1a2ab729908355879ab58c1e962890b","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"}],\"hop\":11,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-4a27aa88d17eaae90265","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v0f\n- v0e --refutes--> v06\n- v01 --supports--> v05\n- v03 --supports--> v0b\n- v02 --refutes--> v0e\n- v0d --refutes--> v10\n- v0f --supports--> v04\n- v03 --refutes--> v0c\n- v07 --supports--> v00\n- v00 --refutes--> v0a\n- v03 --refutes--> v09\n- v10 --refutes--> v01\n- v0e --refutes--> v07\n- v08 --supports--> v03\n- v0b --refutes--> v0d\n- v0f --supports--> v02\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"599bb1f3b5d43d9bd229c33934e34e5d10fbb98de42635815af1a940b284a8f0","schema":"livegraph.boolean-propagation.v1","spec_hash":"4a27aa88d17eaae90265edaba38e6067b3768e4cbea1089a4ff18a59361052a4","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-2bd4ff48397cc5030a97","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v00\n- v07 --supports--> v02\n- v00 --refutes--> v10\n- v09 --refutes--> v0a\n- v00 --supports--> v0c\n- v0b --supports--> v04\n- v10 --supports--> v06\n- v04 --refutes--> v0e\n- v04 --supports--> v05\n- v07 --supports--> v08\n- v06 --refutes--> v0b\n- v02 --supports--> v0d\n- v01 --refutes--> v03\n- v0f --supports--> v09\n- v0e --refutes--> v0f\n- v09 --supports--> v01\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3de8ec5ef55a44fb2509bcf076648aac0aec5ccd3a6f2181539acb33115b0817","schema":"livegraph.boolean-propagation.v1","spec_hash":"2bd4ff48397cc5030a975da8dca9fb8ca9b7ad55cea4a09afbd6b7b310858efe","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-02de2302f3c757ff7612","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v05\n- v05 --supports--> v07\n- v09 --refutes--> v0f\n- v01 --refutes--> v0b\n- v0d --supports--> v0a\n- v0f --supports--> v0c\n- v0c --refutes--> v06\n- v06 --refutes--> v10\n- v0a --refutes--> v01\n- v02 --supports--> v09\n- v04 --supports--> v0d\n- v0d --supports--> v03\n- v0c --supports--> v08\n- v07 --supports--> v04\n- v04 --refutes--> v00\n- v01 --supports--> v0e\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e0d1cea42bb6d3be5cb59f989aa745543fad794bfdad20708bcbdb22b4034cfc","schema":"livegraph.boolean-propagation.v1","spec_hash":"02de2302f3c757ff7612cb89f636b9b7549af6e98740ea820b2a722970e4fd40","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-abf368475be652f51413","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v0b\n- v02 --refutes--> v01\n- v09 --supports--> v0a\n- v06 --refutes--> v09\n- v0a --refutes--> v05\n- v0d --supports--> v0c\n- v0c --supports--> v00\n- v07 --supports--> v0e\n- v05 --refutes--> v0d\n- v0b --supports--> v02\n- v07 --refutes--> v06\n- v01 --refutes--> v07\n- v0a --supports--> v04\n- v0a --supports--> v10\n- v0c --refutes--> v03\n- v09 --supports--> v0f\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dde5c69737eb1a89fc076ca4e1b4695db478893542a4587496a531218c979aa0","schema":"livegraph.boolean-propagation.v1","spec_hash":"abf368475be652f51413e8d6701e917fbaedcb4a7294663b6f487ebeec289ead","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":11,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-4abc6072f98793d3d0d9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v0b\n- v05 --refutes--> v02\n- v06 --refutes--> v01\n- v02 --refutes--> v08\n- v03 --refutes--> v04\n- v08 --refutes--> v00\n- v0b --refutes--> v0c\n- v04 --refutes--> v0d\n- v0c --refutes--> v0e\n- v09 --supports--> v10\n- v06 --supports--> v03\n- v09 --refutes--> v06\n- v0c --refutes--> v07\n- v07 --refutes--> v09\n- v09 --supports--> v0f\n- v09 --supports--> v0a\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"93a1b6954b5ede275d3fc7917cac630c68c44862a18ced4287b83efe4ca9c5fb","schema":"livegraph.boolean-propagation.v1","spec_hash":"4abc6072f98793d3d0d9b0d90ebcaab74a22046546c629ac554e13750094e16a","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":11,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-c514d8917a1a2bba23fc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v05\n- v06 --refutes--> v10\n- v00 --supports--> v02\n- v0c --refutes--> v0d\n- v09 --refutes--> v00\n- v10 --supports--> v04\n- v08 --refutes--> v0e\n- v0f --supports--> v07\n- v00 --supports--> v0c\n- v08 --refutes--> v06\n- v0c --supports--> v01\n- v10 --supports--> v03\n- v0d --supports--> v0a\n- v04 --supports--> v0f\n- v0b --refutes--> v09\n- v0a --supports--> v08\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c4002e27ce9d8503ead653d850e3130fa75cb3bda18b64a945275a248f8aa276","schema":"livegraph.boolean-propagation.v1","spec_hash":"c514d8917a1a2bba23fcaf58ba72b932ef3c5c4189e3408c213199508a320702","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-ca7040a69e88cf503d4a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v07\n- v10 --refutes--> v00\n- v06 --supports--> v0f\n- v06 --supports--> v02\n- v0a --refutes--> v08\n- v0e --refutes--> v0d\n- v08 --refutes--> v01\n- v05 --supports--> v10\n- v10 --refutes--> v03\n- v09 --supports--> v05\n- v10 --supports--> v0e\n- v0c --refutes--> v04\n- v0c --refutes--> v0a\n- v07 --refutes--> v06\n- v0f --refutes--> v0b\n- v0b --supports--> v0c\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"db578362026dd90308bcaa65d59c50d58bb37d01b929438c9b965ee92d7e0526","schema":"livegraph.boolean-propagation.v1","spec_hash":"ca7040a69e88cf503d4acce932420bdf5ec5b5f17ec41740c2f65f4a4582842b","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-201b91bab861507a9cb3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v03\n- v07 --supports--> v02\n- v02 --refutes--> v0b\n- v0d --supports--> v10\n- v05 --refutes--> v0e\n- v04 --supports--> v0a\n- v0b --supports--> v01\n- v09 --supports--> v08\n- v05 --supports--> v06\n- v10 --supports--> v09\n- v0e --refutes--> v0f\n- v0f --refutes--> v07\n- v0e --refutes--> v00\n- v01 --supports--> v04\n- v0e --supports--> v0c\n- v09 --refutes--> v05\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"20ffb89c1ae5c67e379679f8bffce22d3bf6c92d50af8f01e96484a62be8f1e3","schema":"livegraph.boolean-propagation.v1","spec_hash":"201b91bab861507a9cb3c003e683ea6d6fc9fe98f0cd0c9ced39e759cf3d4170","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-0851eb3bee80211d6d0a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v02\n- v02 --supports--> v01\n- v00 --supports--> v09\n- v03 --supports--> v10\n- v10 --supports--> v06\n- v02 --refutes--> v08\n- v0e --supports--> v03\n- v0f --refutes--> v0a\n- v0a --supports--> v0d\n- v04 --supports--> v0f\n- v0d --supports--> v05\n- v0a --refutes--> v0c\n- v06 --refutes--> v04\n- v02 --supports--> v0b\n- v0f --refutes--> v07\n- v05 --refutes--> v00\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8815d40f2010da2677afc237e3f1687120bd7b6fa3e74c8850192b8c5ab48314","schema":"livegraph.boolean-propagation.v1","spec_hash":"0851eb3bee80211d6d0a31bbaba49a09640ff7868e72776beb44dd5b751a44fc","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-6834ea1be774ac34cf70","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v07\n- v02 --supports--> v0c\n- v00 --supports--> v0d\n- v07 --refutes--> v08\n- v0a --refutes--> v03\n- v0f --refutes--> v01\n- v0f --refutes--> v10\n- v06 --supports--> v0a\n- v05 --supports--> v09\n- v10 --refutes--> v06\n- v06 --refutes--> v0e\n- v0d --supports--> v05\n- v08 --supports--> v00\n- v02 --supports--> v04\n- v07 --supports--> v0b\n- v09 --supports--> v0f\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3f652250c5093bd1b91d0c22352e45144c5a689509f0f5b30a48561cc8150259","schema":"livegraph.boolean-propagation.v1","spec_hash":"6834ea1be774ac34cf705cd6e747d39123fc9598c4bfacfdcfb2fae8f1640179","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-0c37cc6033ee9f67e0cd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v0b\n- v09 --supports--> v07\n- v0b --supports--> v05\n- v00 --refutes--> v0e\n- v06 --supports--> v04\n- v00 --supports--> v01\n- v01 --supports--> v06\n- v04 --supports--> v08\n- v09 --supports--> v03\n- v04 --refutes--> v0d\n- v0f --refutes--> v00\n- v05 --refutes--> v0f\n- v02 --refutes--> v0a\n- v0d --refutes--> v09\n- v09 --refutes--> v10\n- v0f --refutes--> v0c\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3bca6c86bfa8322a754c08a32ec44dcaaf01b9760a84d8bc1d86dddcfd4d05a6","schema":"livegraph.boolean-propagation.v1","spec_hash":"0c37cc6033ee9f67e0cdc89a848da41cceb21d7c56e4b3008ca201e18a44ca34","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-ea188a6b7765752555aa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v07\n- v10 --supports--> v06\n- v08 --refutes--> v00\n- v0a --supports--> v02\n- v03 --supports--> v0d\n- v0f --refutes--> v0e\n- v06 --supports--> v01\n- v02 --supports--> v04\n- v0b --refutes--> v10\n- v0e --refutes--> v08\n- v04 --refutes--> v09\n- v08 --refutes--> v0c\n- v06 --refutes--> v03\n- v08 --refutes--> v0a\n- v04 --supports--> v05\n- v09 --refutes--> v0b\n\n[VERIFIED ROOT]\nv0f = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"881b291002edc9db233e4eb54648d25e3c59ac166c87a4e89c1855232e2b2881","schema":"livegraph.boolean-propagation.v1","spec_hash":"ea188a6b7765752555aa4b4c0a16cade687aec36b08dce1c7c5315250403934b","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-eb07d910825e351f8cbf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --supports--> v00\n- v08 --refutes--> v06\n- v03 --refutes--> v09\n- v0b --supports--> v10\n- v0c --supports--> v02\n- v0b --supports--> v0d\n- v01 --refutes--> v0c\n- v0e --refutes--> v01\n- v0a --refutes--> v03\n- v08 --refutes--> v0b\n- v08 --refutes--> v04\n- v10 --refutes--> v0e\n- v0f --supports--> v05\n- v09 --supports--> v08\n- v0c --supports--> v07\n- v07 --supports--> v0f\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1c9fe756b183226daed5947a10fa2d160f1cfbc2d47e5068e4d5f6d5a8216d14","schema":"livegraph.boolean-propagation.v1","spec_hash":"eb07d910825e351f8cbf34b8e13cbb2b799544173b59bc1313935d07235e5108","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-4ace9cede9b7aefa927a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v0c\n- v0b --refutes--> v09\n- v0d --supports--> v04\n- v0a --supports--> v03\n- v02 --refutes--> v0f\n- v0f --refutes--> v01\n- v0b --refutes--> v08\n- v0d --refutes--> v10\n- v01 --refutes--> v07\n- v06 --supports--> v0d\n- v04 --refutes--> v05\n- v0e --supports--> v0b\n- v03 --refutes--> v02\n- v0d --supports--> v00\n- v01 --refutes--> v0e\n- v09 --refutes--> v06\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bd7f370f0352f773718f2ccde249d125641a2c8a91f3a8a752e94d3656d11888","schema":"livegraph.boolean-propagation.v1","spec_hash":"4ace9cede9b7aefa927af78da29ecaba0d7505b6aa0e806042dbfd946840c03d","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-2ed641117194fad27dc7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v02\n- v0e --supports--> v0f\n- v0f --supports--> v09\n- v09 --refutes--> v01\n- v10 --supports--> v04\n- v06 --refutes--> v10\n- v0c --refutes--> v0d\n- v0f --supports--> v07\n- v0a --refutes--> v0e\n- v08 --refutes--> v0b\n- v0d --refutes--> v08\n- v08 --refutes--> v05\n- v0b --refutes--> v0a\n- v04 --refutes--> v00\n- v04 --supports--> v0c\n- v0c --refutes--> v03\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f917914581c4dee54593479ea2054454df6db620f5cd453409db8019bdea6793","schema":"livegraph.boolean-propagation.v1","spec_hash":"2ed641117194fad27dc7be852770a2d365d7fa430514acfe52eeb6173d01b03c","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-e64a39370b530f2ec886","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v0e\n- v0c --refutes--> v10\n- v01 --supports--> v05\n- v0b --supports--> v00\n- v09 --supports--> v07\n- v0e --supports--> v02\n- v04 --supports--> v06\n- v03 --supports--> v0f\n- v0e --supports--> v09\n- v05 --refutes--> v0d\n- v03 --supports--> v0a\n- v10 --refutes--> v04\n- v0b --refutes--> v0c\n- v0d --supports--> v0b\n- v0f --refutes--> v01\n- v09 --supports--> v08\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bd630539425d39fb44e6858978cd177d1ddf3acc7df1838ff7318a0a8e78345f","schema":"livegraph.boolean-propagation.v1","spec_hash":"e64a39370b530f2ec886e2f135f9ccfb30337c57f16a52432b5c1b3141cdab41","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-5e3f41c55c5e96f8ca3d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v00\n- v00 --refutes--> v08\n- v05 --supports--> v0c\n- v10 --refutes--> v09\n- v0b --refutes--> v03\n- v00 --refutes--> v0e\n- v0e --refutes--> v07\n- v0d --refutes--> v0a\n- v0b --refutes--> v04\n- v07 --supports--> v0d\n- v10 --refutes--> v0f\n- v0d --refutes--> v0b\n- v02 --supports--> v06\n- v04 --refutes--> v10\n- v05 --supports--> v01\n- v09 --supports--> v05\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"401e7bec1d503ea5b05fb620dc9459cf101af4728e0b663363f7935852663dcd","schema":"livegraph.boolean-propagation.v1","spec_hash":"5e3f41c55c5e96f8ca3df5e057145b716bbd2650f8a0eae82b16a6dcbd14d97a","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-4e37334f038f3e3a11df","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v01\n- v02 --supports--> v0f\n- v0f --refutes--> v06\n- v04 --supports--> v0e\n- v0d --supports--> v08\n- v01 --refutes--> v03\n- v07 --supports--> v10\n- v05 --supports--> v00\n- v09 --refutes--> v0c\n- v03 --supports--> v0d\n- v04 --refutes--> v0b\n- v08 --refutes--> v0a\n- v08 --supports--> v02\n- v0f --refutes--> v09\n- v09 --refutes--> v07\n- v07 --refutes--> v05\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cf4127e814e4e47788c55011f6ab3f4df1d39adc1da62b0dd9156d406ebeea3d","schema":"livegraph.boolean-propagation.v1","spec_hash":"4e37334f038f3e3a11df5a3581e70672519fc32748a1292cd1624f773b353173","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-4aea240b2ec2f5be11bd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v07\n- v01 --refutes--> v05\n- v0d --refutes--> v0f\n- v09 --refutes--> v02\n- v0b --supports--> v03\n- v0e --supports--> v0a\n- v07 --supports--> v06\n- v07 --refutes--> v01\n- v0c --refutes--> v0d\n- v07 --supports--> v08\n- v03 --refutes--> v04\n- v0e --refutes--> v0b\n- v00 --refutes--> v0e\n- v09 --refutes--> v00\n- v04 --supports--> v0c\n- v03 --supports--> v10\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0920ad6dd09fa226df6462053a55eb92941e210b3c35947296de38e607b93a0c","schema":"livegraph.boolean-propagation.v1","spec_hash":"4aea240b2ec2f5be11bd5dce88cf66accf97b20f0899ab8d8949866055864770","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-0c90866a47c908aae679","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v07\n- v10 --refutes--> v04\n- v09 --refutes--> v00\n- v01 --supports--> v02\n- v07 --refutes--> v05\n- v0c --refutes--> v10\n- v02 --supports--> v06\n- v06 --refutes--> v09\n- v02 --supports--> v0d\n- v03 --refutes--> v08\n- v07 --supports--> v0a\n- v00 --supports--> v03\n- v02 --refutes--> v0b\n- v05 --refutes--> v0f\n- v04 --refutes--> v01\n- v10 --refutes--> v0e\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d84f2103885d611c6a5a7010cbed16891d3eb35a1656dc356e87c4dbf4d61d96","schema":"livegraph.boolean-propagation.v1","spec_hash":"0c90866a47c908aae679bd8a1f0c5b9c6371a78f4c8cf16389d1663949345875","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-39638c6496df13fe744c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v09\n- v07 --supports--> v03\n- v00 --refutes--> v0e\n- v0b --supports--> v01\n- v04 --supports--> v08\n- v0c --refutes--> v07\n- v0e --supports--> v0d\n- v02 --refutes--> v10\n- v07 --refutes--> v06\n- v04 --supports--> v0a\n- v02 --refutes--> v0b\n- v0d --refutes--> v0c\n- v0b --refutes--> v04\n- v00 --supports--> v0f\n- v05 --refutes--> v00\n- v09 --supports--> v05\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f7c79bcff4e787e9d4a3ef66a3d5f4cf35d2c8606372789d17461606b40d0b75","schema":"livegraph.boolean-propagation.v1","spec_hash":"39638c6496df13fe744cd3d64b31647fcbca03aafd2441305f5980d17772b8a2","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":11,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-d749253745f590ff92e1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v05\n- v06 --refutes--> v09\n- v05 --supports--> v01\n- v0a --supports--> v0d\n- v01 --supports--> v03\n- v02 --supports--> v07\n- v03 --refutes--> v0e\n- v0e --refutes--> v0f\n- v0c --refutes--> v08\n- v00 --supports--> v0c\n- v06 --supports--> v0b\n- v09 --supports--> v00\n- v06 --supports--> v04\n- v0a --refutes--> v02\n- v01 --supports--> v10\n- v0e --refutes--> v0a\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0d53e687463d72bad5cad8927252fee9cfb52acd1e3b6c449bc7c0ad70e13ece","schema":"livegraph.boolean-propagation.v1","spec_hash":"d749253745f590ff92e1cdb53e25025f4d076ff3ce1dc5ae3d9459909aec5b63","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"}],\"hop\":11,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-7a9c107e2472eb22380b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v04\n- v09 --supports--> v00\n- v0b --supports--> v05\n- v08 --supports--> v0c\n- v09 --refutes--> v0d\n- v0e --supports--> v0f\n- v01 --refutes--> v03\n- v0d --refutes--> v10\n- v01 --supports--> v09\n- v0e --refutes--> v02\n- v06 --supports--> v07\n- v0a --supports--> v0b\n- v10 --supports--> v08\n- v05 --supports--> v01\n- v0e --supports--> v06\n- v07 --refutes--> v0a\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f3705e99884ce8a60d0c47c42c290caabb6779c2f2a072b0ded050a28009cbde","schema":"livegraph.boolean-propagation.v1","spec_hash":"7a9c107e2472eb22380b71e0fc805b5bdb1c82dfec69c7f66b2cecffeec134f0","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-af85d684c5c0bc64ad52","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v03\n- v0a --supports--> v0b\n- v07 --refutes--> v0d\n- v04 --supports--> v06\n- v0d --refutes--> v0e\n- v05 --refutes--> v08\n- v0f --refutes--> v10\n- v0d --supports--> v01\n- v0b --refutes--> v05\n- v06 --supports--> v0a\n- v10 --refutes--> v04\n- v08 --supports--> v07\n- v0b --refutes--> v00\n- v0c --supports--> v0f\n- v04 --refutes--> v02\n- v0f --refutes--> v09\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b4cd4b55390fd27c1ddd56f520b7cdb6f4db1cf7eb8b7480a81a37ec8f042323","schema":"livegraph.boolean-propagation.v1","spec_hash":"af85d684c5c0bc64ad52cf1f05ff4fd36eb5b5594aeb5268bba2e319dd4ae4d5","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-1f656061af9141f55cf3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v06\n- v06 --supports--> v08\n- v00 --supports--> v03\n- v0d --refutes--> v10\n- v0d --refutes--> v01\n- v0e --supports--> v04\n- v01 --supports--> v0e\n- v0e --supports--> v07\n- v03 --supports--> v0b\n- v0f --supports--> v05\n- v07 --refutes--> v0f\n- v08 --supports--> v0d\n- v03 --refutes--> v0a\n- v0f --supports--> v0c\n- v09 --refutes--> v00\n- v0d --refutes--> v02\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d18911e0382a9e4e44af4fbece6b2fe24aadd39edf45d1036d847b49aa8b6d4d","schema":"livegraph.boolean-propagation.v1","spec_hash":"1f656061af9141f55cf3997536f3f30a5dca5c61aff2b81b4695ef0cf211d215","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-adef2924098ba8c89ea0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v00\n- v05 --refutes--> v07\n- v03 --supports--> v09\n- v00 --refutes--> v06\n- v08 --refutes--> v02\n- v03 --supports--> v0f\n- v06 --refutes--> v04\n- v0c --supports--> v0a\n- v0a --refutes--> v05\n- v06 --refutes--> v0b\n- v10 --supports--> v0d\n- v07 --refutes--> v01\n- v01 --supports--> v08\n- v10 --supports--> v0c\n- v06 --refutes--> v10\n- v0c --supports--> v0e\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5a51371549cc1081a4a57abd3afe915a7196e49c4a286774e7499cb7441d2486","schema":"livegraph.boolean-propagation.v1","spec_hash":"adef2924098ba8c89ea0697d859ac602b5fd27552a2e1dfbfd4e0dccc0bccf95","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-1cc360ea4cb2b569c7b1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v10\n- v10 --supports--> v04\n- v10 --supports--> v06\n- v08 --supports--> v00\n- v09 --supports--> v0d\n- v0b --supports--> v08\n- v10 --supports--> v0a\n- v0e --refutes--> v09\n- v02 --supports--> v07\n- v07 --supports--> v0f\n- v0d --supports--> v01\n- v00 --supports--> v0e\n- v0b --supports--> v03\n- v01 --refutes--> v05\n- v10 --refutes--> v02\n- v05 --refutes--> v0c\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"19d097f7115739aaa118b2e87a07f1d4901f9d297baf98626147234b4d0a0e70","schema":"livegraph.boolean-propagation.v1","spec_hash":"1cc360ea4cb2b569c7b194b5543005ca78d4b405f8883180a81bf608091fc194","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-eaefea3eef4945338105","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v0e\n- v00 --supports--> v0d\n- v0a --refutes--> v05\n- v09 --refutes--> v0c\n- v00 --supports--> v07\n- v01 --refutes--> v0f\n- v04 --refutes--> v08\n- v04 --supports--> v02\n- v0e --refutes--> v04\n- v09 --refutes--> v03\n- v04 --supports--> v06\n- v04 --supports--> v10\n- v0b --supports--> v0a\n- v0d --refutes--> v09\n- v08 --refutes--> v00\n- v03 --refutes--> v0b\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2b2c45cddef8d2106cac7d3aaa794a3272cff203e7d4fb71f72652de7666087d","schema":"livegraph.boolean-propagation.v1","spec_hash":"eaefea3eef49453381059ee25b1829a7f2460f07616aa8bbafd64b538ee13d83","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-9001c9599fffe1f6bb72","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v03\n- v01 --refutes--> v0c\n- v0b --supports--> v0a\n- v0d --supports--> v00\n- v0a --refutes--> v04\n- v0f --supports--> v01\n- v0a --supports--> v07\n- v01 --refutes--> v0e\n- v04 --supports--> v02\n- v10 --supports--> v09\n- v00 --refutes--> v0b\n- v02 --refutes--> v05\n- v10 --supports--> v0f\n- v05 --supports--> v06\n- v0a --refutes--> v08\n- v06 --refutes--> v10\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"23f1cb170c02a56fbe50e2b258324428390091857afba04bf1fda47331830a69","schema":"livegraph.boolean-propagation.v1","spec_hash":"9001c9599fffe1f6bb72ba57505ba3afae5f3e1e8c3f69c1e929eaadb8b9d035","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-aba9b0de376e1377725a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --supports--> v04\n- v0c --supports--> v08\n- v0e --supports--> v0d\n- v09 --supports--> v0a\n- v05 --supports--> v0f\n- v0d --refutes--> v07\n- v0d --supports--> v0c\n- v0c --supports--> v0b\n- v03 --refutes--> v01\n- v00 --refutes--> v0e\n- v01 --supports--> v00\n- v02 --supports--> v09\n- v08 --supports--> v06\n- v08 --supports--> v10\n- v05 --refutes--> v02\n- v09 --supports--> v03\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"318f15998eb2ebe1ebb44852b38f2a12994a6f58ae9aa2f0c9e0cb1f0d610281","schema":"livegraph.boolean-propagation.v1","spec_hash":"aba9b0de376e1377725a518aa251e11dfd88ed82b98083dea1d64e96dc749836","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-b91770625bde6908e3a7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v05\n- v0b --supports--> v04\n- v0d --refutes--> v0c\n- v07 --supports--> v0f\n- v10 --supports--> v08\n- v0d --refutes--> v09\n- v02 --supports--> v0d\n- v0b --refutes--> v06\n- v02 --refutes--> v0e\n- v00 --refutes--> v07\n- v04 --supports--> v0a\n- v07 --supports--> v0b\n- v03 --refutes--> v02\n- v01 --refutes--> v00\n- v0a --supports--> v03\n- v08 --refutes--> v01\n\n[VERIFIED ROOT]\nv10 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b86d92b50947d082a4f197211cff637d7c3e1a9d7760eaf50925367510f41cb5","schema":"livegraph.boolean-propagation.v1","spec_hash":"b91770625bde6908e3a7b5a7c97a6fb28fce12500fd45abec5f5427e24805e21","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-fc08b084320bf3e75ae8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v07\n- v01 --supports--> v00\n- v05 --refutes--> v03\n- v04 --refutes--> v09\n- v03 --refutes--> v04\n- v0e --supports--> v0a\n- v00 --supports--> v02\n- v0a --supports--> v0c\n- v10 --supports--> v05\n- v02 --refutes--> v06\n- v0a --refutes--> v0b\n- v04 --supports--> v01\n- v0d --refutes--> v0f\n- v04 --supports--> v08\n- v0c --refutes--> v0d\n- v02 --supports--> v0e\n\n[VERIFIED ROOT]\nv10 = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f56516babdac1c52a2ecf7dffe6bfe470811f7c54201f61f64d1bb33e73d2a1f","schema":"livegraph.boolean-propagation.v1","spec_hash":"fc08b084320bf3e75ae8368a69a16fd898a82b8652ca6a4d039bf66ab5268649","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-47e0575c6d320f3ec20e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v05\n- v0e --supports--> v09\n- v02 --supports--> v0f\n- v06 --supports--> v0d\n- v08 --supports--> v0e\n- v06 --supports--> v07\n- v10 --refutes--> v04\n- v0d --refutes--> v0b\n- v0c --refutes--> v03\n- v0b --refutes--> v08\n- v01 --refutes--> v0a\n- v05 --refutes--> v02\n- v10 --supports--> v0c\n- v03 --supports--> v01\n- v02 --supports--> v06\n- v0b --supports--> v00\n\n[VERIFIED ROOT]\nv10 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"13964a10093b01510481214ae4a1a0f0493f1849150ce7952f2e24a361f022ed","schema":"livegraph.boolean-propagation.v1","spec_hash":"47e0575c6d320f3ec20eae232c249f1acde9a71c84414c0e7a0d8a35533d194e","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-7ee032de1ee2791d01d9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v00\n- v0b --refutes--> v05\n- v07 --supports--> v0a\n- v10 --supports--> v07\n- v0d --refutes--> v02\n- v0b --refutes--> v0c\n- v04 --supports--> v0f\n- v07 --supports--> v08\n- v0d --supports--> v0b\n- v01 --supports--> v0e\n- v0e --refutes--> v10\n- v0d --supports--> v03\n- v0f --supports--> v06\n- v09 --refutes--> v0d\n- v06 --refutes--> v01\n- v08 --supports--> v09\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c88c185a2a64f95ae98097e41487c3370eedbe048b54f7a11e718b789bb30091","schema":"livegraph.boolean-propagation.v1","spec_hash":"7ee032de1ee2791d01d94bb3bca32782a8a69d3eee6921513a147facada46f24","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-d692fbc612ab3d784856","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v0b\n- v09 --supports--> v04\n- v0d --supports--> v06\n- v0f --supports--> v10\n- v09 --supports--> v0a\n- v03 --refutes--> v0e\n- v0e --refutes--> v02\n- v05 --refutes--> v0f\n- v02 --refutes--> v01\n- v0d --refutes--> v07\n- v09 --supports--> v00\n- v01 --supports--> v0c\n- v0a --refutes--> v05\n- v08 --supports--> v03\n- v0c --supports--> v0d\n- v0d --supports--> v09\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6f709f09e6174f6bd5067e68c2cf15bbb89710dd4046c15a10167fc46ed8a2bb","schema":"livegraph.boolean-propagation.v1","spec_hash":"d692fbc612ab3d7848561c8d5b354edc1a225aaf0cf37894323dcc2061312ec0","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-599b3977fef284bb6484","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v02\n- v08 --refutes--> v0a\n- v02 --supports--> v0d\n- v09 --supports--> v10\n- v04 --supports--> v08\n- v0b --supports--> v0f\n- v06 --supports--> v04\n- v05 --refutes--> v09\n- v03 --refutes--> v05\n- v0a --refutes--> v0c\n- v02 --refutes--> v07\n- v0e --refutes--> v03\n- v10 --refutes--> v06\n- v09 --supports--> v01\n- v0b --refutes--> v0e\n- v06 --supports--> v00\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"726a78b70537bc117eb442c6a490b653c3064426dfaf916ae5b394d0fcf07955","schema":"livegraph.boolean-propagation.v1","spec_hash":"599b3977fef284bb6484af9133219c83223359876f47887bdfa4e412a08c45b6","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-fdcf37dc793deb376a53","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v00\n- v00 --refutes--> v0c\n- v03 --refutes--> v06\n- v0a --supports--> v0b\n- v08 --supports--> v0e\n- v10 --refutes--> v09\n- v0d --supports--> v10\n- v04 --supports--> v0f\n- v03 --supports--> v0a\n- v0a --supports--> v02\n- v0a --refutes--> v07\n- v0e --refutes--> v03\n- v04 --supports--> v05\n- v07 --refutes--> v01\n- v10 --refutes--> v04\n- v0c --refutes--> v08\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0d6a97948bdb737dab31becca5e148dcc5c450e35033435c13393174cfab814b","schema":"livegraph.boolean-propagation.v1","spec_hash":"fdcf37dc793deb376a539cbb7e507bc5b24d448e7536516a12293121d3735daf","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-34da2f285089dde530a1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v0a\n- v0d --supports--> v02\n- v0c --supports--> v06\n- v04 --refutes--> v05\n- v10 --supports--> v04\n- v03 --supports--> v0c\n- v09 --supports--> v08\n- v0e --supports--> v09\n- v0b --supports--> v0d\n- v0f --supports--> v07\n- v06 --refutes--> v0f\n- v04 --refutes--> v0e\n- v06 --supports--> v01\n- v0f --refutes--> v00\n- v07 --supports--> v0b\n- v09 --supports--> v03\n\n[VERIFIED ROOT]\nv10 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"26dd609345949541272f2ccf29a490ac3018c3a280d673556aea9bbe03efed76","schema":"livegraph.boolean-propagation.v1","spec_hash":"34da2f285089dde530a15ed9cb10924a0e01b1ca4482cde7621f5b3d65974b88","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-0d914f5462c90e70d760","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v01\n- v07 --supports--> v08\n- v05 --refutes--> v0e\n- v10 --supports--> v09\n- v10 --refutes--> v0b\n- v08 --supports--> v10\n- v00 --supports--> v05\n- v0c --refutes--> v07\n- v08 --refutes--> v06\n- v04 --refutes--> v00\n- v0b --supports--> v04\n- v0f --refutes--> v02\n- v0b --refutes--> v0a\n- v07 --refutes--> v0d\n- v03 --supports--> v0f\n- v0e --refutes--> v03\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4109688037fff9bd19066680f81bd1cd9ea5423e9cbcaf748543a6a4b94c4bde","schema":"livegraph.boolean-propagation.v1","spec_hash":"0d914f5462c90e70d76052dfc07c6b9467fe97cac752195dec3d7b052ef1f619","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-9152549be369629a91fb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v0b\n- v0c --refutes--> v0d\n- v09 --refutes--> v06\n- v0c --refutes--> v03\n- v0d --supports--> v09\n- v0e --refutes--> v02\n- v0a --supports--> v10\n- v0d --supports--> v04\n- v01 --supports--> v0c\n- v02 --refutes--> v07\n- v02 --refutes--> v05\n- v0b --refutes--> v08\n- v10 --supports--> v01\n- v08 --refutes--> v0f\n- v01 --supports--> v00\n- v08 --refutes--> v0a\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b5ac391d915666024773413677aa51ee46ece0daaa49c09eee7fc9ca446151ef","schema":"livegraph.boolean-propagation.v1","spec_hash":"9152549be369629a91fbbcce5022a0b79786ec3a4b81126fc52b8f4585e3a2d7","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-93ed562121bcfe57452d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v00\n- v0e --refutes--> v10\n- v01 --refutes--> v09\n- v0b --supports--> v01\n- v00 --supports--> v06\n- v0e --refutes--> v0b\n- v0f --refutes--> v05\n- v09 --refutes--> v0a\n- v02 --refutes--> v0e\n- v0e --supports--> v03\n- v01 --refutes--> v07\n- v0a --refutes--> v08\n- v02 --refutes--> v0d\n- v08 --refutes--> v04\n- v04 --refutes--> v0f\n- v00 --refutes--> v0c\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e64c65353e89ef16dd5cb7621d2e79cab88566ec70f57aedb6d97cab276d3a86","schema":"livegraph.boolean-propagation.v1","spec_hash":"93ed562121bcfe57452d1ab86fa2c2ab277e2e7e0827d3f46d8cc2f5ad3944b4","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-7b716920f7aa006475ad","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v0e\n- v03 --supports--> v10\n- v10 --refutes--> v0d\n- v0d --refutes--> v07\n- v00 --refutes--> v04\n- v01 --refutes--> v00\n- v05 --refutes--> v0a\n- v0e --refutes--> v0c\n- v0b --refutes--> v0f\n- v0d --refutes--> v0b\n- v00 --supports--> v05\n- v0b --supports--> v09\n- v06 --supports--> v08\n- v0a --supports--> v06\n- v0d --refutes--> v02\n- v0c --refutes--> v01\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c0b38dc22c974b62fbfbd6a4ac39b33ba9c319ac126679b7790833c528a6b415","schema":"livegraph.boolean-propagation.v1","spec_hash":"7b716920f7aa006475adc63419177bae1913bbfde8ff1ebbfb3a290d6eef1dfa","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-b8b454bcc1606d8b1604","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v03\n- v0f --supports--> v09\n- v07 --supports--> v01\n- v0c --refutes--> v07\n- v07 --refutes--> v0a\n- v07 --refutes--> v08\n- v0b --supports--> v10\n- v10 --supports--> v0e\n- v06 --supports--> v0f\n- v01 --supports--> v02\n- v03 --refutes--> v05\n- v00 --supports--> v0b\n- v02 --refutes--> v00\n- v0c --refutes--> v04\n- v01 --supports--> v0d\n- v0f --supports--> v0c\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fbfd4683b8daa74ff9e5a581cf13d50a770d656e2b0938ed67a8831d440bc686","schema":"livegraph.boolean-propagation.v1","spec_hash":"b8b454bcc1606d8b1604fb8f879d8739cd35a5cc325a1467cea7ce90cb1571c0","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-c625d74762c941e671ec","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v0e\n- v03 --supports--> v09\n- v0a --supports--> v02\n- v07 --refutes--> v0d\n- v10 --refutes--> v07\n- v0e --refutes--> v10\n- v00 --refutes--> v03\n- v0d --refutes--> v01\n- v0e --supports--> v0c\n- v0a --refutes--> v08\n- v0f --refutes--> v05\n- v06 --refutes--> v00\n- v03 --supports--> v0a\n- v0e --supports--> v04\n- v08 --refutes--> v0b\n- v05 --supports--> v06\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"035c0275b1c35ac59af7a00ae5658b0cf1d37a223a257af87d3143deda4f253b","schema":"livegraph.boolean-propagation.v1","spec_hash":"c625d74762c941e671ec0af34220f001741bb6fc83b53345b17b17eb3cf753ca","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-d75a71b72d362c2816eb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v0d\n- v0d --refutes--> v04\n- v10 --supports--> v05\n- v08 --refutes--> v0b\n- v0d --supports--> v09\n- v00 --refutes--> v08\n- v0b --supports--> v0a\n- v00 --refutes--> v03\n- v0e --refutes--> v0f\n- v01 --supports--> v00\n- v04 --refutes--> v01\n- v0d --supports--> v07\n- v05 --supports--> v06\n- v0e --refutes--> v0c\n- v0b --refutes--> v02\n- v0c --refutes--> v10\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"15fda9eb47797ca4c5b63d589ccab32afbbbe2c589f7f6958edc4198c8c9fa3d","schema":"livegraph.boolean-propagation.v1","spec_hash":"d75a71b72d362c2816eb29a2bfa788e6eb8812cd05cee78b826d8ad8175292d7","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-6cec88f2ef12a11b547c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v08\n- v0c --refutes--> v04\n- v01 --supports--> v0e\n- v08 --supports--> v0d\n- v05 --supports--> v01\n- v01 --refutes--> v07\n- v04 --refutes--> v0a\n- v02 --supports--> v0f\n- v09 --refutes--> v03\n- v04 --supports--> v09\n- v09 --supports--> v02\n- v05 --supports--> v06\n- v0d --supports--> v00\n- v0b --supports--> v0c\n- v0d --refutes--> v0b\n- v0f --supports--> v10\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"db62733285c1831eda40f344102116c90395e748d74db349537fb044270f350a","schema":"livegraph.boolean-propagation.v1","spec_hash":"6cec88f2ef12a11b547c03a72e0e6da11c92329498af186604e00dd925cac8ee","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-f71c31dd71082c29bcb2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v10\n- v08 --refutes--> v07\n- v06 --refutes--> v01\n- v02 --supports--> v08\n- v07 --refutes--> v0f\n- v04 --supports--> v0a\n- v09 --refutes--> v0d\n- v0e --supports--> v06\n- v04 --supports--> v09\n- v08 --refutes--> v0c\n- v01 --refutes--> v05\n- v10 --supports--> v0e\n- v04 --refutes--> v03\n- v05 --supports--> v02\n- v02 --refutes--> v00\n- v0d --supports--> v0b\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0aefdc7110b658a266706d31e4686237dc4fc53204b1d3919c98d9764e5e0514","schema":"livegraph.boolean-propagation.v1","spec_hash":"f71c31dd71082c29bcb29836128d0b11c4f0a98416cb4d00b2bdd90b8ef0a4f3","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-a9b8f8226ed1ec009571","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v08\n- v0a --refutes--> v10\n- v04 --supports--> v05\n- v02 --refutes--> v01\n- v0f --supports--> v09\n- v05 --refutes--> v0b\n- v0c --supports--> v07\n- v0e --refutes--> v0f\n- v0e --supports--> v00\n- v10 --refutes--> v02\n- v0b --refutes--> v0a\n- v0c --supports--> v06\n- v01 --refutes--> v0c\n- v06 --supports--> v0d\n- v04 --supports--> v03\n- v06 --supports--> v0e\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bc33771033eec2da640763a61c6c4a6dbbdedc9e50f2b6a5be712ce79346e336","schema":"livegraph.boolean-propagation.v1","spec_hash":"a9b8f8226ed1ec009571de880968e7907740421fd903b1abfd72f030b27f9090","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-ee9b0d5ecbf646ceee95","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v0c\n- v0d --supports--> v01\n- v0c --refutes--> v00\n- v10 --supports--> v02\n- v01 --supports--> v03\n- v0e --refutes--> v08\n- v0e --supports--> v0a\n- v0f --supports--> v09\n- v01 --supports--> v0f\n- v0b --refutes--> v04\n- v0e --supports--> v05\n- v09 --supports--> v10\n- v08 --supports--> v06\n- v06 --refutes--> v0b\n- v0e --refutes--> v07\n- v00 --supports--> v0d\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"48c73755e0c933f4c6912cfb30cd96ca3b0efe852b6202bebdd73a4e22f70a47","schema":"livegraph.boolean-propagation.v1","spec_hash":"ee9b0d5ecbf646ceee955d6ec22638165836b28093179a8a8f1e3a1a0c224cb2","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-2dc0d4ea50a4e22571a5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v05\n- v04 --supports--> v0e\n- v0d --supports--> v01\n- v06 --supports--> v0d\n- v06 --supports--> v07\n- v0f --supports--> v0c\n- v04 --refutes--> v03\n- v0a --supports--> v10\n- v0b --refutes--> v0f\n- v0c --supports--> v00\n- v09 --supports--> v0a\n- v0c --supports--> v08\n- v02 --supports--> v0b\n- v05 --supports--> v02\n- v0d --refutes--> v04\n- v04 --refutes--> v09\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"112271fff3199671f3ddca3c2660d5f89bf49513924049dcd0d040e51c6cdf56","schema":"livegraph.boolean-propagation.v1","spec_hash":"2dc0d4ea50a4e22571a53a8cab21f66ac5be07eb59e7eabc230eddf8a29d3b18","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-eb041765cb9a759b8e49","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v07\n- v0c --supports--> v01\n- v00 --refutes--> v02\n- v07 --supports--> v0f\n- v06 --refutes--> v0d\n- v07 --refutes--> v04\n- v0e --supports--> v06\n- v0c --refutes--> v00\n- v10 --refutes--> v08\n- v05 --refutes--> v10\n- v02 --supports--> v0a\n- v04 --refutes--> v0b\n- v02 --supports--> v03\n- v06 --refutes--> v05\n- v0b --refutes--> v0e\n- v06 --refutes--> v09\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"731e866d26dc7c39b93977f98b458bf3260b95d03f071f70b18d5e21e6980831","schema":"livegraph.boolean-propagation.v1","spec_hash":"eb041765cb9a759b8e492502a930ba8803d9906210beffa832740ebf2994fa93","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-4e0dd0e8ae82dd626d51","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v0c\n- v0d --supports--> v08\n- v02 --refutes--> v00\n- v0d --refutes--> v07\n- v02 --supports--> v03\n- v09 --supports--> v10\n- v04 --refutes--> v06\n- v07 --supports--> v05\n- v00 --refutes--> v01\n- v04 --refutes--> v09\n- v07 --supports--> v0a\n- v01 --supports--> v0e\n- v0a --supports--> v04\n- v0f --supports--> v0d\n- v0a --refutes--> v0b\n- v0e --refutes--> v0f\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7df216d24b48bc390eb1283480d117cb9c190f191518455015c285d21bdd6732","schema":"livegraph.boolean-propagation.v1","spec_hash":"4e0dd0e8ae82dd626d51e86e6d35d9b219ee396000f23f2f38f73eeaccc1c964","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-9ab5617c76ec0e0dc603","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v0e\n- v07 --refutes--> v04\n- v07 --supports--> v10\n- v00 --supports--> v02\n- v06 --refutes--> v0b\n- v0c --supports--> v00\n- v0b --supports--> v0c\n- v02 --refutes--> v09\n- v08 --supports--> v03\n- v0b --refutes--> v01\n- v09 --refutes--> v05\n- v0d --refutes--> v0a\n- v0e --refutes--> v06\n- v08 --supports--> v07\n- v05 --refutes--> v0d\n- v05 --refutes--> v0f\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e0c5177e491e05525d02e33cd83392a105174f3ecb3b7d6a3f3c9725d99fce1d","schema":"livegraph.boolean-propagation.v1","spec_hash":"9ab5617c76ec0e0dc603fc11845bd2b0dfe4b97dfb2144550f8774aee70c6c15","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"}],\"hop\":11,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-72895c5473134cf0c1a2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v0a\n- v10 --supports--> v01\n- v01 --supports--> v0e\n- v09 --supports--> v00\n- v08 --supports--> v06\n- v0b --supports--> v09\n- v04 --refutes--> v05\n- v05 --refutes--> v0c\n- v08 --supports--> v0f\n- v03 --refutes--> v0d\n- v08 --supports--> v10\n- v01 --supports--> v0b\n- v09 --refutes--> v02\n- v0c --supports--> v08\n- v02 --supports--> v07\n- v07 --refutes--> v03\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"81757a35bfeae990f5a89b6967f12ce59458256097395ecc991cca6724c7964b","schema":"livegraph.boolean-propagation.v1","spec_hash":"72895c5473134cf0c1a29bb2abccd5255f2c051223202e82ca0d389d28e5e763","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-c3db6afe7b59a3f3fbba","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v10\n- v03 --refutes--> v02\n- v10 --supports--> v0b\n- v09 --refutes--> v01\n- v0b --supports--> v0c\n- v0e --refutes--> v04\n- v05 --supports--> v0d\n- v00 --supports--> v05\n- v06 --refutes--> v00\n- v08 --supports--> v03\n- v04 --refutes--> v07\n- v09 --supports--> v08\n- v03 --supports--> v0a\n- v04 --refutes--> v09\n- v04 --refutes--> v0f\n- v0c --refutes--> v06\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"37079969b67799e502d277e7b240cf8684551267fc8f58dcea029e994b321090","schema":"livegraph.boolean-propagation.v1","spec_hash":"c3db6afe7b59a3f3fbba5e24f734cdefd0b6f133dea15e1f2436537fae3fc943","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-fd74cf276ff91f423fc9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v04\n- v04 --supports--> v0a\n- v09 --supports--> v08\n- v01 --supports--> v05\n- v07 --refutes--> v09\n- v04 --supports--> v03\n- v0e --refutes--> v01\n- v02 --refutes--> v0b\n- v0a --refutes--> v07\n- v02 --refutes--> v00\n- v0f --supports--> v0e\n- v0c --refutes--> v06\n- v01 --supports--> v02\n- v08 --refutes--> v0f\n- v07 --supports--> v10\n- v0c --supports--> v0d\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c6f567eb8e6f498ace154dbfe0233131e55bc78188781ea617e5720e96be500b","schema":"livegraph.boolean-propagation.v1","spec_hash":"fd74cf276ff91f423fc9aaf9f90c81d73f8093c562870b74d00e2fd6de7b1485","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-1ecf9e72d6089d74c634","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v0d\n- v0c --supports--> v01\n- v0a --refutes--> v03\n- v01 --supports--> v07\n- v02 --supports--> v09\n- v0d --supports--> v05\n- v09 --supports--> v0c\n- v09 --supports--> v06\n- v01 --supports--> v0b\n- v09 --refutes--> v00\n- v0f --supports--> v10\n- v10 --supports--> v04\n- v0b --refutes--> v0a\n- v03 --refutes--> v0f\n- v0a --refutes--> v08\n- v0b --refutes--> v0e\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ec3ff467a951f14bd2c3d0b7b9fdbe154f9d2009c745991d0e7e5a53587b5cee","schema":"livegraph.boolean-propagation.v1","spec_hash":"1ecf9e72d6089d74c634489a53efae1126c9f157869233ad6e7879290182ef8d","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-4f3f1981321d0befa425","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v03\n- v0f --refutes--> v08\n- v01 --refutes--> v0a\n- v02 --supports--> v06\n- v03 --refutes--> v0c\n- v09 --refutes--> v01\n- v0d --supports--> v0e\n- v0e --supports--> v05\n- v05 --refutes--> v04\n- v10 --refutes--> v00\n- v07 --refutes--> v10\n- v10 --supports--> v0d\n- v03 --refutes--> v07\n- v05 --supports--> v09\n- v0f --refutes--> v02\n- v09 --supports--> v0b\n\n[VERIFIED ROOT]\nv0f = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"77b26fc9a0e2ff7313e53ac32fce3412a883678bc0662cc1067e080b206281b8","schema":"livegraph.boolean-propagation.v1","spec_hash":"4f3f1981321d0befa4253942ec9e2c1747d4a76f278d651aa2334ab1fa5bb0b8","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-21916277f0bf1b96fba0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --supports--> v06\n- v0d --refutes--> v03\n- v09 --refutes--> v01\n- v01 --refutes--> v0f\n- v09 --refutes--> v05\n- v0f --supports--> v10\n- v02 --refutes--> v04\n- v0e --refutes--> v0b\n- v0a --refutes--> v0c\n- v00 --refutes--> v0e\n- v0e --refutes--> v02\n- v10 --refutes--> v07\n- v0a --supports--> v0d\n- v00 --refutes--> v08\n- v07 --supports--> v0a\n- v04 --refutes--> v09\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"95bbfcfa5037cc52e0e7d3cd9ef0284812418de27f83b42fe775a198fa19f719","schema":"livegraph.boolean-propagation.v1","spec_hash":"21916277f0bf1b96fba0849bda66dd3c8ccf8fbe77c64e41f014a63c99ce7879","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-b380b1a91f04ffba97b3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v0d\n- v09 --supports--> v0e\n- v0c --supports--> v08\n- v10 --refutes--> v00\n- v06 --supports--> v0f\n- v0e --refutes--> v03\n- v0e --refutes--> v0c\n- v08 --supports--> v10\n- v01 --supports--> v07\n- v10 --refutes--> v04\n- v08 --refutes--> v0b\n- v02 --supports--> v01\n- v0f --refutes--> v02\n- v00 --supports--> v05\n- v07 --refutes--> v09\n- v07 --refutes--> v0a\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ddc833942d1945571cb829821cd4d23b5804ea29e28ca05fd1516fbad6860a88","schema":"livegraph.boolean-propagation.v1","spec_hash":"b380b1a91f04ffba97b363b52080fac747eb256e25c381a8d4219b4bb7a5f42c","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-b4ce65e9ccd692fbaef2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v10\n- v05 --supports--> v0c\n- v0a --supports--> v0f\n- v10 --supports--> v00\n- v01 --refutes--> v0b\n- v10 --refutes--> v09\n- v00 --supports--> v06\n- v00 --refutes--> v03\n- v04 --refutes--> v05\n- v07 --supports--> v04\n- v0f --supports--> v01\n- v0d --refutes--> v02\n- v0c --supports--> v0d\n- v0b --refutes--> v07\n- v0b --refutes--> v0e\n- v0d --supports--> v08\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"00e6e44af7a5da69dff23e2869f32db628d331fba29d8603ef8dcc06d7c6a771","schema":"livegraph.boolean-propagation.v1","spec_hash":"b4ce65e9ccd692fbaef2099a98146f1e90a5431adaa455ffae2d864e4578e80c","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-7cea4d722e2da8b8e443","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v0c\n- v0c --refutes--> v05\n- v01 --refutes--> v0e\n- v03 --refutes--> v0a\n- v05 --refutes--> v10\n- v0a --refutes--> v0b\n- v0b --supports--> v01\n- v01 --supports--> v07\n- v01 --supports--> v00\n- v05 --refutes--> v08\n- v0e --refutes--> v09\n- v09 --supports--> v06\n- v0f --supports--> v0d\n- v0d --supports--> v03\n- v0c --supports--> v02\n- v05 --supports--> v0f\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b79d512a1f7a5f7d344763f994d6cf1b05eddce19f0b36deb74e0fa247cfbb76","schema":"livegraph.boolean-propagation.v1","spec_hash":"7cea4d722e2da8b8e443f1a279fa38f09f35cfd8a3a96668c9a306e7c20723d6","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-3ede9035e2be6b560e04","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v08\n- v03 --refutes--> v0d\n- v02 --supports--> v09\n- v0a --refutes--> v0b\n- v07 --refutes--> v01\n- v02 --supports--> v03\n- v04 --refutes--> v10\n- v04 --refutes--> v0e\n- v0d --refutes--> v04\n- v10 --supports--> v0f\n- v0b --supports--> v0c\n- v07 --refutes--> v02\n- v00 --refutes--> v0a\n- v05 --refutes--> v07\n- v0f --supports--> v00\n- v10 --refutes--> v06\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"38d2151a28e3bd9bff67e90eac2e5679ba91f16a888736f832e2824997fb38d9","schema":"livegraph.boolean-propagation.v1","spec_hash":"3ede9035e2be6b560e0437f3c158441ca1b428ffda59018f0abf4cdec0399c33","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-caf4ce21af02f866b542","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v05\n- v07 --refutes--> v03\n- v0b --refutes--> v0e\n- v07 --supports--> v10\n- v0e --refutes--> v00\n- v0f --supports--> v0d\n- v04 --refutes--> v01\n- v05 --supports--> v06\n- v03 --refutes--> v0c\n- v0b --refutes--> v0a\n- v04 --refutes--> v0b\n- v00 --supports--> v0f\n- v0c --supports--> v02\n- v08 --supports--> v04\n- v0c --supports--> v08\n- v03 --supports--> v09\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"57fa2728a4d3146c75de2d53d1bb219be1b6dc03c8843260b7726bd4a6938d3e","schema":"livegraph.boolean-propagation.v1","spec_hash":"caf4ce21af02f866b542342f927da52bc03b560c4755139db383b6c21c1210a5","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-a3b792664fbd55b11882","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v03\n- v02 --supports--> v09\n- v0d --supports--> v06\n- v0c --supports--> v0f\n- v0b --refutes--> v02\n- v0e --supports--> v10\n- v01 --refutes--> v0b\n- v10 --refutes--> v0d\n- v02 --refutes--> v05\n- v0c --refutes--> v04\n- v0b --supports--> v08\n- v0d --refutes--> v01\n- v07 --refutes--> v00\n- v0f --refutes--> v0e\n- v09 --refutes--> v07\n- v00 --supports--> v0a\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2c95957b8d3b362a1a4c419013f43150fd5dffe69432e3eaf526d1bdb86e22e0","schema":"livegraph.boolean-propagation.v1","spec_hash":"a3b792664fbd55b1188277ac2765e98c3a4c0d34c777f64c057d9fb653c4fa0e","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-f44114906189c181b142","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v0d\n- v00 --supports--> v0e\n- v0f --supports--> v09\n- v0f --supports--> v03\n- v04 --supports--> v05\n- v09 --supports--> v01\n- v0e --refutes--> v10\n- v0b --refutes--> v04\n- v05 --refutes--> v0a\n- v07 --refutes--> v06\n- v0a --supports--> v00\n- v08 --supports--> v0b\n- v0e --refutes--> v07\n- v01 --supports--> v08\n- v07 --refutes--> v02\n- v08 --refutes--> v0c\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"97f951aa9083b2c70a86fe7f3d11e67903ffe25154b73f02c41dd392d2741078","schema":"livegraph.boolean-propagation.v1","spec_hash":"f44114906189c181b1420bc48a498890c0cd9e8ece10e373b0edbb3ab477aae4","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-af4c4cfcbc22d24dfcf7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v0f\n- v12 --supports--> v00\n- v0f --refutes--> v0a\n- v04 --supports--> v12\n- v01 --refutes--> v09\n- v0f --refutes--> v05\n- v0c --supports--> v08\n- v0c --refutes--> v06\n- v12 --refutes--> v01\n- v0b --supports--> v04\n- v07 --refutes--> v0e\n- v09 --refutes--> v11\n- v03 --refutes--> v0c\n- v0f --supports--> v10\n- v07 --supports--> v02\n- v11 --supports--> v03\n- v10 --refutes--> v0b\n- v04 --refutes--> v0d\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9fd6d9552f1de1c1b042a8674a8384bb4c0720940a95124328fbb4d4c22b241b","schema":"livegraph.boolean-propagation.v1","spec_hash":"af4c4cfcbc22d24dfcf7780c5817329ba4207ae29b25b49c2a14601126629e05","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-f89b802ccc844e178349","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --supports--> v0d\n- v11 --refutes--> v0f\n- v0b --refutes--> v06\n- v10 --refutes--> v03\n- v11 --refutes--> v05\n- v05 --refutes--> v00\n- v06 --supports--> v0e\n- v0a --supports--> v12\n- v02 --refutes--> v11\n- v05 --refutes--> v07\n- v01 --supports--> v0b\n- v10 --refutes--> v0c\n- v0d --refutes--> v04\n- v04 --supports--> v02\n- v09 --supports--> v10\n- v06 --supports--> v09\n- v10 --refutes--> v08\n- v05 --supports--> v01\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dae752fd81bb9e5f66ec28a4218a5f4105a093ae68fb0af823fb75cd77474056","schema":"livegraph.boolean-propagation.v1","spec_hash":"f89b802ccc844e178349daed03675550edc1373f59222a3cbca0486acb38928b","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-b33077c80edb74c0e6a6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v08\n- v12 --refutes--> v07\n- v10 --supports--> v02\n- v0a --supports--> v05\n- v01 --refutes--> v0b\n- v0a --supports--> v12\n- v0b --refutes--> v0a\n- v06 --supports--> v09\n- v0d --supports--> v06\n- v12 --refutes--> v10\n- v10 --refutes--> v0f\n- v03 --supports--> v11\n- v10 --supports--> v00\n- v08 --supports--> v0d\n- v03 --refutes--> v0c\n- v0d --refutes--> v0e\n- v02 --refutes--> v04\n- v04 --supports--> v03\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"40cffb25ee536dfe9f31e7f50cf13f4d8469096a3aca087d1b031d4e2dfdfbf0","schema":"livegraph.boolean-propagation.v1","spec_hash":"b33077c80edb74c0e6a644339234b31b8b6d39190821136f9c0b7210eeafd5ae","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-5720e17484815a2f3ee9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v0f\n- v0c --refutes--> v03\n- v0d --supports--> v07\n- v05 --supports--> v0b\n- v0a --refutes--> v0c\n- v07 --refutes--> v00\n- v06 --refutes--> v12\n- v08 --refutes--> v0a\n- v0b --refutes--> v01\n- v12 --refutes--> v10\n- v01 --supports--> v08\n- v0c --supports--> v0e\n- v00 --supports--> v06\n- v0e --refutes--> v0d\n- v06 --supports--> v02\n- v08 --refutes--> v09\n- v0a --supports--> v04\n- v0e --refutes--> v11\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2aedeee5f76cfe328d4f0f74ac061e23fac9bb2674bf284b5f4c71b085baf5f5","schema":"livegraph.boolean-propagation.v1","spec_hash":"5720e17484815a2f3ee952dedeb25024c4f54e665372b38415bf207fd5cff8d3","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-76a7d58861543b928ecf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v04\n- v03 --refutes--> v01\n- v0c --refutes--> v0e\n- v05 --supports--> v0a\n- v0a --supports--> v06\n- v0c --refutes--> v02\n- v03 --supports--> v11\n- v0c --refutes--> v12\n- v06 --refutes--> v08\n- v08 --refutes--> v07\n- v0e --refutes--> v10\n- v07 --refutes--> v0b\n- v01 --refutes--> v0d\n- v09 --refutes--> v05\n- v09 --supports--> v0f\n- v0b --supports--> v03\n- v10 --refutes--> v09\n- v08 --refutes--> v00\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c1545d5eaa5b6277b7489512ca702365f84cfbf715b45f700e1b296fe73d32c6","schema":"livegraph.boolean-propagation.v1","spec_hash":"76a7d58861543b928ecfb50bab6c6b9ce92e2f3cc98907e8e4f0fbc18b62b6fc","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-61a8edf87189f408b7a2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v0a\n- v01 --supports--> v10\n- v11 --supports--> v06\n- v0d --supports--> v11\n- v0b --supports--> v05\n- v01 --supports--> v09\n- v05 --supports--> v00\n- v0b --refutes--> v04\n- v06 --supports--> v02\n- v10 --supports--> v07\n- v0e --refutes--> v01\n- v10 --supports--> v0b\n- v0a --supports--> v0c\n- v0c --supports--> v03\n- v0b --refutes--> v12\n- v01 --supports--> v08\n- v0c --refutes--> v0f\n- v00 --refutes--> v0d\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"59e0c0141e95f1ff0b8340e45ea90aea62fbbb5e5806530f81ab94d94a9c94f6","schema":"livegraph.boolean-propagation.v1","spec_hash":"61a8edf87189f408b7a28b3e898aae4aa86ba1d553f9a9aa1f1d5c97ba555073","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-d7be4db5c90dfe58ce8f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v08\n- v12 --refutes--> v0f\n- v04 --supports--> v11\n- v0d --supports--> v0a\n- v0a --refutes--> v12\n- v0e --supports--> v00\n- v01 --refutes--> v09\n- v04 --supports--> v10\n- v0b --supports--> v03\n- v0e --supports--> v0b\n- v06 --supports--> v04\n- v0f --supports--> v01\n- v12 --supports--> v02\n- v07 --supports--> v05\n- v09 --supports--> v0c\n- v04 --refutes--> v0e\n- v07 --supports--> v0d\n- v08 --supports--> v07\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0396af97ab00ed64fbdfacab6018801a3b38af0896ad3fd16f0581bc14e36cf3","schema":"livegraph.boolean-propagation.v1","spec_hash":"d7be4db5c90dfe58ce8f62e91e438d6f406a6c629de5a3b3acfca5d02d6be800","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-8d4c78055a359af4e6df","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v05\n- v07 --supports--> v06\n- v12 --supports--> v08\n- v0a --refutes--> v0e\n- v03 --refutes--> v11\n- v09 --supports--> v00\n- v07 --refutes--> v0d\n- v0e --supports--> v01\n- v12 --supports--> v07\n- v0c --refutes--> v0a\n- v09 --refutes--> v0f\n- v04 --supports--> v0b\n- v03 --supports--> v12\n- v00 --supports--> v04\n- v0b --supports--> v10\n- v01 --refutes--> v03\n- v09 --refutes--> v02\n- v06 --supports--> v09\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f8edb6a39d97d94fefb15941aeb64c97d11474478190a1f145799b9111155467","schema":"livegraph.boolean-propagation.v1","spec_hash":"8d4c78055a359af4e6df2d2bbe7e5faa85786251bdb33efb89cbba35d42e802f","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-356d4a8549cf79bbb521","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v0c\n- v00 --supports--> v0b\n- v09 --supports--> v05\n- v01 --supports--> v00\n- v04 --refutes--> v07\n- v0f --supports--> v02\n- v0b --supports--> v0d\n- v04 --supports--> v10\n- v0a --refutes--> v01\n- v07 --supports--> v09\n- v0c --supports--> v04\n- v09 --refutes--> v11\n- v0b --refutes--> v06\n- v0f --supports--> v12\n- v0b --supports--> v08\n- v12 --refutes--> v0e\n- v0e --refutes--> v0a\n- v09 --refutes--> v0f\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7036cc21623c224b2303406ef8cf7d7d1b150fbe115840dc5cbd8962d543f987","schema":"livegraph.boolean-propagation.v1","spec_hash":"356d4a8549cf79bbb5215b79c4d3111dc26c8560a8a60c6e12c879dea8653495","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-c50cafa9658c201d7fa0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v0c\n- v0a --refutes--> v03\n- v0b --refutes--> v0e\n- v00 --refutes--> v04\n- v0b --refutes--> v06\n- v0e --refutes--> v07\n- v10 --refutes--> v0a\n- v00 --supports--> v01\n- v0c --supports--> v05\n- v07 --refutes--> v00\n- v08 --refutes--> v09\n- v05 --supports--> v08\n- v09 --refutes--> v10\n- v03 --supports--> v02\n- v07 --supports--> v0d\n- v02 --refutes--> v11\n- v10 --refutes--> v0f\n- v07 --refutes--> v12\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ab6418b85d2ac8496adbf200897e749dde83141dda203d3dc9048faa2285838b","schema":"livegraph.boolean-propagation.v1","spec_hash":"c50cafa9658c201d7fa039781489d674033af618e4652de4df46697560609984","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-fe6c928cfbecad5e45bf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v05\n- v12 --supports--> v0f\n- v0a --supports--> v0e\n- v08 --supports--> v09\n- v02 --supports--> v0b\n- v03 --refutes--> v02\n- v04 --supports--> v08\n- v09 --refutes--> v03\n- v0f --supports--> v0a\n- v08 --supports--> v11\n- v0b --supports--> v01\n- v08 --supports--> v10\n- v0f --refutes--> v0d\n- v0e --supports--> v04\n- v08 --refutes--> v07\n- v0b --refutes--> v00\n- v0b --refutes--> v06\n- v0c --supports--> v12\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"31e4dc504759daee31c8977ef222c61c68471d7fae612485774abcf30dd56c5b","schema":"livegraph.boolean-propagation.v1","spec_hash":"fe6c928cfbecad5e45bfb22cf9af9ec95b480a467908b95b911bf9489d7a5b73","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-802bc2ca890660802705","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v0d\n- v01 --refutes--> v06\n- v00 --supports--> v03\n- v0d --supports--> v11\n- v04 --refutes--> v09\n- v02 --refutes--> v01\n- v11 --supports--> v0b\n- v01 --supports--> v07\n- v08 --refutes--> v05\n- v02 --refutes--> v0f\n- v10 --supports--> v08\n- v0b --supports--> v0a\n- v04 --supports--> v00\n- v03 --supports--> v02\n- v04 --supports--> v0c\n- v06 --refutes--> v10\n- v0b --refutes--> v0e\n- v10 --supports--> v12\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d28a8d0ef1b2135eee4c0abaf6d56466fd6b53e173dcc38b35a8d343b7d69801","schema":"livegraph.boolean-propagation.v1","spec_hash":"802bc2ca890660802705b948e626f8a305b3658ad7d119f29cb9f17f990de617","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-4e77e40dcde6ae7b0904","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v08\n- v03 --refutes--> v01\n- v02 --supports--> v0c\n- v00 --refutes--> v12\n- v12 --refutes--> v03\n- v04 --supports--> v11\n- v09 --supports--> v06\n- v05 --refutes--> v02\n- v09 --supports--> v0d\n- v04 --refutes--> v0e\n- v02 --supports--> v07\n- v0c --refutes--> v00\n- v0a --refutes--> v09\n- v0d --refutes--> v05\n- v11 --refutes--> v0a\n- v03 --supports--> v0f\n- v02 --refutes--> v0b\n- v10 --refutes--> v04\n\n[VERIFIED ROOT]\nv10 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"77a2966e3a1cb086d6315c8f50b477c9aaaf31fddd3517dbc366db18a87642a7","schema":"livegraph.boolean-propagation.v1","spec_hash":"4e77e40dcde6ae7b0904ea3de8b039d34646da610528e5b43df8e1b68cc1c321","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-2efa48e361ec4f622c99","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --supports--> v0f\n- v02 --supports--> v03\n- v0b --supports--> v12\n- v10 --refutes--> v02\n- v0c --refutes--> v09\n- v0e --refutes--> v00\n- v07 --supports--> v10\n- v0e --supports--> v06\n- v01 --supports--> v0c\n- v03 --supports--> v01\n- v00 --supports--> v04\n- v0c --refutes--> v05\n- v0b --supports--> v0d\n- v11 --refutes--> v0e\n- v00 --supports--> v0b\n- v0f --supports--> v07\n- v12 --refutes--> v0a\n- v02 --refutes--> v08\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ae0ed4155f10e5548e2153ec95754d9e2c33f4825af104d316bb8c545c976dd5","schema":"livegraph.boolean-propagation.v1","spec_hash":"2efa48e361ec4f622c99380be6218b160ea52fdf1129f34019a8af2d0d94a2bb","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-1c5f8a7caffbe1ba030f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v0c\n- v04 --refutes--> v0e\n- v0d --supports--> v04\n- v00 --refutes--> v0b\n- v07 --refutes--> v0a\n- v08 --refutes--> v05\n- v12 --refutes--> v07\n- v06 --refutes--> v11\n- v0c --refutes--> v0d\n- v07 --refutes--> v06\n- v0f --refutes--> v00\n- v02 --refutes--> v12\n- v08 --refutes--> v03\n- v02 --refutes--> v10\n- v08 --refutes--> v09\n- v01 --supports--> v08\n- v11 --refutes--> v0f\n- v04 --refutes--> v01\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1724e45eeb4e993d2a60a4df025995812207de0a87cd2d93cb8d966b1b7d7d23","schema":"livegraph.boolean-propagation.v1","spec_hash":"1c5f8a7caffbe1ba030feb557b29ae343892b3290a172fd4be50681e3c0954e7","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-eab777acfce49b446b91","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --supports--> v0a\n- v11 --supports--> v02\n- v05 --refutes--> v03\n- v04 --refutes--> v0e\n- v0b --supports--> v07\n- v0f --supports--> v04\n- v00 --supports--> v0d\n- v0c --supports--> v00\n- v0a --refutes--> v12\n- v0d --supports--> v0f\n- v0a --supports--> v0b\n- v11 --refutes--> v0c\n- v03 --supports--> v01\n- v0f --refutes--> v08\n- v03 --supports--> v09\n- v0d --supports--> v06\n- v0e --refutes--> v05\n- v03 --supports--> v10\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1e18d0ca9aa98fadd91797059ff06322d81fa3c8555eaadce2d42ce7dc432b90","schema":"livegraph.boolean-propagation.v1","spec_hash":"eab777acfce49b446b912872a0beb3ad953a944e1fa7df2536466b726be5b62d","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-7bf6d8ac806f6bd100b1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --supports--> v07\n- v0c --refutes--> v11\n- v04 --supports--> v0d\n- v09 --refutes--> v0c\n- v07 --supports--> v12\n- v0c --refutes--> v01\n- v0a --supports--> v0b\n- v0d --supports--> v06\n- v0f --refutes--> v05\n- v0c --refutes--> v02\n- v0c --supports--> v10\n- v00 --refutes--> v04\n- v03 --supports--> v0a\n- v09 --supports--> v08\n- v11 --supports--> v0e\n- v0a --supports--> v00\n- v05 --supports--> v03\n- v12 --supports--> v0f\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"20aeaa1ed34def2483b29d70b5f706fc9433bf1340f41d39ba8b2ab45d5557d9","schema":"livegraph.boolean-propagation.v1","spec_hash":"7bf6d8ac806f6bd100b1710b95f5057e2d0768f022dcd0904fda86031ea5ac0b","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-865a3cf4fe63e9820e25","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v0c\n- v0b --refutes--> v09\n- v10 --supports--> v12\n- v04 --refutes--> v05\n- v06 --supports--> v07\n- v0f --refutes--> v06\n- v09 --supports--> v04\n- v10 --refutes--> v03\n- v0f --refutes--> v00\n- v02 --supports--> v08\n- v07 --refutes--> v0e\n- v04 --supports--> v0a\n- v11 --supports--> v10\n- v12 --refutes--> v02\n- v02 --supports--> v0d\n- v04 --supports--> v11\n- v0e --supports--> v0b\n- v01 --refutes--> v0f\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"05564a546515abd35c40e851c181840b83406193c73e6db99b45b2c6cd87377c","schema":"livegraph.boolean-propagation.v1","spec_hash":"865a3cf4fe63e9820e2514f6641b820f8c5ce988542d7eca942daf0edf1d5d89","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-12160cb80a05e9ba2463","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v0d\n- v0f --refutes--> v07\n- v10 --refutes--> v0f\n- v07 --supports--> v08\n- v0b --supports--> v00\n- v00 --refutes--> v02\n- v02 --refutes--> v04\n- v0b --refutes--> v09\n- v07 --refutes--> v01\n- v0d --supports--> v0b\n- v0e --refutes--> v0a\n- v00 --supports--> v0c\n- v04 --supports--> v03\n- v02 --refutes--> v06\n- v0d --refutes--> v12\n- v03 --supports--> v11\n- v05 --supports--> v10\n- v03 --refutes--> v05\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"35c832b33f102ae5599fda4d525fbba7400d00d3484adbdab4ffc90137d7dfe0","schema":"livegraph.boolean-propagation.v1","spec_hash":"12160cb80a05e9ba2463142a9e8ecc1559b8b43c87c2146091f5f6cb51d3fa8a","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-164f10613da4ee758b7a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v0f\n- v05 --supports--> v0e\n- v0b --supports--> v08\n- v07 --refutes--> v12\n- v00 --refutes--> v0d\n- v08 --supports--> v04\n- v0e --refutes--> v02\n- v0b --refutes--> v0a\n- v05 --supports--> v01\n- v0b --supports--> v11\n- v06 --refutes--> v03\n- v04 --refutes--> v05\n- v0f --refutes--> v06\n- v02 --refutes--> v10\n- v07 --supports--> v0c\n- v07 --refutes--> v00\n- v02 --supports--> v07\n- v03 --refutes--> v0b\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cca740276bb6083b2a1dae4b268fa7ffcf8054b609847b4ef824b11cb3b260f4","schema":"livegraph.boolean-propagation.v1","spec_hash":"164f10613da4ee758b7a10ca0a36ad64c25771ce360495c0f8c13dfa93322507","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":12,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-44c044d60965e23c6c66","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v02\n- v08 --supports--> v09\n- v12 --supports--> v01\n- v11 --refutes--> v0e\n- v08 --supports--> v00\n- v0d --supports--> v10\n- v07 --supports--> v0f\n- v06 --supports--> v0d\n- v0e --refutes--> v06\n- v0f --refutes--> v03\n- v07 --supports--> v04\n- v0d --supports--> v0c\n- v01 --refutes--> v05\n- v0c --refutes--> v08\n- v0f --refutes--> v11\n- v02 --supports--> v0a\n- v05 --supports--> v07\n- v06 --supports--> v0b\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e08789c0fd2cdc2a979e053d615efcce1c9500bb0069d6d9367e90b7194e8b39","schema":"livegraph.boolean-propagation.v1","spec_hash":"44c044d60965e23c6c66f1efe387a11a885c5d7baf03ac09fc54bc34292fc136","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-97596a3764f4d0977272","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v05\n- v0d --supports--> v0b\n- v0f --supports--> v04\n- v0c --supports--> v02\n- v02 --refutes--> v0f\n- v05 --refutes--> v06\n- v0a --supports--> v09\n- v11 --supports--> v0d\n- v04 --supports--> v11\n- v0b --refutes--> v0a\n- v01 --supports--> v03\n- v0f --supports--> v08\n- v0f --supports--> v0e\n- v06 --refutes--> v00\n- v05 --refutes--> v07\n- v0a --refutes--> v01\n- v06 --supports--> v12\n- v05 --refutes--> v10\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fdd1d244eddf4fc8fb1abcff87ddfaf391c262e1fc0b41fdbdc3212ad5e3c1b8","schema":"livegraph.boolean-propagation.v1","spec_hash":"97596a3764f4d097727274dbba9a38b9347be7bd5118decb6661038e28b8a06c","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-e2a87c40cc2ad422fd48","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v0e\n- v10 --supports--> v07\n- v0b --supports--> v0f\n- v0a --supports--> v12\n- v0f --refutes--> v05\n- v12 --supports--> v01\n- v07 --supports--> v04\n- v09 --supports--> v00\n- v12 --refutes--> v11\n- v11 --refutes--> v06\n- v0d --supports--> v0a\n- v02 --refutes--> v10\n- v0c --refutes--> v09\n- v02 --supports--> v03\n- v10 --supports--> v08\n- v07 --supports--> v0c\n- v0f --refutes--> v02\n- v06 --supports--> v0b\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cd11b588ed167806d12627cc6b2ecf756d50456da551e35ebd5e56ce281b2114","schema":"livegraph.boolean-propagation.v1","spec_hash":"e2a87c40cc2ad422fd4847187f1cf59434f596eec35c3ecf4e32262d69d8de62","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-eea40175d596bddc4d74","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v0b\n- v00 --supports--> v02\n- v0b --supports--> v08\n- v01 --supports--> v03\n- v08 --refutes--> v00\n- v11 --refutes--> v04\n- v05 --refutes--> v06\n- v09 --refutes--> v01\n- v10 --refutes--> v12\n- v06 --refutes--> v09\n- v10 --supports--> v11\n- v0c --refutes--> v0e\n- v11 --refutes--> v0f\n- v07 --refutes--> v10\n- v02 --refutes--> v07\n- v11 --refutes--> v0a\n- v0c --supports--> v05\n- v10 --supports--> v0d\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"65155014d998910e5326e2a4908bdf2115c5f173ca3d16c213e9e81f2cb9f1a7","schema":"livegraph.boolean-propagation.v1","spec_hash":"eea40175d596bddc4d746cc8d464546e3715d75485514b0571be37249c93928f","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-7206abcec7276d9723ad","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v0c\n- v03 --refutes--> v0f\n- v06 --supports--> v0b\n- v01 --refutes--> v02\n- v03 --refutes--> v0e\n- v02 --refutes--> v05\n- v01 --refutes--> v07\n- v0a --refutes--> v0d\n- v0b --refutes--> v10\n- v0b --supports--> v08\n- v0f --refutes--> v06\n- v02 --supports--> v11\n- v12 --supports--> v00\n- v00 --supports--> v0a\n- v0d --supports--> v01\n- v06 --supports--> v04\n- v0c --refutes--> v12\n- v06 --supports--> v09\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7743d9506269c1a95c4e0bb263caeee1c6df0788d153f5f3dc9e741031a5fb80","schema":"livegraph.boolean-propagation.v1","spec_hash":"7206abcec7276d9723ada998ef6a487189d5fa2447b7af1ec78deb7784570f20","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-a99e8b285fe498bf8f01","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v0f\n- v08 --supports--> v01\n- v0b --refutes--> v0e\n- v11 --refutes--> v0b\n- v02 --refutes--> v05\n- v00 --supports--> v11\n- v01 --supports--> v03\n- v10 --supports--> v0a\n- v0e --supports--> v0c\n- v0e --supports--> v02\n- v0f --refutes--> v10\n- v09 --refutes--> v12\n- v06 --refutes--> v04\n- v02 --refutes--> v08\n- v06 --refutes--> v0d\n- v07 --refutes--> v00\n- v09 --supports--> v06\n- v01 --supports--> v09\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2a4da1414ad95821932161659d965bdd3b64c8d9fd42fa9a405e88eb7530ff59","schema":"livegraph.boolean-propagation.v1","spec_hash":"a99e8b285fe498bf8f0165a24bd31b0beffeed920251afd66239eafd46e0e5e4","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-6ff4082b8488cb85578d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --supports--> v00\n- v00 --supports--> v09\n- v01 --refutes--> v02\n- v08 --refutes--> v12\n- v12 --supports--> v05\n- v01 --supports--> v0f\n- v10 --refutes--> v03\n- v09 --supports--> v01\n- v0d --refutes--> v0b\n- v05 --refutes--> v07\n- v0c --refutes--> v04\n- v06 --supports--> v10\n- v02 --supports--> v0d\n- v07 --refutes--> v06\n- v0d --refutes--> v0e\n- v02 --refutes--> v0a\n- v02 --refutes--> v11\n- v0c --supports--> v08\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3ed76a513666cb40bdfb11ec360c06aedee20eeaef6b3bbbc9cb9ed38fc524c7","schema":"livegraph.boolean-propagation.v1","spec_hash":"6ff4082b8488cb85578deefe6a30f54657f1662f96c2f119facab67e0e124b25","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-849a38a450e14c965d5e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v03\n- v05 --refutes--> v10\n- v06 --supports--> v0e\n- v0f --supports--> v11\n- v07 --refutes--> v0b\n- v08 --refutes--> v01\n- v05 --supports--> v07\n- v0e --supports--> v12\n- v06 --supports--> v04\n- v11 --supports--> v06\n- v0d --supports--> v00\n- v0e --refutes--> v02\n- v0c --supports--> v05\n- v0b --refutes--> v0f\n- v02 --supports--> v0d\n- v0d --supports--> v09\n- v08 --supports--> v0c\n- v00 --supports--> v0a\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b0b0f70a7858530d56be103217bfe3484a3fb5894fa4431c33023970dc80ddb9","schema":"livegraph.boolean-propagation.v1","spec_hash":"849a38a450e14c965d5ec3a5cffc7e936cee8d4577e626b3c4d3ef7e7839ecf2","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-d26d157d91183c5bd7fd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v01\n- v05 --supports--> v10\n- v12 --refutes--> v09\n- v0e --supports--> v02\n- v09 --supports--> v0a\n- v03 --refutes--> v12\n- v0a --supports--> v04\n- v05 --supports--> v06\n- v06 --supports--> v0f\n- v0b --refutes--> v03\n- v04 --supports--> v0e\n- v02 --refutes--> v08\n- v08 --refutes--> v07\n- v0e --refutes--> v0d\n- v06 --refutes--> v0c\n- v06 --supports--> v00\n- v0c --refutes--> v0b\n- v0e --supports--> v11\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"57e64bbb187ac3ef511eb5ba1a000bfb2a9c98beb209ff52d3f3e29a79a61239","schema":"livegraph.boolean-propagation.v1","spec_hash":"d26d157d91183c5bd7fda75681ce500af46be3c6f2c38a3e0ad2bde5207e0690","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-e1cc600a87d14dc25613","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --supports--> v01\n- v0e --refutes--> v0b\n- v07 --refutes--> v06\n- v08 --supports--> v03\n- v09 --supports--> v08\n- v05 --refutes--> v02\n- v0c --refutes--> v0a\n- v04 --refutes--> v12\n- v02 --supports--> v0d\n- v0c --supports--> v04\n- v08 --supports--> v0c\n- v04 --supports--> v0e\n- v07 --supports--> v10\n- v04 --refutes--> v00\n- v0f --refutes--> v09\n- v02 --refutes--> v07\n- v0b --supports--> v11\n- v11 --refutes--> v05\n\n[VERIFIED ROOT]\nv0f = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9778841e66e7b14376bfa8a846e9394c76814ef88729b938245cb8d22a6fa886","schema":"livegraph.boolean-propagation.v1","spec_hash":"e1cc600a87d14dc25613911a626059cd0433ef81dcf244b846325f07f2f89d62","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-3dca46b86a2ec363da10","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v0f\n- v08 --refutes--> v0c\n- v04 --refutes--> v02\n- v0a --supports--> v08\n- v0d --refutes--> v07\n- v0f --refutes--> v12\n- v0d --refutes--> v0b\n- v06 --refutes--> v10\n- v06 --supports--> v0d\n- v02 --refutes--> v09\n- v12 --refutes--> v01\n- v0e --supports--> v0a\n- v08 --supports--> v06\n- v01 --refutes--> v0e\n- v07 --refutes--> v00\n- v0a --supports--> v03\n- v0e --supports--> v05\n- v0d --supports--> v11\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a16a52e6e548850293860c1d0b76fe00b294ec478aceca04fff3c976eff87298","schema":"livegraph.boolean-propagation.v1","spec_hash":"3dca46b86a2ec363da103be52376e29d2173d58de5e1cfc8749c1300e446ef88","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-1e44591616937fca3de6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v0c\n- v0f --supports--> v0d\n- v08 --refutes--> v0f\n- v02 --supports--> v07\n- v09 --supports--> v06\n- v03 --refutes--> v04\n- v04 --refutes--> v0b\n- v0f --refutes--> v03\n- v10 --supports--> v01\n- v12 --refutes--> v02\n- v10 --supports--> v11\n- v0b --supports--> v10\n- v07 --supports--> v09\n- v03 --supports--> v05\n- v09 --supports--> v00\n- v12 --refutes--> v0a\n- v01 --refutes--> v12\n- v04 --supports--> v0e\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a29a69691a1608a49e8847cab6de5cd7ec7e94319b797ff3946f159789f06dbd","schema":"livegraph.boolean-propagation.v1","spec_hash":"1e44591616937fca3de6a627004f9223064729f53a4088697dffd3faadb1d440","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-b48576a32670b19cff82","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v06\n- v0b --supports--> v12\n- v03 --supports--> v00\n- v09 --refutes--> v11\n- v0b --refutes--> v02\n- v0a --refutes--> v10\n- v0c --supports--> v03\n- v07 --refutes--> v08\n- v03 --supports--> v0b\n- v0b --supports--> v07\n- v0d --supports--> v01\n- v10 --refutes--> v04\n- v01 --refutes--> v05\n- v04 --refutes--> v0d\n- v11 --refutes--> v0c\n- v0b --supports--> v0f\n- v01 --supports--> v09\n- v08 --supports--> v0e\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"53ec4d455b0d9fc2a9b3fc7246107539ca017a71721bae5301c81d3a0bbae633","schema":"livegraph.boolean-propagation.v1","spec_hash":"b48576a32670b19cff8247e1e85c61fb9947c0804437ac68dc41857cfa51ba76","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-971199c3be950acb5faf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v05\n- v06 --refutes--> v0c\n- v03 --supports--> v01\n- v00 --supports--> v0b\n- v0f --refutes--> v10\n- v01 --refutes--> v08\n- v12 --refutes--> v00\n- v10 --supports--> v12\n- v05 --refutes--> v0a\n- v06 --refutes--> v02\n- v10 --refutes--> v04\n- v0f --supports--> v09\n- v05 --refutes--> v03\n- v0d --refutes--> v06\n- v0d --supports--> v11\n- v01 --supports--> v0f\n- v0e --supports--> v07\n- v0b --supports--> v0e\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f8067c13d856e8cd6380581c95a921d9a6390d8667342b5e8a5d66e5eecee273","schema":"livegraph.boolean-propagation.v1","spec_hash":"971199c3be950acb5fafec124610a35fc3dfce4218138e3ce3206c147d69b011","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-54140368ea1707ee8e90","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v12\n- v05 --refutes--> v0e\n- v09 --supports--> v0c\n- v04 --supports--> v0b\n- v08 --refutes--> v11\n- v10 --refutes--> v05\n- v12 --refutes--> v0d\n- v02 --supports--> v09\n- v11 --refutes--> v07\n- v09 --refutes--> v0a\n- v05 --supports--> v04\n- v07 --supports--> v10\n- v01 --supports--> v02\n- v0c --refutes--> v0f\n- v01 --supports--> v03\n- v0c --supports--> v00\n- v0c --supports--> v06\n- v12 --supports--> v01\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9d433e11953c9a6f14b92ecfe2e1c72636087b8d544f1fc97893493cc69b7e31","schema":"livegraph.boolean-propagation.v1","spec_hash":"54140368ea1707ee8e90c8ad24c9d82038106f31c2f98e099a2f88746d2b3758","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-c4d405dd3bd12572dc93","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v02\n- v10 --refutes--> v05\n- v0d --supports--> v10\n- v0e --supports--> v0a\n- v11 --supports--> v07\n- v0a --supports--> v0c\n- v03 --supports--> v0b\n- v06 --supports--> v04\n- v06 --refutes--> v0f\n- v0c --supports--> v06\n- v08 --refutes--> v0e\n- v0b --refutes--> v08\n- v10 --supports--> v00\n- v0f --refutes--> v12\n- v06 --refutes--> v01\n- v10 --supports--> v03\n- v0f --supports--> v09\n- v09 --supports--> v11\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"41bb80e0e76d0e8bf8ff028447d730741e14875ec42b761386a1443c1ab0130b","schema":"livegraph.boolean-propagation.v1","spec_hash":"c4d405dd3bd12572dc934c5119f9917a3072c7ab52788687b716ca09f5a6dea9","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-2175da0f3438ec72c925","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v12\n- v0d --refutes--> v0a\n- v07 --supports--> v01\n- v0b --supports--> v0d\n- v06 --supports--> v08\n- v0e --refutes--> v03\n- v0f --supports--> v04\n- v03 --supports--> v0c\n- v10 --supports--> v0b\n- v0e --refutes--> v09\n- v08 --refutes--> v0e\n- v07 --supports--> v05\n- v12 --refutes--> v07\n- v10 --supports--> v11\n- v07 --refutes--> v00\n- v0a --supports--> v06\n- v03 --refutes--> v02\n- v0f --supports--> v10\n\n[VERIFIED ROOT]\nv0f = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"937952f82b1d437e091fbf4e606029b678e4d8bb06918ef6e1fff0880606f007","schema":"livegraph.boolean-propagation.v1","spec_hash":"2175da0f3438ec72c925effce370e3aca636c19cc8f718cb4831797c65d9f830","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-77dbd885cb24731bb9a3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v0a\n- v01 --refutes--> v07\n- v00 --supports--> v09\n- v05 --supports--> v06\n- v08 --refutes--> v0e\n- v0f --supports--> v02\n- v10 --supports--> v0d\n- v0e --refutes--> v05\n- v0e --refutes--> v0c\n- v0d --supports--> v04\n- v04 --refutes--> v08\n- v00 --refutes--> v0f\n- v01 --refutes--> v00\n- v06 --supports--> v01\n- v11 --supports--> v03\n- v03 --refutes--> v10\n- v00 --refutes--> v0b\n- v0f --refutes--> v12\n\n[VERIFIED ROOT]\nv11 = FALSE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dad8e20b99f82b37d3ec7740f71e5f080c8159daa2d826190c64a1bf61d9a7ba","schema":"livegraph.boolean-propagation.v1","spec_hash":"77dbd885cb24731bb9a32eaf85f26dfd3b218ceac2e8b99d7361b88526d8b58f","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-0fd5a0ff28db058f4b30","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v0e\n- v08 --supports--> v11\n- v04 --refutes--> v0c\n- v0b --refutes--> v08\n- v00 --supports--> v03\n- v07 --refutes--> v12\n- v0a --refutes--> v0b\n- v08 --supports--> v01\n- v07 --refutes--> v00\n- v02 --supports--> v09\n- v05 --refutes--> v07\n- v02 --refutes--> v10\n- v0b --supports--> v0f\n- v10 --refutes--> v04\n- v0c --supports--> v06\n- v03 --refutes--> v0d\n- v11 --refutes--> v02\n- v03 --refutes--> v0a\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b130b7be6ec45853a3a533def580c792cd49bb9c9926fba0ddf329ee213f6ae0","schema":"livegraph.boolean-propagation.v1","spec_hash":"0fd5a0ff28db058f4b304f22ffcdc29a11b2e231f8c08559d129e7e7b4152bbc","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-0dbbb4707e11a2a57139","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v12\n- v06 --supports--> v09\n- v04 --supports--> v03\n- v0c --refutes--> v08\n- v11 --supports--> v01\n- v08 --refutes--> v05\n- v05 --supports--> v0e\n- v0f --refutes--> v06\n- v01 --supports--> v07\n- v07 --refutes--> v04\n- v0b --supports--> v0d\n- v05 --refutes--> v02\n- v03 --refutes--> v0b\n- v0f --refutes--> v10\n- v01 --refutes--> v0a\n- v0b --refutes--> v0f\n- v09 --supports--> v0c\n- v05 --supports--> v00\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b3bb951f9377eb544eb13c38c81e59af2ee69a1035255e52d4f394e3e1634398","schema":"livegraph.boolean-propagation.v1","spec_hash":"0dbbb4707e11a2a5713921b7e13d1d7417ca30425c0c74008afeeee766da6fa1","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-48ada53ad8bfa76e046c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v0a\n- v0a --supports--> v11\n- v0e --refutes--> v08\n- v00 --refutes--> v10\n- v12 --supports--> v07\n- v0e --refutes--> v03\n- v08 --refutes--> v05\n- v05 --refutes--> v00\n- v02 --refutes--> v06\n- v0f --supports--> v0d\n- v04 --supports--> v12\n- v07 --refutes--> v02\n- v0e --supports--> v01\n- v0b --supports--> v0e\n- v00 --supports--> v09\n- v09 --refutes--> v04\n- v02 --supports--> v0f\n- v0a --supports--> v0c\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e2b53d965e5553435b7316c21dc752b6feabdbc5bac86eb7b9c43c0ab42e3462","schema":"livegraph.boolean-propagation.v1","spec_hash":"48ada53ad8bfa76e046ca49b70062afc83ef3883f026f2215703cbb7106b03ee","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-c39fd66f556d50aa3229","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v0a\n- v0e --supports--> v06\n- v01 --refutes--> v03\n- v09 --refutes--> v02\n- v0b --refutes--> v04\n- v0a --supports--> v07\n- v04 --supports--> v00\n- v00 --refutes--> v0e\n- v04 --refutes--> v0c\n- v09 --refutes--> v05\n- v05 --refutes--> v0d\n- v0e --refutes--> v11\n- v07 --refutes--> v12\n- v08 --refutes--> v0b\n- v01 --refutes--> v0f\n- v11 --supports--> v09\n- v07 --supports--> v01\n- v0f --supports--> v10\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7da0b2e4ea9244baa0e8e35230f1d44a804f7b3a1f9946621c85af386a7d71e3","schema":"livegraph.boolean-propagation.v1","spec_hash":"c39fd66f556d50aa3229d3127d0db579419782a3f3c0b1ba030e375eb572b7d7","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-237bf58f5ca028c720ad","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v03\n- v01 --supports--> v0b\n- v0a --refutes--> v05\n- v05 --refutes--> v0e\n- v08 --supports--> v0f\n- v10 --refutes--> v0d\n- v0b --refutes--> v09\n- v0d --supports--> v04\n- v0f --refutes--> v10\n- v04 --supports--> v11\n- v09 --supports--> v02\n- v0a --refutes--> v12\n- v06 --supports--> v08\n- v05 --supports--> v06\n- v06 --supports--> v0c\n- v09 --supports--> v0a\n- v07 --supports--> v00\n- v04 --supports--> v07\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2f8d2533748a4af991b9ce01d62103f47d4e6e2f2ac638c2733f8f22eedfec5d","schema":"livegraph.boolean-propagation.v1","spec_hash":"237bf58f5ca028c720ada5fee1a46ab0019a1ff4643dd92ba26f48dd0e71067c","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-06b9ee2ff8ef16c2f435","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v05\n- v08 --refutes--> v09\n- v02 --refutes--> v04\n- v0f --supports--> v07\n- v03 --refutes--> v0e\n- v0b --supports--> v06\n- v09 --supports--> v11\n- v03 --supports--> v0c\n- v0c --supports--> v01\n- v07 --refutes--> v0b\n- v0c --refutes--> v10\n- v11 --supports--> v03\n- v0f --supports--> v0d\n- v0c --refutes--> v12\n- v06 --refutes--> v08\n- v11 --refutes--> v0a\n- v05 --refutes--> v02\n- v01 --refutes--> v00\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"76d1ff8367c6beb18d2c3c5d1a87ab49085a1c362a8c39ae7875ff809fcd19e4","schema":"livegraph.boolean-propagation.v1","spec_hash":"06b9ee2ff8ef16c2f4353d16dc2d5356dd7f857d56894c126b84b90404c4893a","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-e6a42b028ebea31d0445","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v05\n- v07 --refutes--> v12\n- v0c --supports--> v0e\n- v0d --supports--> v0c\n- v09 --refutes--> v0b\n- v0f --refutes--> v03\n- v03 --supports--> v01\n- v02 --supports--> v0d\n- v08 --refutes--> v0a\n- v0e --refutes--> v04\n- v03 --supports--> v11\n- v08 --supports--> v10\n- v06 --refutes--> v0f\n- v01 --refutes--> v09\n- v09 --supports--> v00\n- v0b --supports--> v02\n- v07 --refutes--> v08\n- v0e --refutes--> v07\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f71ef13c8a353763cd2f5182186d4f12f028fb1a417e329e513a7f112c4e779c","schema":"livegraph.boolean-propagation.v1","spec_hash":"e6a42b028ebea31d044510a43e7f4a69f4a5b6a33405dda0d8993f633b6184b9","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":12,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-45479c1b700a73ddfc88","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v06\n- v00 --refutes--> v0a\n- v02 --supports--> v0c\n- v06 --refutes--> v0e\n- v0e --supports--> v10\n- v02 --supports--> v0b\n- v07 --supports--> v0f\n- v0c --supports--> v09\n- v01 --refutes--> v11\n- v07 --supports--> v05\n- v01 --refutes--> v04\n- v06 --supports--> v03\n- v12 --refutes--> v02\n- v0a --supports--> v01\n- v0a --supports--> v0d\n- v07 --refutes--> v12\n- v09 --refutes--> v00\n- v11 --supports--> v08\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cda68fc5c6ff45e159fe9792659e21c188f4ec1f9670897fbf4f2c43d906cb3a","schema":"livegraph.boolean-propagation.v1","spec_hash":"45479c1b700a73ddfc889b1888fbf05795942a2b94ab0907564c6dafce5b6d31","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-3c65690d27fe8e553163","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --supports--> v08\n- v08 --supports--> v0b\n- v07 --refutes--> v0d\n- v08 --refutes--> v11\n- v0e --refutes--> v03\n- v05 --refutes--> v01\n- v01 --supports--> v0a\n- v12 --refutes--> v06\n- v0a --refutes--> v0f\n- v0b --supports--> v02\n- v0a --refutes--> v0c\n- v0b --supports--> v09\n- v0e --supports--> v04\n- v12 --refutes--> v00\n- v0c --refutes--> v0e\n- v00 --supports--> v05\n- v0e --supports--> v10\n- v0d --supports--> v12\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9339cc9e74cafaccf409250a3493f79b2c99dea1feb7b56037d1ba70ecf27393","schema":"livegraph.boolean-propagation.v1","spec_hash":"3c65690d27fe8e553163460873dd9443c8e2ed5e63e5b0f4eb4964105c836b92","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-2efd78ce4c8287eae878","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --supports--> v04\n- v00 --supports--> v0e\n- v01 --supports--> v02\n- v0b --supports--> v08\n- v03 --supports--> v0f\n- v06 --refutes--> v0c\n- v0c --refutes--> v01\n- v02 --refutes--> v12\n- v02 --refutes--> v0a\n- v0e --refutes--> v06\n- v01 --supports--> v07\n- v04 --supports--> v0b\n- v02 --supports--> v09\n- v12 --refutes--> v10\n- v11 --refutes--> v05\n- v0c --supports--> v0d\n- v11 --supports--> v00\n- v0f --refutes--> v11\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"15d0c726cbbe8fd2273da4e0cf351cb12fed7abca545d2660cccbbdd214a7fe6","schema":"livegraph.boolean-propagation.v1","spec_hash":"2efd78ce4c8287eae87861ea87133256fa11396ebb21c4e170451312d73b0a4a","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-88a6279183a2c06d7bfa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v09\n- v05 --refutes--> v11\n- v0e --supports--> v00\n- v0c --refutes--> v0e\n- v11 --refutes--> v02\n- v11 --refutes--> v04\n- v01 --refutes--> v12\n- v0d --refutes--> v0a\n- v07 --refutes--> v0d\n- v06 --refutes--> v05\n- v06 --refutes--> v10\n- v0b --supports--> v0c\n- v0f --supports--> v07\n- v01 --supports--> v08\n- v04 --supports--> v01\n- v01 --refutes--> v0b\n- v0a --supports--> v06\n- v0c --supports--> v03\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"02ba8ab800d3e858c872a50772aa8b6bffd93460411df1fa100db4c8d7f9c4f6","schema":"livegraph.boolean-propagation.v1","spec_hash":"88a6279183a2c06d7bfaf23d12a58de9b12dca5413b62ea5dad535df66d72168","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-1f5c14aba58f8dd728c7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v05\n- v0a --refutes--> v10\n- v0a --supports--> v09\n- v01 --supports--> v07\n- v0c --refutes--> v08\n- v07 --supports--> v02\n- v02 --refutes--> v00\n- v08 --refutes--> v0e\n- v05 --refutes--> v11\n- v02 --refutes--> v0c\n- v08 --refutes--> v03\n- v08 --supports--> v04\n- v11 --refutes--> v12\n- v0b --refutes--> v0d\n- v09 --refutes--> v0b\n- v0c --refutes--> v0f\n- v0d --supports--> v01\n- v11 --refutes--> v06\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2c2639cf1cb811170f6db084ca15a7e16312f70b2806647eb601082bd7860eff","schema":"livegraph.boolean-propagation.v1","spec_hash":"1f5c14aba58f8dd728c7cdbf7ae1da435ea55796eac7cb120a0708971b0ff4ff","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-f832016df090b783c2ea","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v12\n- v03 --refutes--> v0a\n- v07 --supports--> v05\n- v08 --refutes--> v0c\n- v0c --supports--> v03\n- v01 --supports--> v11\n- v06 --refutes--> v00\n- v01 --supports--> v0e\n- v05 --supports--> v02\n- v07 --refutes--> v10\n- v05 --refutes--> v04\n- v02 --refutes--> v0b\n- v0a --supports--> v0d\n- v00 --supports--> v01\n- v01 --supports--> v08\n- v0f --supports--> v07\n- v0b --refutes--> v09\n- v0a --supports--> v0f\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4994bdf0c1a14d7f56013caa0d3a12542c39b32f2426aa4384a393d396f6d261","schema":"livegraph.boolean-propagation.v1","spec_hash":"f832016df090b783c2ead1f4138f87185b6dc34b1ad42209495c75965c298383","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":12,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-a4c2ea3b9029180e4c9c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --supports--> v07\n- v03 --supports--> v09\n- v05 --refutes--> v0e\n- v03 --refutes--> v0d\n- v09 --supports--> v01\n- v0a --supports--> v05\n- v0f --refutes--> v0a\n- v0f --refutes--> v12\n- v08 --refutes--> v0c\n- v00 --refutes--> v0f\n- v07 --refutes--> v08\n- v10 --refutes--> v11\n- v01 --refutes--> v00\n- v10 --supports--> v02\n- v04 --supports--> v0b\n- v04 --refutes--> v06\n- v08 --refutes--> v04\n- v05 --supports--> v10\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"73c23ed215adee038d7748b54518eb96e31df4dc2ff5bc2ee717aedd7df9143a","schema":"livegraph.boolean-propagation.v1","spec_hash":"a4c2ea3b9029180e4c9c6177bb5ee5517edeec384cf9f4131b7ad161e2e05020","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-71f7defd84499fe6ec66","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --supports--> v11\n- v0f --supports--> v06\n- v0a --supports--> v00\n- v0b --refutes--> v01\n- v12 --supports--> v0d\n- v0e --supports--> v03\n- v10 --refutes--> v0c\n- v10 --refutes--> v02\n- v07 --supports--> v04\n- v0b --refutes--> v0e\n- v09 --supports--> v05\n- v0f --refutes--> v09\n- v02 --refutes--> v12\n- v08 --refutes--> v10\n- v04 --refutes--> v08\n- v00 --supports--> v0b\n- v09 --supports--> v07\n- v12 --refutes--> v0a\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"59479081d1bef7703c04e37491896c77824d4b738839311d90320afd11680eed","schema":"livegraph.boolean-propagation.v1","spec_hash":"71f7defd84499fe6ec66acc949ebc2c57cbd7bdf0b2d481116c7e1682fdb4f4f","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-35c45219086eddb69c61","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v12\n- v00 --supports--> v0f\n- v0d --refutes--> v10\n- v05 --refutes--> v06\n- v0a --refutes--> v02\n- v10 --refutes--> v04\n- v11 --refutes--> v05\n- v01 --refutes--> v0b\n- v07 --refutes--> v03\n- v04 --supports--> v08\n- v03 --refutes--> v11\n- v0f --supports--> v09\n- v01 --supports--> v00\n- v0f --refutes--> v0c\n- v0a --supports--> v07\n- v0c --refutes--> v0a\n- v04 --refutes--> v01\n- v04 --supports--> v0e\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4d8969361fd16ec31d0abe39a39684c2cd2e054eb5b94a755e5f023e33530e62","schema":"livegraph.boolean-propagation.v1","spec_hash":"35c45219086eddb69c610bf2ba837efb31f8a1c6d31d82e52ee1442e4890a6c3","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-6b2f85f6c264b49ad01a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v08\n- v06 --supports--> v03\n- v0f --refutes--> v0e\n- v12 --refutes--> v0b\n- v03 --supports--> v00\n- v04 --refutes--> v0a\n- v0c --refutes--> v09\n- v03 --supports--> v11\n- v03 --refutes--> v12\n- v0b --supports--> v07\n- v0d --refutes--> v0f\n- v05 --refutes--> v0d\n- v0a --refutes--> v06\n- v04 --refutes--> v02\n- v04 --supports--> v01\n- v08 --refutes--> v05\n- v07 --refutes--> v0c\n- v05 --refutes--> v10\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f42f285cee02b2ad4dc2e54f52a3f64ae167697ef58d178697331a608f04855f","schema":"livegraph.boolean-propagation.v1","spec_hash":"6b2f85f6c264b49ad01a04d81f0e0dc17ee6323996c40633400cba75bc6063a6","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-c0840012e30bed5a291b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v06\n- v06 --refutes--> v0b\n- v08 --refutes--> v0d\n- v08 --refutes--> v0f\n- v04 --supports--> v05\n- v04 --refutes--> v09\n- v03 --supports--> v12\n- v02 --refutes--> v01\n- v09 --supports--> v10\n- v0e --supports--> v11\n- v11 --refutes--> v04\n- v10 --refutes--> v07\n- v0b --supports--> v00\n- v02 --supports--> v0c\n- v03 --supports--> v08\n- v10 --refutes--> v02\n- v00 --supports--> v03\n- v03 --supports--> v0a\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bcf8561384b579f652f6aa1a393a3a54ac5f39733534be159f74b4bad439202a","schema":"livegraph.boolean-propagation.v1","spec_hash":"c0840012e30bed5a291b31d79f569024ad8f6f82d4cc23865ec36c5771876470","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-09205a07d3b844a118f0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v06\n- v11 --supports--> v0a\n- v03 --refutes--> v08\n- v06 --supports--> v00\n- v10 --supports--> v04\n- v0a --supports--> v03\n- v0d --supports--> v10\n- v02 --supports--> v01\n- v0a --refutes--> v0f\n- v0b --refutes--> v09\n- v06 --refutes--> v11\n- v09 --supports--> v02\n- v03 --refutes--> v05\n- v05 --refutes--> v0c\n- v06 --refutes--> v0e\n- v02 --supports--> v0d\n- v0d --supports--> v12\n- v0c --refutes--> v07\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"01a1e2362924743c790419c009bec56c9794d8434481fefe60ee1b2253d37fd7","schema":"livegraph.boolean-propagation.v1","spec_hash":"09205a07d3b844a118f0450ebf26f6a074937a1d0780dbc60085ddfcea3d78d1","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-4101dd4fc7bb823b9b73","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v0f\n- v02 --supports--> v0e\n- v00 --supports--> v08\n- v05 --supports--> v09\n- v09 --supports--> v0b\n- v02 --refutes--> v01\n- v09 --supports--> v11\n- v0e --refutes--> v03\n- v05 --refutes--> v0a\n- v10 --supports--> v0d\n- v08 --refutes--> v0c\n- v0b --supports--> v06\n- v03 --supports--> v04\n- v0d --supports--> v00\n- v06 --refutes--> v07\n- v08 --refutes--> v02\n- v04 --refutes--> v05\n- v09 --refutes--> v12\n\n[VERIFIED ROOT]\nv10 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"801678fb03072d6c3abf76639a45d80454bdab602ddfbcd32883a9ecfe5ab601","schema":"livegraph.boolean-propagation.v1","spec_hash":"4101dd4fc7bb823b9b73c33a32b2bd06d54b4bb9275c67aa48c141c43ba693ff","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-2c45bc22b82dc084051d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v02\n- v0b --supports--> v08\n- v0a --refutes--> v03\n- v05 --supports--> v06\n- v00 --refutes--> v04\n- v00 --supports--> v0f\n- v09 --refutes--> v00\n- v03 --supports--> v01\n- v07 --refutes--> v0a\n- v0e --refutes--> v11\n- v11 --supports--> v09\n- v0f --supports--> v0c\n- v01 --refutes--> v0e\n- v05 --refutes--> v0b\n- v10 --supports--> v0d\n- v11 --supports--> v12\n- v0b --refutes--> v10\n- v0f --refutes--> v05\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b3bf3784be8c53bb3ea085a078613b017688ea17fcc602c860010ac2873a4373","schema":"livegraph.boolean-propagation.v1","spec_hash":"2c45bc22b82dc084051d767c67cef6ea2bd50cd0fcd921f8bbf7d50dbeb224d2","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-85d4f942740811e8c337","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v09\n- v01 --refutes--> v02\n- v08 --refutes--> v0a\n- v01 --refutes--> v12\n- v02 --refutes--> v03\n- v06 --supports--> v0d\n- v0a --refutes--> v0c\n- v02 --refutes--> v06\n- v09 --refutes--> v05\n- v05 --supports--> v0f\n- v09 --refutes--> v0b\n- v0e --refutes--> v04\n- v05 --refutes--> v01\n- v0e --supports--> v00\n- v0c --refutes--> v10\n- v00 --refutes--> v08\n- v10 --refutes--> v07\n- v07 --supports--> v11\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d99830ca47f88935f0c6b689c4fb86a83ed43131b757714fe2b0f794721ba14d","schema":"livegraph.boolean-propagation.v1","spec_hash":"85d4f942740811e8c3374abeaf90ff167879585a4133c4ee4012bb16a6c2e470","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-03cc68a071fd6750a204","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v0e\n- v05 --refutes--> v0f\n- v05 --supports--> v12\n- v0d --refutes--> v02\n- v0b --supports--> v11\n- v08 --supports--> v0b\n- v07 --supports--> v0a\n- v05 --refutes--> v06\n- v02 --refutes--> v09\n- v00 --supports--> v0d\n- v02 --supports--> v10\n- v11 --refutes--> v07\n- v00 --refutes--> v0c\n- v00 --supports--> v03\n- v0a --supports--> v00\n- v09 --refutes--> v05\n- v05 --refutes--> v01\n- v0e --refutes--> v08\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0c313033e2b36b1e217c0022b8d1e558f1e995e5db88d162b0a34c00f484a552","schema":"livegraph.boolean-propagation.v1","spec_hash":"03cc68a071fd6750a204ce02d3997f182f09a54fc930a495133d768c3c2ab53a","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-e4ce9f339b9d277cbf31","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v08\n- v0b --supports--> v01\n- v0c --refutes--> v04\n- v01 --supports--> v10\n- v05 --refutes--> v03\n- v0b --refutes--> v0a\n- v12 --supports--> v0b\n- v00 --supports--> v09\n- v04 --refutes--> v11\n- v06 --refutes--> v12\n- v01 --refutes--> v05\n- v11 --refutes--> v07\n- v03 --supports--> v0c\n- v0b --supports--> v0f\n- v09 --refutes--> v0e\n- v0c --supports--> v0d\n- v0e --refutes--> v06\n- v00 --supports--> v02\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d1176daec112e6cd677bc52fba79869d036797b27ecd1edc14c454e48462400e","schema":"livegraph.boolean-propagation.v1","spec_hash":"e4ce9f339b9d277cbf3158b0e17441dbfe3ac3b98f6ec00f6cb96547d84d6bfc","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-f7db3fc54543f8996643","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v07\n- v08 --supports--> v0c\n- v0b --supports--> v08\n- v00 --supports--> v11\n- v0f --refutes--> v0e\n- v0f --refutes--> v09\n- v07 --supports--> v01\n- v0d --supports--> v02\n- v01 --supports--> v0a\n- v02 --refutes--> v12\n- v04 --supports--> v0f\n- v0f --refutes--> v05\n- v12 --supports--> v04\n- v01 --supports--> v0d\n- v01 --supports--> v10\n- v11 --refutes--> v0b\n- v0d --supports--> v06\n- v08 --refutes--> v03\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"18ef3faabb549079c1da3637e926c47f0640715a2c104b24c10fed6e69cb157c","schema":"livegraph.boolean-propagation.v1","spec_hash":"f7db3fc54543f899664354591762414e62d2a4fbb090c25c43723ac4599e1915","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-7a11445bb35ccc393cb7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v0e\n- v08 --supports--> v11\n- v12 --refutes--> v0d\n- v05 --supports--> v01\n- v02 --refutes--> v04\n- v03 --supports--> v10\n- v10 --refutes--> v07\n- v11 --refutes--> v05\n- v0d --supports--> v00\n- v0d --refutes--> v0c\n- v07 --supports--> v0a\n- v0a --supports--> v08\n- v0c --supports--> v0f\n- v0e --supports--> v02\n- v05 --supports--> v12\n- v08 --supports--> v09\n- v0d --refutes--> v0b\n- v05 --refutes--> v06\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3a3b4fa10024c38a08d69d6c67be2575bb1576f97109f43a78ac83ea61d123e4","schema":"livegraph.boolean-propagation.v1","spec_hash":"7a11445bb35ccc393cb72c8263fd23973a34de02aa9c2926091e89a032e7da08","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-7d20cabffbae2d70053e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v09\n- v0d --refutes--> v0b\n- v02 --refutes--> v07\n- v09 --refutes--> v04\n- v06 --refutes--> v05\n- v04 --refutes--> v0d\n- v02 --refutes--> v08\n- v10 --refutes--> v01\n- v12 --refutes--> v0f\n- v06 --refutes--> v10\n- v11 --refutes--> v0c\n- v12 --refutes--> v03\n- v01 --supports--> v11\n- v11 --supports--> v0a\n- v02 --supports--> v06\n- v12 --supports--> v00\n- v0b --supports--> v0e\n- v0e --refutes--> v12\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"480a7a50f534c486589c6618b0892c527b8e0037e113f4d4aadc9151dcd1a13a","schema":"livegraph.boolean-propagation.v1","spec_hash":"7d20cabffbae2d70053e7c8a05493acedc5fc64d17129a87b08ff578f86cf1c9","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-f6c2cbe23a083b745f48","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v10\n- v08 --supports--> v01\n- v05 --supports--> v0b\n- v05 --supports--> v12\n- v09 --supports--> v05\n- v01 --supports--> v0a\n- v0c --supports--> v06\n- v02 --supports--> v07\n- v10 --supports--> v09\n- v0e --refutes--> v03\n- v0c --supports--> v00\n- v0c --refutes--> v0f\n- v0a --supports--> v02\n- v04 --refutes--> v0c\n- v0c --supports--> v11\n- v0c --refutes--> v0d\n- v12 --supports--> v04\n- v03 --refutes--> v08\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0121dc09b384ddd33dd23b40bbea0238954b3e8a7bd33d523dcac86c3ed43858","schema":"livegraph.boolean-propagation.v1","spec_hash":"f6c2cbe23a083b745f483d800beaee2ebc5075231aa4a46209890f6c13a04653","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-ead4b1da1dfce48607b4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v07\n- v0d --supports--> v05\n- v08 --refutes--> v06\n- v02 --supports--> v09\n- v0e --supports--> v0f\n- v01 --supports--> v04\n- v07 --refutes--> v11\n- v11 --refutes--> v02\n- v0a --refutes--> v12\n- v0d --refutes--> v03\n- v0a --supports--> v08\n- v0a --supports--> v0c\n- v0e --refutes--> v00\n- v10 --supports--> v01\n- v02 --supports--> v0b\n- v0e --refutes--> v10\n- v06 --refutes--> v0d\n- v0b --refutes--> v0a\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ecbdccdbbad1524dc142e8e356b886ef3a463251e80b7ceb5329c946c9b68658","schema":"livegraph.boolean-propagation.v1","spec_hash":"ead4b1da1dfce48607b46ff93374f9941fa751df686889e8b8f3b167beb50a25","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-540cd8261e7dff85aa9d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v06\n- v11 --supports--> v02\n- v02 --refutes--> v0b\n- v0b --supports--> v10\n- v0b --refutes--> v03\n- v09 --refutes--> v11\n- v00 --refutes--> v0d\n- v04 --supports--> v0f\n- v0e --supports--> v07\n- v12 --refutes--> v00\n- v01 --supports--> v04\n- v07 --refutes--> v05\n- v0b --supports--> v01\n- v04 --refutes--> v0c\n- v04 --supports--> v08\n- v0a --refutes--> v12\n- v05 --refutes--> v09\n- v00 --supports--> v0e\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c072fbabcada083d5e992b76623f8f546370259c0635c1053cefb8f168a7d480","schema":"livegraph.boolean-propagation.v1","spec_hash":"540cd8261e7dff85aa9d9ef938b2fd9a8c1537ed0104bf7fb6d1d8a331978ead","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-776a43254a5d08410833","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v05\n- v09 --refutes--> v00\n- v02 --supports--> v12\n- v0b --supports--> v01\n- v08 --supports--> v0f\n- v0d --supports--> v08\n- v0a --supports--> v10\n- v07 --supports--> v0d\n- v11 --supports--> v0b\n- v01 --refutes--> v0e\n- v08 --refutes--> v06\n- v09 --supports--> v07\n- v03 --refutes--> v11\n- v02 --refutes--> v09\n- v04 --refutes--> v03\n- v0a --refutes--> v04\n- v0f --refutes--> v0a\n- v04 --supports--> v0c\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fe8c742b0c8735714aee2ea7c3df58d657a9f4c6578edf46d6a80456c10f5a4f","schema":"livegraph.boolean-propagation.v1","spec_hash":"776a43254a5d084108331a52f7388eebb1463244c8822162540c3d73a2fb240a","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-e19a97c932269ff7eddc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v12\n- v11 --supports--> v00\n- v0f --supports--> v0c\n- v04 --refutes--> v10\n- v09 --supports--> v11\n- v11 --refutes--> v06\n- v10 --refutes--> v08\n- v03 --supports--> v0a\n- v0e --supports--> v02\n- v0f --supports--> v0b\n- v03 --supports--> v07\n- v03 --supports--> v05\n- v0c --refutes--> v04\n- v09 --supports--> v01\n- v12 --supports--> v09\n- v0d --supports--> v0f\n- v10 --refutes--> v0e\n- v07 --refutes--> v0d\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3409d58c15075979a333846da88580faa45dce814aa44c911e8a0108c2ed8e56","schema":"livegraph.boolean-propagation.v1","spec_hash":"e19a97c932269ff7eddc886526338694f1ed90f85efadd611c02f243786a26c2","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-78d2e16dfc2e2c285ae6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v0b\n- v0f --refutes--> v09\n- v0c --refutes--> v07\n- v05 --supports--> v0d\n- v0a --refutes--> v02\n- v05 --refutes--> v10\n- v0a --supports--> v11\n- v06 --supports--> v0f\n- v0a --refutes--> v08\n- v08 --refutes--> v01\n- v05 --supports--> v04\n- v10 --supports--> v06\n- v0b --supports--> v0a\n- v09 --refutes--> v03\n- v03 --supports--> v12\n- v0c --supports--> v05\n- v0e --supports--> v0c\n- v0e --refutes--> v00\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a8f240a3cae0866e3b897a349f1bd45b0fa963471af504911a4a1dfd0e8b25fc","schema":"livegraph.boolean-propagation.v1","spec_hash":"78d2e16dfc2e2c285ae6e57c653b7faf0c0de6d36d251c7c53748103afa6bb16","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-9b4b98c0d313aac2f6bc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v0c\n- v00 --refutes--> v03\n- v0a --supports--> v11\n- v0a --refutes--> v06\n- v0d --refutes--> v02\n- v0b --refutes--> v0d\n- v02 --supports--> v00\n- v08 --supports--> v05\n- v11 --refutes--> v01\n- v02 --supports--> v04\n- v05 --supports--> v07\n- v02 --supports--> v12\n- v01 --refutes--> v0e\n- v03 --supports--> v08\n- v01 --refutes--> v09\n- v03 --supports--> v0f\n- v07 --supports--> v0a\n- v0e --refutes--> v10\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b1da7065161a960916f80c6f01a7819e5126ba654f745b1092e8559c59be0027","schema":"livegraph.boolean-propagation.v1","spec_hash":"9b4b98c0d313aac2f6bc14b685173634ef8bdac3e67ded8603b36dc971eb3ab6","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-a6a182af72fa43c37148","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v0f\n- v0c --supports--> v12\n- v07 --supports--> v0c\n- v0b --refutes--> v0d\n- v0c --supports--> v08\n- v11 --supports--> v01\n- v0f --supports--> v10\n- v04 --supports--> v11\n- v0f --refutes--> v00\n- v08 --refutes--> v0b\n- v07 --supports--> v0a\n- v09 --supports--> v02\n- v0d --refutes--> v09\n- v0d --refutes--> v05\n- v02 --refutes--> v03\n- v03 --supports--> v06\n- v01 --supports--> v07\n- v09 --refutes--> v0e\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7b1860f2469ace4f9ad69983df6509c2480f0fb4093ddca2c9d51a352314150e","schema":"livegraph.boolean-propagation.v1","spec_hash":"a6a182af72fa43c37148623b775f6c0a031b60ac45db39575a96760d96d56b3d","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-7f40edbdc881f631307a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v0c\n- v0d --refutes--> v0b\n- v0d --supports--> v03\n- v06 --refutes--> v05\n- v02 --supports--> v04\n- v11 --supports--> v0d\n- v12 --supports--> v06\n- v0e --refutes--> v11\n- v06 --refutes--> v0a\n- v04 --supports--> v12\n- v05 --refutes--> v00\n- v0d --refutes--> v10\n- v04 --refutes--> v08\n- v0d --refutes--> v0f\n- v01 --refutes--> v02\n- v09 --supports--> v0e\n- v0c --supports--> v01\n- v02 --supports--> v07\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0e9d5ba2ad1c977f1bfa6c1f64234a1245b77355d53ffc9c685c91a39cbbff30","schema":"livegraph.boolean-propagation.v1","spec_hash":"7f40edbdc881f631307a64ff0996b895629b745e209a44e335b23c755dbad907","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-afb54254ace17f16177e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v11\n- v02 --refutes--> v08\n- v03 --refutes--> v0f\n- v0b --refutes--> v02\n- v12 --refutes--> v04\n- v12 --supports--> v03\n- v01 --refutes--> v09\n- v07 --refutes--> v06\n- v06 --refutes--> v0b\n- v0e --refutes--> v0d\n- v02 --supports--> v0a\n- v08 --supports--> v0e\n- v07 --refutes--> v10\n- v0e --supports--> v05\n- v05 --supports--> v0c\n- v00 --supports--> v12\n- v0f --refutes--> v01\n- v05 --supports--> v00\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"93a8a6647b9eea2d77bf7f7889e8999fc9fd2da2fd32eb9a80f019d52b6042e6","schema":"livegraph.boolean-propagation.v1","spec_hash":"afb54254ace17f16177eceeaadc00c51530a953f96b309e8bf0becea9909d6ac","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-bc727921dab56a3825b1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v08\n- v0f --refutes--> v09\n- v0e --supports--> v11\n- v0d --refutes--> v06\n- v10 --refutes--> v0c\n- v0e --supports--> v01\n- v0c --supports--> v02\n- v0c --supports--> v00\n- v02 --refutes--> v07\n- v05 --refutes--> v0a\n- v07 --supports--> v03\n- v0d --refutes--> v04\n- v01 --supports--> v05\n- v02 --refutes--> v12\n- v10 --supports--> v0b\n- v09 --refutes--> v10\n- v08 --refutes--> v0f\n- v07 --supports--> v0e\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"76dd104805341b6cadb473544ee8cca08efa3706fb183df652f546a7f51ddc31","schema":"livegraph.boolean-propagation.v1","spec_hash":"bc727921dab56a3825b1e179da848eab4eb5f68323f19ffc913e682fecc36de4","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-f195858f04ff0214ebe2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v00\n- v10 --refutes--> v04\n- v09 --supports--> v0f\n- v0f --supports--> v0c\n- v0b --refutes--> v09\n- v09 --supports--> v03\n- v10 --refutes--> v12\n- v11 --supports--> v01\n- v08 --supports--> v0e\n- v0f --supports--> v0d\n- v0e --refutes--> v02\n- v06 --supports--> v0a\n- v04 --refutes--> v06\n- v11 --supports--> v10\n- v04 --refutes--> v05\n- v07 --refutes--> v11\n- v0c --refutes--> v08\n- v02 --refutes--> v07\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"99e15cc868d4df0b89244efc6c0ac56a1e864992ee72ca3e2d221d5f56b2d178","schema":"livegraph.boolean-propagation.v1","spec_hash":"f195858f04ff0214ebe240b4919991c51a7be55ab697f0937baec852f49f57a0","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-13a3712cfe9216344193","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v08\n- v02 --supports--> v06\n- v03 --supports--> v0d\n- v0a --refutes--> v00\n- v07 --refutes--> v04\n- v0c --supports--> v02\n- v01 --refutes--> v11\n- v0e --supports--> v0a\n- v01 --refutes--> v0f\n- v01 --refutes--> v12\n- v09 --refutes--> v10\n- v0d --refutes--> v05\n- v07 --supports--> v01\n- v06 --supports--> v0e\n- v10 --refutes--> v0c\n- v0d --supports--> v07\n- v02 --supports--> v0b\n- v00 --supports--> v03\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"48621377e3f1698a972da907dc0399a372bee2fc7b1267a45a07b5039d878e29","schema":"livegraph.boolean-propagation.v1","spec_hash":"13a3712cfe9216344193077da650e98bc18fcb2cb13354268d6e59f4299eb2c8","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-4230bf26051171a97418","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v01\n- v06 --refutes--> v08\n- v02 --refutes--> v09\n- v0b --refutes--> v0a\n- v02 --refutes--> v0c\n- v10 --supports--> v11\n- v11 --supports--> v12\n- v07 --refutes--> v0b\n- v03 --refutes--> v04\n- v08 --refutes--> v0f\n- v01 --refutes--> v0e\n- v12 --supports--> v07\n- v0b --supports--> v03\n- v11 --refutes--> v0d\n- v0c --supports--> v10\n- v00 --supports--> v02\n- v01 --supports--> v06\n- v10 --supports--> v05\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"81027dccb88c23678f357a748e7773a48fe11f3470d9c8b6728be2abac0ddd4b","schema":"livegraph.boolean-propagation.v1","spec_hash":"4230bf26051171a97418ffcde44ee011bd5d9106bca280ea3bb15591df95d1d4","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-f4602556f9706bbf4d7f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v00\n- v06 --refutes--> v02\n- v10 --supports--> v01\n- v0d --supports--> v06\n- v05 --refutes--> v0f\n- v01 --refutes--> v0e\n- v06 --refutes--> v07\n- v03 --refutes--> v11\n- v11 --refutes--> v09\n- v09 --supports--> v10\n- v10 --supports--> v08\n- v06 --refutes--> v0a\n- v06 --refutes--> v0b\n- v05 --supports--> v03\n- v06 --refutes--> v05\n- v12 --supports--> v04\n- v00 --refutes--> v12\n- v04 --supports--> v0c\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2baa8122ee5a9d6f258f347991b76864d8db6ad8c7c8250d8bc7dcc7acd34c78","schema":"livegraph.boolean-propagation.v1","spec_hash":"f4602556f9706bbf4d7f3d17b2706ed15e14257c90ba13fe67099b7998b4912c","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-769809f354d375d5827e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v05\n- v07 --supports--> v04\n- v0b --refutes--> v0f\n- v02 --supports--> v0b\n- v0c --refutes--> v03\n- v01 --supports--> v0c\n- v0c --supports--> v12\n- v08 --refutes--> v09\n- v0b --supports--> v07\n- v07 --refutes--> v0e\n- v09 --supports--> v00\n- v00 --refutes--> v10\n- v04 --supports--> v08\n- v06 --supports--> v01\n- v03 --refutes--> v02\n- v0c --refutes--> v0d\n- v00 --supports--> v0a\n- v11 --supports--> v06\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"45502f036b5e412103c42c4bd842905546c7383afefa6aca8b4153a0895538de","schema":"livegraph.boolean-propagation.v1","spec_hash":"769809f354d375d5827ec8aced035f74df1662c4845e8bcd17449bc7ac2ebb40","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-a2610efd273a913a4282","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v05\n- v02 --refutes--> v0a\n- v10 --supports--> v0e\n- v0d --supports--> v08\n- v0f --supports--> v11\n- v0c --supports--> v0f\n- v0d --refutes--> v06\n- v0a --refutes--> v01\n- v12 --supports--> v09\n- v11 --refutes--> v0d\n- v09 --supports--> v02\n- v08 --refutes--> v10\n- v0f --supports--> v03\n- v09 --refutes--> v04\n- v07 --refutes--> v00\n- v05 --refutes--> v12\n- v0a --supports--> v0b\n- v10 --refutes--> v07\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"521ad9a8e13aab931613ad061ef169b2cd8285cc9d3b7302f0f76519dbaa4dc8","schema":"livegraph.boolean-propagation.v1","spec_hash":"a2610efd273a913a42823cc3aa9a1b43a2cfb89b4938429d610f591605608cc7","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-66b1965c20d2c60a19da","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v0c\n- v0e --supports--> v0a\n- v03 --supports--> v0f\n- v0c --supports--> v0e\n- v08 --supports--> v01\n- v06 --supports--> v12\n- v06 --refutes--> v11\n- v02 --refutes--> v06\n- v05 --supports--> v00\n- v12 --supports--> v05\n- v03 --refutes--> v07\n- v0e --refutes--> v03\n- v01 --refutes--> v10\n- v09 --refutes--> v02\n- v05 --supports--> v0d\n- v00 --supports--> v08\n- v0e --refutes--> v04\n- v0b --supports--> v09\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"041cb7e8ec007f057485ae62bfc109f907ff01ae8e7f03b2a3f3f0149cbc7138","schema":"livegraph.boolean-propagation.v1","spec_hash":"66b1965c20d2c60a19da85c8d25264131419fade8ce94bc77068ce3bfdd950c1","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-7cd4241977cd2953f8b1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v0c\n- v07 --refutes--> v0a\n- v05 --refutes--> v08\n- v06 --refutes--> v02\n- v11 --supports--> v0e\n- v0f --supports--> v06\n- v01 --supports--> v05\n- v06 --supports--> v09\n- v09 --supports--> v03\n- v10 --supports--> v12\n- v0e --refutes--> v0d\n- v03 --supports--> v07\n- v0b --refutes--> v00\n- v07 --refutes--> v01\n- v0e --supports--> v0b\n- v05 --supports--> v10\n- v0f --refutes--> v04\n- v00 --refutes--> v0f\n\n[VERIFIED ROOT]\nv11 = FALSE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9317c3b7f8932fcce9088d79c8ffb227457c423449eb1add1d4024b96170e2f2","schema":"livegraph.boolean-propagation.v1","spec_hash":"7cd4241977cd2953f8b177abffa0b2022f54831f80e75e647e0c7b6078401444","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-3f9a22f829a6844f7931","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v0c\n- v06 --refutes--> v0e\n- v00 --refutes--> v10\n- v0e --refutes--> v03\n- v0f --supports--> v04\n- v11 --refutes--> v07\n- v01 --supports--> v0f\n- v0b --supports--> v02\n- v08 --refutes--> v01\n- v0f --supports--> v06\n- v0a --supports--> v00\n- v10 --supports--> v0b\n- v0e --refutes--> v0a\n- v02 --refutes--> v11\n- v0b --supports--> v12\n- v06 --supports--> v09\n- v07 --refutes--> v0d\n- v0b --refutes--> v05\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9b7f3458a8eb80ece5846f04fb80eaa7775701c6038a7567e82ce65f4d4d520e","schema":"livegraph.boolean-propagation.v1","spec_hash":"3f9a22f829a6844f7931e6e8de0487aafcdd1c4bf258cad3c8dc0728e5a21644","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-e537266158caa1cfbecc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v09\n- v02 --refutes--> v0a\n- v0d --supports--> v0e\n- v03 --supports--> v10\n- v0c --refutes--> v0d\n- v02 --supports--> v06\n- v09 --refutes--> v12\n- v0d --refutes--> v07\n- v0b --supports--> v03\n- v02 --refutes--> v00\n- v10 --refutes--> v02\n- v01 --refutes--> v0f\n- v04 --refutes--> v0b\n- v0f --supports--> v0c\n- v09 --refutes--> v01\n- v0b --refutes--> v11\n- v08 --refutes--> v05\n- v08 --refutes--> v04\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5e49f6a9ebfd9bdff4fd4f6f8db6ed5edcc73a37571b39ba880960ccfa91d05a","schema":"livegraph.boolean-propagation.v1","spec_hash":"e537266158caa1cfbecc560954082f46debf1b04ff06e852740f70fb2330f00a","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-191241668cfbd1d4ea0f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v0a\n- v08 --supports--> v0d\n- v05 --supports--> v06\n- v0a --supports--> v11\n- v10 --refutes--> v09\n- v11 --supports--> v07\n- v01 --supports--> v0e\n- v07 --supports--> v01\n- v10 --refutes--> v02\n- v03 --refutes--> v12\n- v0e --refutes--> v10\n- v0d --refutes--> v0c\n- v0c --refutes--> v04\n- v0e --refutes--> v00\n- v02 --supports--> v05\n- v0c --supports--> v03\n- v0a --refutes--> v0b\n- v0a --refutes--> v0f\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5449ea04e9fd46ec6eef7690166a39293755ff39c06fb94fd9ae374cab314d1a","schema":"livegraph.boolean-propagation.v1","spec_hash":"191241668cfbd1d4ea0fc559e29a14786d0cbf609a78b0da797cc7ef8da99aa5","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-fb5254bf8ea098b20736","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v10\n- v04 --refutes--> v0b\n- v10 --refutes--> v0a\n- v01 --refutes--> v08\n- v02 --refutes--> v03\n- v0b --refutes--> v05\n- v12 --refutes--> v0e\n- v01 --supports--> v07\n- v0f --refutes--> v12\n- v0a --refutes--> v0c\n- v03 --supports--> v00\n- v04 --supports--> v0d\n- v08 --refutes--> v09\n- v05 --refutes--> v01\n- v04 --refutes--> v06\n- v08 --refutes--> v02\n- v10 --supports--> v11\n- v0c --refutes--> v04\n\n[VERIFIED ROOT]\nv0f = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"124495c7fbcd731b84ff994e3b4fe9da080e97f9e0035cc166e8310d2407be24","schema":"livegraph.boolean-propagation.v1","spec_hash":"fb5254bf8ea098b2073667d1df493e0a1f3ea54cbb26a49c44f852ec012bc9f4","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-34075b7ee8d486a10a38","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v10\n- v00 --refutes--> v03\n- v02 --refutes--> v07\n- v10 --supports--> v0b\n- v12 --refutes--> v0e\n- v0c --refutes--> v0d\n- v04 --refutes--> v11\n- v03 --supports--> v08\n- v03 --refutes--> v0c\n- v0b --refutes--> v0f\n- v05 --supports--> v09\n- v0e --supports--> v00\n- v06 --supports--> v02\n- v0e --supports--> v01\n- v0d --supports--> v04\n- v0b --refutes--> v05\n- v05 --refutes--> v12\n- v00 --refutes--> v0a\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"37490f00bd62f7acd7efc7e0a16dd058554b6e432a898dbcec4c3a2e1d8baf8b","schema":"livegraph.boolean-propagation.v1","spec_hash":"34075b7ee8d486a10a386b2a91f71613ae52a0eea59b2617538fdd58174f8f92","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-820acde2f68074e341de","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v06\n- v0e --refutes--> v0d\n- v00 --refutes--> v02\n- v05 --refutes--> v07\n- v0e --supports--> v10\n- v03 --refutes--> v00\n- v05 --supports--> v0f\n- v0d --refutes--> v01\n- v09 --supports--> v08\n- v02 --supports--> v0b\n- v11 --supports--> v0c\n- v0d --supports--> v05\n- v03 --refutes--> v0a\n- v02 --supports--> v04\n- v0c --refutes--> v0e\n- v0c --refutes--> v12\n- v08 --refutes--> v03\n- v06 --refutes--> v09\n\n[VERIFIED ROOT]\nv11 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c80e9992d5e147e9cf546e8c970dd9a7d231ebd774498f31863119653951f740","schema":"livegraph.boolean-propagation.v1","spec_hash":"820acde2f68074e341def193753de3904d36779782ea26aaa7d25d47384cb797","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-98fe09f7cd1ae4892e05","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v12\n- v01 --supports--> v02\n- v0b --supports--> v0d\n- v10 --refutes--> v0b\n- v0c --supports--> v0a\n- v08 --refutes--> v0e\n- v0b --refutes--> v0f\n- v02 --supports--> v0c\n- v07 --supports--> v06\n- v0d --supports--> v08\n- v06 --refutes--> v03\n- v11 --refutes--> v01\n- v03 --refutes--> v10\n- v08 --supports--> v11\n- v0d --supports--> v00\n- v0c --refutes--> v05\n- v0a --refutes--> v09\n- v0d --refutes--> v04\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5c869fc03ec52fc832b786728276134d594f7947e2c7daab4dfaf2e5440d4c4a","schema":"livegraph.boolean-propagation.v1","spec_hash":"98fe09f7cd1ae4892e0572e92140471e3dc66a29fef2bf848d6cbe0e6ef852de","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-8a46ea986a258067642c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v02\n- v01 --supports--> v0b\n- v08 --refutes--> v03\n- v0b --supports--> v06\n- v10 --refutes--> v11\n- v0d --supports--> v08\n- v0c --refutes--> v0d\n- v0b --refutes--> v12\n- v0c --refutes--> v0a\n- v0f --refutes--> v10\n- v10 --supports--> v05\n- v0e --supports--> v07\n- v07 --supports--> v01\n- v0f --refutes--> v04\n- v11 --supports--> v0c\n- v06 --supports--> v00\n- v11 --supports--> v09\n- v00 --refutes--> v0f\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"baf4887137f4d020ab8b0a8e2c53525b5693a1f7845e5cca6689c69358523b8a","schema":"livegraph.boolean-propagation.v1","spec_hash":"8a46ea986a258067642ca395dce6908ce4f60e634c45dd250c42b7b3b25d2d16","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-2bbfcbf1d0daa76bb85b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v04\n- v01 --refutes--> v0f\n- v12 --supports--> v10\n- v09 --refutes--> v01\n- v0c --refutes--> v06\n- v07 --supports--> v0e\n- v08 --refutes--> v00\n- v02 --refutes--> v0c\n- v12 --supports--> v03\n- v11 --supports--> v05\n- v10 --supports--> v0a\n- v0e --refutes--> v11\n- v10 --supports--> v07\n- v02 --supports--> v0b\n- v0c --supports--> v12\n- v05 --refutes--> v0d\n- v01 --supports--> v08\n- v00 --supports--> v02\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3fe5245e2f1fca10fb57027973b92ed7b51a88be0506b8161a7ac4cc6771eb7e","schema":"livegraph.boolean-propagation.v1","spec_hash":"2bbfcbf1d0daa76bb85ba0a184b5b201514489bd857c8dd37fb8a5b3d4a73b58","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-c413001a1e001e48c1ca","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v06\n- v04 --supports--> v0c\n- v04 --supports--> v01\n- v0a --supports--> v09\n- v01 --supports--> v12\n- v11 --supports--> v02\n- v08 --supports--> v0d\n- v0a --supports--> v03\n- v0d --supports--> v05\n- v10 --refutes--> v07\n- v11 --refutes--> v0e\n- v12 --refutes--> v0b\n- v10 --refutes--> v11\n- v03 --refutes--> v08\n- v0f --supports--> v0a\n- v0d --refutes--> v10\n- v06 --refutes--> v04\n- v12 --refutes--> v00\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"342460930d322d9ef0a6eabf78ff1560e98c1f67958ede8003df5332a546b892","schema":"livegraph.boolean-propagation.v1","spec_hash":"c413001a1e001e48c1caf335f1be868f8d17f08eda4f7139b1d8246b693f31d3","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-c53c52863656b0c7f593","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v06\n- v0a --refutes--> v07\n- v08 --supports--> v10\n- v04 --refutes--> v01\n- v05 --supports--> v04\n- v07 --supports--> v00\n- v11 --refutes--> v0e\n- v06 --supports--> v0a\n- v00 --supports--> v08\n- v10 --supports--> v11\n- v12 --refutes--> v03\n- v12 --refutes--> v09\n- v11 --refutes--> v02\n- v02 --refutes--> v05\n- v02 --supports--> v0f\n- v04 --refutes--> v12\n- v04 --refutes--> v0b\n- v08 --refutes--> v0d\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7b03bc0756143ab020f1dbc683d6a4075c13a3d8828580ae4b6313a3c4540f7a","schema":"livegraph.boolean-propagation.v1","spec_hash":"c53c52863656b0c7f593a5ce12c3dccc0862f865094f5ee1e486b9460ed95b66","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-f2a43c41f2aa41618551","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v0e\n- v06 --supports--> v0a\n- v0e --supports--> v05\n- v0c --supports--> v08\n- v02 --supports--> v11\n- v04 --refutes--> v02\n- v04 --refutes--> v00\n- v09 --supports--> v10\n- v07 --supports--> v0d\n- v11 --supports--> v0f\n- v09 --refutes--> v06\n- v04 --supports--> v0b\n- v0d --supports--> v04\n- v05 --supports--> v09\n- v09 --supports--> v12\n- v0e --supports--> v03\n- v0a --supports--> v07\n- v07 --supports--> v01\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1f48d23cd393a7daab58bd345da66560c0b1d853fcd8dc0ac82da6108b938bf1","schema":"livegraph.boolean-propagation.v1","spec_hash":"f2a43c41f2aa416185513aa57e1d17f6dafe9c80812b5c151bde5617d56a81ba","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-07b2c8dac5cf9651a3f0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v01\n- v05 --refutes--> v0b\n- v0e --refutes--> v03\n- v01 --refutes--> v12\n- v10 --refutes--> v02\n- v02 --refutes--> v0e\n- v0a --refutes--> v11\n- v0a --supports--> v0d\n- v02 --supports--> v04\n- v09 --supports--> v00\n- v06 --refutes--> v08\n- v12 --refutes--> v0a\n- v03 --supports--> v09\n- v06 --refutes--> v10\n- v11 --supports--> v06\n- v11 --supports--> v0c\n- v01 --refutes--> v07\n- v11 --supports--> v0f\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"440cf60d6b5b36d8cf0087fe14c903d91e5b92a1507eee077938ee5b5b1d27bf","schema":"livegraph.boolean-propagation.v1","spec_hash":"07b2c8dac5cf9651a3f0f860d75424341dade34c821c9c9cb07d7b1e5626cb70","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-7fe738d7aef0c9ee312b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v0c\n- v08 --supports--> v0d\n- v04 --supports--> v0e\n- v0d --refutes--> v01\n- v04 --supports--> v09\n- v0d --refutes--> v03\n- v0c --refutes--> v0b\n- v09 --supports--> v11\n- v0f --supports--> v08\n- v03 --refutes--> v12\n- v0b --supports--> v0a\n- v11 --supports--> v05\n- v08 --refutes--> v10\n- v12 --supports--> v02\n- v07 --refutes--> v00\n- v0a --refutes--> v04\n- v07 --supports--> v0f\n- v12 --refutes--> v06\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a589046c63f6a9f7a269cf2b73b98ba57746d016380595e974dd5e121799311b","schema":"livegraph.boolean-propagation.v1","spec_hash":"7fe738d7aef0c9ee312b51826bdcf3e6026d0c5eb473e806491198db91c79879","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-a9d6fd98e56fefbfdb68","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v04\n- v08 --supports--> v0d\n- v00 --supports--> v0b\n- v0a --supports--> v01\n- v0f --refutes--> v07\n- v03 --refutes--> v06\n- v0d --supports--> v10\n- v0f --refutes--> v12\n- v05 --supports--> v02\n- v09 --refutes--> v11\n- v0a --refutes--> v0f\n- v10 --supports--> v0c\n- v0c --supports--> v05\n- v06 --supports--> v09\n- v00 --refutes--> v0e\n- v09 --refutes--> v00\n- v07 --supports--> v03\n- v05 --supports--> v0a\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e3a56f695d413a9c6fd19065fd4d35fe83040516f489396cd11546ad82dec5c6","schema":"livegraph.boolean-propagation.v1","spec_hash":"a9d6fd98e56fefbfdb689e88f90af06f728a4721a6d0c39f2b24f4de45381f6e","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-5119516ebc97a7297a54","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v01\n- v02 --refutes--> v07\n- v11 --refutes--> v02\n- v00 --supports--> v11\n- v03 --supports--> v0d\n- v00 --supports--> v10\n- v05 --supports--> v12\n- v01 --refutes--> v05\n- v12 --refutes--> v0f\n- v05 --refutes--> v04\n- v0f --refutes--> v00\n- v0e --refutes--> v0b\n- v09 --supports--> v0e\n- v0d --refutes--> v09\n- v02 --refutes--> v03\n- v0a --refutes--> v08\n- v02 --refutes--> v06\n- v02 --supports--> v0c\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5f8ba443a8facdd988b7ee9961059a0549d2e15bf055846fda0d9f868026111e","schema":"livegraph.boolean-propagation.v1","spec_hash":"5119516ebc97a7297a54293b3a4e7b8bf208db08ff53397ae89600691b1135fa","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-973cccba77d15bed6162","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v0f\n- v0d --supports--> v00\n- v0b --supports--> v11\n- v07 --refutes--> v02\n- v0d --refutes--> v05\n- v08 --supports--> v04\n- v08 --supports--> v03\n- v0e --supports--> v0b\n- v07 --supports--> v0a\n- v0d --refutes--> v10\n- v05 --refutes--> v09\n- v09 --supports--> v07\n- v01 --supports--> v06\n- v07 --supports--> v12\n- v0c --refutes--> v0d\n- v01 --refutes--> v0c\n- v04 --supports--> v01\n- v02 --supports--> v0e\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7727c608af17302d97858b9473b1139f378bf1597f324bdb3fd8b941a9110518","schema":"livegraph.boolean-propagation.v1","spec_hash":"973cccba77d15bed6162af0dc4b7dcbcfd07c02375a96fc133bb8c158940464a","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-2fc4986fdd509dcc4974","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v06\n- v09 --refutes--> v12\n- v01 --refutes--> v0d\n- v09 --supports--> v04\n- v0e --refutes--> v11\n- v0d --refutes--> v0c\n- v0a --refutes--> v02\n- v03 --refutes--> v10\n- v0f --refutes--> v03\n- v0e --supports--> v09\n- v10 --supports--> v0e\n- v0b --refutes--> v0a\n- v0d --refutes--> v05\n- v10 --supports--> v00\n- v0c --refutes--> v0b\n- v02 --supports--> v0f\n- v09 --supports--> v08\n- v04 --refutes--> v07\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"08ac7a2eacc318f33d306e9cfdccdbd9895e50df543d0a21900b834cefc08c6c","schema":"livegraph.boolean-propagation.v1","spec_hash":"2fc4986fdd509dcc4974281b9e138a369c216392611902bf0e4f3123a89b464f","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-5ac888a07c88d8c14231","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v03\n- v0c --supports--> v02\n- v12 --supports--> v00\n- v02 --supports--> v0d\n- v05 --supports--> v0f\n- v11 --supports--> v04\n- v0a --refutes--> v0b\n- v08 --supports--> v06\n- v0d --refutes--> v10\n- v04 --refutes--> v05\n- v0d --supports--> v11\n- v0a --supports--> v09\n- v06 --refutes--> v01\n- v06 --refutes--> v0c\n- v03 --supports--> v08\n- v0c --refutes--> v07\n- v05 --refutes--> v12\n- v05 --supports--> v0e\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"57fbb51730d44b94c44f32e900a11bc15d47e995293e7596f077d24a86640262","schema":"livegraph.boolean-propagation.v1","spec_hash":"5ac888a07c88d8c14231bab47dffb19ccc93ddaa9cbfb1c4500cf94ba43e3a84","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-0065aa02a5bb5a1ac589","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v01\n- v12 --supports--> v05\n- v10 --refutes--> v0a\n- v11 --refutes--> v08\n- v03 --supports--> v0f\n- v0a --refutes--> v03\n- v01 --supports--> v00\n- v06 --refutes--> v12\n- v03 --refutes--> v0c\n- v0e --refutes--> v02\n- v03 --refutes--> v0b\n- v08 --supports--> v06\n- v06 --supports--> v0d\n- v08 --refutes--> v04\n- v0f --supports--> v11\n- v00 --supports--> v10\n- v0e --refutes--> v09\n- v03 --refutes--> v07\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"51aa5d8c680f21ef56a671224644934163a534237f3a4984896e5244707f3c3d","schema":"livegraph.boolean-propagation.v1","spec_hash":"0065aa02a5bb5a1ac5895ecf003f1f5034c0082cb33fbc6f223dbe3f6ea80ff8","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-40bb2216369a53c1be99","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v04\n- v0e --supports--> v10\n- v03 --supports--> v09\n- v0d --refutes--> v12\n- v06 --refutes--> v11\n- v06 --supports--> v08\n- v04 --supports--> v01\n- v08 --supports--> v07\n- v12 --supports--> v0f\n- v0d --refutes--> v00\n- v09 --supports--> v0e\n- v07 --supports--> v0d\n- v0b --refutes--> v06\n- v12 --supports--> v0a\n- v01 --refutes--> v03\n- v0e --refutes--> v0b\n- v0c --supports--> v05\n- v0b --supports--> v02\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1438251213a3cd0c02e59b325d43f830d145e9fb19a5bd15cf5359c2d6035178","schema":"livegraph.boolean-propagation.v1","spec_hash":"40bb2216369a53c1be993650c4d54b7d3660506f26d4e4d48be40f11f8fac069","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-ff1c94d719378f7645ea","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v12\n- v00 --supports--> v09\n- v00 --supports--> v01\n- v11 --refutes--> v0f\n- v0a --supports--> v00\n- v12 --supports--> v02\n- v11 --supports--> v0b\n- v06 --supports--> v0d\n- v00 --supports--> v0c\n- v12 --supports--> v07\n- v0a --supports--> v04\n- v11 --refutes--> v0e\n- v0f --refutes--> v03\n- v10 --supports--> v06\n- v01 --supports--> v08\n- v0d --supports--> v11\n- v02 --supports--> v0a\n- v08 --refutes--> v05\n\n[VERIFIED ROOT]\nv10 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cf17bf32ce92a17b9c6534a7481dffb630857eaefa054f7ff299aec7264c747b","schema":"livegraph.boolean-propagation.v1","spec_hash":"ff1c94d719378f7645ea5659f264a42a5d7dc065c777cd2540a057dfdf772371","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-6400d523ddbdec6a666b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v0a\n- v09 --supports--> v12\n- v08 --supports--> v10\n- v08 --supports--> v00\n- v0e --supports--> v02\n- v0a --refutes--> v08\n- v04 --supports--> v06\n- v04 --refutes--> v0f\n- v12 --supports--> v0e\n- v03 --supports--> v07\n- v03 --supports--> v0c\n- v02 --refutes--> v05\n- v00 --refutes--> v11\n- v08 --refutes--> v0d\n- v11 --refutes--> v09\n- v02 --supports--> v0b\n- v02 --refutes--> v01\n- v0f --supports--> v03\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"659e69ba0358cbd5218dee5432a53ba18b58b40edfe9efa71bf3de98c0967a1d","schema":"livegraph.boolean-propagation.v1","spec_hash":"6400d523ddbdec6a666b847218501f8e10529b2b1e25e48e0f9c036bce620c07","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-6ddccd671c05f6a3ffec","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v04\n- v05 --supports--> v0f\n- v0b --refutes--> v0e\n- v0e --refutes--> v07\n- v00 --refutes--> v11\n- v10 --supports--> v09\n- v05 --refutes--> v03\n- v0e --refutes--> v01\n- v10 --refutes--> v05\n- v0c --supports--> v12\n- v05 --supports--> v06\n- v04 --refutes--> v0a\n- v10 --refutes--> v08\n- v0a --refutes--> v00\n- v02 --supports--> v0b\n- v0b --supports--> v0d\n- v12 --supports--> v10\n- v11 --supports--> v02\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f73d62e090a9a0537eb4841fc2e81823d529e59ea4793954aa0571b0f98fe41c","schema":"livegraph.boolean-propagation.v1","spec_hash":"6ddccd671c05f6a3ffec19011e6ecc1ba299a25e69e38755d2273e9d267fc037","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-4bd402426564a8a5f54e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --supports--> v06\n- v0c --refutes--> v00\n- v03 --refutes--> v02\n- v11 --supports--> v10\n- v06 --supports--> v05\n- v0b --supports--> v04\n- v0c --supports--> v0d\n- v02 --supports--> v0a\n- v10 --supports--> v08\n- v08 --refutes--> v01\n- v10 --supports--> v0f\n- v0b --supports--> v12\n- v01 --refutes--> v0c\n- v01 --refutes--> v0e\n- v05 --refutes--> v03\n- v0d --refutes--> v0b\n- v0d --refutes--> v09\n- v10 --supports--> v07\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e01f729091974d89f733286e3c5ff9674c35666640fe47a58f11e7f527ff8b9e","schema":"livegraph.boolean-propagation.v1","spec_hash":"4bd402426564a8a5f54e39391b1112fee804851f661f6a942ae07b500dde937a","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-5d658c4dfc6fa181eb36","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v01\n- v0b --supports--> v0c\n- v01 --refutes--> v06\n- v11 --refutes--> v08\n- v0b --supports--> v09\n- v09 --supports--> v0a\n- v00 --supports--> v0e\n- v06 --supports--> v0f\n- v12 --supports--> v02\n- v00 --refutes--> v12\n- v10 --supports--> v11\n- v12 --supports--> v04\n- v04 --supports--> v03\n- v06 --refutes--> v05\n- v06 --refutes--> v10\n- v08 --refutes--> v07\n- v09 --refutes--> v0d\n- v03 --supports--> v0b\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f50a02e89ecade8bff9d7717226f9a7588b663f0f1eb5348d50457029a795e28","schema":"livegraph.boolean-propagation.v1","spec_hash":"5d658c4dfc6fa181eb36ad1603531b84180f28f3806f953c0e79323b98513c3f","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-a47cb144e5b535c0d2b5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v11\n- v0c --supports--> v02\n- v05 --supports--> v06\n- v09 --refutes--> v03\n- v10 --refutes--> v07\n- v04 --supports--> v09\n- v03 --supports--> v01\n- v11 --refutes--> v00\n- v02 --refutes--> v12\n- v02 --supports--> v0f\n- v11 --supports--> v08\n- v05 --supports--> v0c\n- v01 --refutes--> v0e\n- v00 --refutes--> v0b\n- v02 --supports--> v04\n- v0c --refutes--> v0a\n- v0e --refutes--> v10\n- v03 --refutes--> v0d\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c1a40cf56af12e3ea11f10fbfc9e86d56c3d04774d2780253d72a738072ad91a","schema":"livegraph.boolean-propagation.v1","spec_hash":"a47cb144e5b535c0d2b525c693c35a24999b20a055e0de4047b7ed82876dff50","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-9d145fc96af9c84e69ee","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v0b\n- v08 --refutes--> v0a\n- v0b --supports--> v01\n- v0b --refutes--> v0d\n- v03 --refutes--> v05\n- v12 --refutes--> v06\n- v0d --refutes--> v09\n- v0c --refutes--> v03\n- v0d --refutes--> v11\n- v05 --supports--> v0e\n- v12 --supports--> v00\n- v07 --supports--> v02\n- v09 --supports--> v04\n- v04 --supports--> v12\n- v0a --supports--> v0c\n- v0c --supports--> v10\n- v12 --supports--> v07\n- v04 --refutes--> v0f\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"23b67b28ecd5b1a2b69de4662c45eec8ba32d65e44ff9e8cc3bc3610a1d9fa2c","schema":"livegraph.boolean-propagation.v1","spec_hash":"9d145fc96af9c84e69eeeb0704ddd40db48ec2d2d6509a0d2eee418f19d280d6","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-1c99f4452393bd534521","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v0f\n- v00 --refutes--> v0b\n- v01 --supports--> v04\n- v12 --refutes--> v07\n- v01 --refutes--> v10\n- v03 --supports--> v11\n- v07 --supports--> v0c\n- v09 --supports--> v0a\n- v08 --supports--> v05\n- v0b --supports--> v09\n- v07 --refutes--> v0e\n- v01 --supports--> v08\n- v0a --refutes--> v02\n- v05 --refutes--> v12\n- v0c --refutes--> v0d\n- v0d --supports--> v03\n- v0a --supports--> v01\n- v12 --supports--> v06\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"870138252e602891bc00578aadcc7b7a54476e05227d26a48a098a662c0eac9c","schema":"livegraph.boolean-propagation.v1","spec_hash":"1c99f4452393bd53452192faa396b338b433e044e541e49bdae1a174a6ed4b97","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-deadd798a61e667ad624","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v00\n- v12 --refutes--> v01\n- v09 --supports--> v10\n- v0c --refutes--> v0a\n- v0b --refutes--> v0c\n- v02 --refutes--> v06\n- v04 --refutes--> v0d\n- v06 --refutes--> v12\n- v0f --refutes--> v05\n- v0d --supports--> v11\n- v00 --supports--> v02\n- v12 --supports--> v0f\n- v0f --supports--> v0e\n- v11 --supports--> v08\n- v0a --refutes--> v03\n- v0c --supports--> v07\n- v0f --supports--> v09\n- v11 --supports--> v0b\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ebff262b91dc6bd1fdc741bc6c37dc0b736c9bf9d9c8204bda59320f41c16f71","schema":"livegraph.boolean-propagation.v1","spec_hash":"deadd798a61e667ad6247945b3ba68df6454af8858d01d935886eff5aa8190cc","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-b5197d7b564757ba5d19","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v0d\n- v0f --supports--> v08\n- v0d --supports--> v0c\n- v0d --refutes--> v02\n- v0e --refutes--> v04\n- v0e --refutes--> v10\n- v04 --refutes--> v0a\n- v0d --supports--> v01\n- v0f --refutes--> v00\n- v06 --supports--> v07\n- v0c --refutes--> v0f\n- v07 --supports--> v03\n- v0f --refutes--> v11\n- v09 --supports--> v05\n- v12 --refutes--> v06\n- v09 --supports--> v0b\n- v08 --refutes--> v0e\n- v05 --refutes--> v12\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"02205c9d7b6df5e299ca27952b7f3560f5a3afc337cb5852367fd3852c26a629","schema":"livegraph.boolean-propagation.v1","spec_hash":"b5197d7b564757ba5d1905deb7a6a06ef6d223164da3aaaa19dd8557795acc42","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-64908e65db22747c3023","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v06\n- v09 --supports--> v0f\n- v01 --supports--> v10\n- v11 --supports--> v0c\n- v0b --supports--> v03\n- v0b --supports--> v07\n- v12 --refutes--> v0b\n- v06 --supports--> v04\n- v06 --supports--> v0d\n- v11 --refutes--> v00\n- v0e --supports--> v12\n- v05 --supports--> v09\n- v08 --supports--> v0a\n- v08 --refutes--> v11\n- v0f --supports--> v08\n- v11 --refutes--> v0e\n- v03 --supports--> v02\n- v02 --supports--> v01\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"00fc458e3bb914081054cb9ac095dab88a16b805a58d653ee4035d3b430b2d0f","schema":"livegraph.boolean-propagation.v1","spec_hash":"64908e65db22747c3023b926b9e758eb9c49f2238f2d408673c4eba5114adf28","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-257e04e6c756f3da872f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v12\n- v11 --refutes--> v0c\n- v0a --refutes--> v05\n- v06 --supports--> v09\n- v11 --supports--> v07\n- v08 --supports--> v02\n- v02 --supports--> v11\n- v0d --refutes--> v01\n- v09 --supports--> v0e\n- v10 --supports--> v06\n- v07 --supports--> v0d\n- v01 --refutes--> v04\n- v0a --supports--> v00\n- v02 --refutes--> v0f\n- v03 --refutes--> v0a\n- v05 --refutes--> v10\n- v0a --refutes--> v0b\n- v01 --refutes--> v03\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"05e85827600a1143a700884066d553a28d0e210455d7f2b584ae6eee39cd7919","schema":"livegraph.boolean-propagation.v1","spec_hash":"257e04e6c756f3da872f7d28c55b6b9db380bad60bdb9980204bc519763587fd","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-7d18613e0b63fbea0510","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v11\n- v09 --supports--> v0b\n- v01 --supports--> v07\n- v07 --supports--> v06\n- v06 --refutes--> v03\n- v10 --supports--> v0a\n- v08 --refutes--> v00\n- v04 --supports--> v0e\n- v0a --supports--> v12\n- v05 --refutes--> v08\n- v09 --supports--> v0c\n- v08 --refutes--> v04\n- v03 --supports--> v09\n- v06 --refutes--> v0d\n- v0b --supports--> v05\n- v10 --refutes--> v0f\n- v0e --supports--> v10\n- v0a --supports--> v02\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f49565519553a1359a83ff1569e15144a64ab5ac0998aac9c9a90e7ad56b1294","schema":"livegraph.boolean-propagation.v1","spec_hash":"7d18613e0b63fbea05103eb3a5e8bcb202c5d6f8425f4e6ecca03d999697d6ac","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-8c97272a3d0ad7648e2d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v12\n- v05 --refutes--> v0b\n- v10 --supports--> v00\n- v09 --supports--> v0c\n- v10 --supports--> v08\n- v02 --supports--> v01\n- v11 --refutes--> v0e\n- v0b --supports--> v0f\n- v0c --supports--> v07\n- v02 --refutes--> v03\n- v10 --refutes--> v09\n- v06 --refutes--> v02\n- v04 --refutes--> v05\n- v0d --refutes--> v0a\n- v0e --refutes--> v04\n- v02 --refutes--> v11\n- v07 --supports--> v06\n- v0b --supports--> v0d\n\n[VERIFIED ROOT]\nv10 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"baba9573e86afdca0a1448130fa9e7537d46ce848ac0a1db3db49a1cda6b8830","schema":"livegraph.boolean-propagation.v1","spec_hash":"8c97272a3d0ad7648e2dd3d5f50755cd156fba272334ee19d1a20c4755a7f5e4","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-e1fb094d57e83f94f3a7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v10\n- v08 --supports--> v05\n- v06 --refutes--> v0e\n- v0c --refutes--> v0a\n- v02 --refutes--> v08\n- v04 --supports--> v07\n- v0f --refutes--> v0d\n- v10 --refutes--> v04\n- v08 --supports--> v0f\n- v0a --supports--> v01\n- v03 --refutes--> v0c\n- v01 --refutes--> v09\n- v0d --refutes--> v0b\n- v04 --refutes--> v00\n- v04 --refutes--> v06\n- v10 --refutes--> v11\n- v06 --supports--> v02\n- v04 --supports--> v12\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0a8c70cfda49106c2b3a0951a7852d5a49c05429a295761b9277cdcec72f909a","schema":"livegraph.boolean-propagation.v1","spec_hash":"e1fb094d57e83f94f3a7aa3c0132ef7f04b73f9f157577c16c48f46ce15af489","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-e684bb596b20ef1276ad","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v03\n- v0c --refutes--> v01\n- v02 --refutes--> v00\n- v0f --refutes--> v02\n- v12 --refutes--> v07\n- v0c --supports--> v09\n- v0a --supports--> v0f\n- v0b --refutes--> v0a\n- v06 --supports--> v05\n- v11 --supports--> v04\n- v0b --supports--> v0d\n- v01 --supports--> v11\n- v05 --refutes--> v08\n- v00 --refutes--> v10\n- v05 --refutes--> v0b\n- v12 --supports--> v06\n- v0b --supports--> v0e\n- v11 --refutes--> v12\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9889384bc76dc8ba1a3afaef3a9a99d041be4054ef8fde05faba85ef8894cde6","schema":"livegraph.boolean-propagation.v1","spec_hash":"e684bb596b20ef1276ad96075d33bb7ad008a432e2123b05fd5702a33103502b","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-80214a826fa1625236ee","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v00\n- v0b --supports--> v05\n- v0d --refutes--> v0a\n- v11 --refutes--> v10\n- v09 --refutes--> v04\n- v0b --refutes--> v0f\n- v0c --supports--> v02\n- v04 --supports--> v11\n- v08 --refutes--> v03\n- v0a --refutes--> v08\n- v0c --supports--> v06\n- v05 --refutes--> v0c\n- v06 --supports--> v09\n- v0a --refutes--> v12\n- v0c --supports--> v01\n- v03 --supports--> v0e\n- v08 --refutes--> v07\n- v0e --refutes--> v0b\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"10d0e4208ebcc19ebcef047f8cd77ddfd13b07022b3f851864e85fc174498fe5","schema":"livegraph.boolean-propagation.v1","spec_hash":"80214a826fa1625236ee97a5393efbe65faa8d1ce2b6513a039b709b613568d2","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-e8b59b9cc9820e2b93e3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v00\n- v01 --supports--> v10\n- v03 --supports--> v07\n- v10 --supports--> v0d\n- v04 --supports--> v06\n- v0f --supports--> v01\n- v08 --supports--> v11\n- v0e --refutes--> v12\n- v04 --refutes--> v0b\n- v07 --supports--> v0c\n- v06 --refutes--> v0a\n- v04 --supports--> v02\n- v0a --refutes--> v03\n- v10 --refutes--> v08\n- v07 --refutes--> v09\n- v12 --refutes--> v04\n- v10 --supports--> v05\n- v11 --supports--> v0e\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"282a276c4a392cce2205f91896012609779017bb8db5e89a1994e1ff10a24f4e","schema":"livegraph.boolean-propagation.v1","spec_hash":"e8b59b9cc9820e2b93e3c08baa07ed858956f385b1f10ef7b1439a792adaa055","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-884edc8185a8f76c5d4a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v08\n- v03 --refutes--> v0a\n- v01 --refutes--> v04\n- v0b --refutes--> v05\n- v0c --supports--> v12\n- v03 --refutes--> v0f\n- v0b --supports--> v09\n- v0f --supports--> v11\n- v08 --refutes--> v02\n- v09 --refutes--> v01\n- v01 --refutes--> v0d\n- v00 --supports--> v07\n- v0c --supports--> v0b\n- v09 --supports--> v0e\n- v02 --supports--> v10\n- v00 --supports--> v03\n- v10 --refutes--> v00\n- v06 --refutes--> v0c\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"44ecb2bb0b0a4bbd4ba172c04f8e891547e71734a1a84b54c1453187d9e19bbe","schema":"livegraph.boolean-propagation.v1","spec_hash":"884edc8185a8f76c5d4acbee061a1b248c5a9f4631ed18ff3ad72f3e68cb18e1","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-532f8f196dd8ec7f6cf6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v12\n- v10 --supports--> v11\n- v10 --refutes--> v06\n- v10 --supports--> v01\n- v0d --supports--> v04\n- v09 --refutes--> v05\n- v12 --supports--> v10\n- v04 --supports--> v08\n- v0a --supports--> v09\n- v05 --refutes--> v07\n- v0b --supports--> v0d\n- v0b --supports--> v03\n- v08 --refutes--> v0f\n- v0c --refutes--> v0a\n- v0c --refutes--> v0e\n- v01 --refutes--> v0c\n- v01 --supports--> v00\n- v0a --supports--> v02\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"007149f115e658028c8f54f97a8f1bf43ac2193465c53f22345f7501ed96b9ab","schema":"livegraph.boolean-propagation.v1","spec_hash":"532f8f196dd8ec7f6cf6c24bf809562a5c2e30e1c18a3cba206e518920d924de","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-94c6ca104d744235797f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v0d\n- v05 --supports--> v00\n- v02 --refutes--> v05\n- v10 --refutes--> v08\n- v0c --supports--> v10\n- v06 --refutes--> v09\n- v12 --supports--> v03\n- v06 --supports--> v02\n- v03 --refutes--> v0b\n- v10 --refutes--> v11\n- v11 --refutes--> v07\n- v10 --supports--> v0e\n- v05 --supports--> v01\n- v07 --supports--> v0f\n- v00 --refutes--> v04\n- v0f --refutes--> v12\n- v0b --refutes--> v06\n- v00 --supports--> v0a\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0848f9976f820233e88ec28410dfb6a451d08e08eb61aa012b459c9a3b2a8e43","schema":"livegraph.boolean-propagation.v1","spec_hash":"94c6ca104d744235797fc4822d0b5adbe0c374c12dcd615ee72b7d90801de119","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-71af722e477edfde6be2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v0d\n- v03 --refutes--> v08\n- v08 --supports--> v04\n- v0d --refutes--> v0e\n- v02 --supports--> v06\n- v0d --refutes--> v03\n- v0b --refutes--> v10\n- v11 --supports--> v02\n- v02 --refutes--> v09\n- v08 --supports--> v0a\n- v10 --refutes--> v11\n- v10 --supports--> v05\n- v0f --supports--> v07\n- v04 --supports--> v0c\n- v12 --refutes--> v0f\n- v07 --refutes--> v0b\n- v0b --supports--> v00\n- v03 --refutes--> v01\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5b4369a8a64e39019b83adc2e2bc25685528e1b683c60a6bed0769e82c97a057","schema":"livegraph.boolean-propagation.v1","spec_hash":"71af722e477edfde6be23171f3e23e684766430173c278030947aefc9b9fc976","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-46c8404ae73010610e7e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v0b\n- v01 --refutes--> v06\n- v07 --supports--> v0a\n- v08 --refutes--> v12\n- v12 --supports--> v01\n- v0d --refutes--> v0f\n- v07 --supports--> v10\n- v06 --supports--> v03\n- v05 --supports--> v11\n- v10 --refutes--> v09\n- v0d --refutes--> v0e\n- v03 --refutes--> v07\n- v01 --supports--> v00\n- v09 --supports--> v0c\n- v02 --refutes--> v05\n- v11 --supports--> v0d\n- v0d --refutes--> v08\n- v01 --refutes--> v04\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"01713f73b13ab64bd0713e9750089d145bd363a8f0b85adfe3dd396ce7a370ba","schema":"livegraph.boolean-propagation.v1","spec_hash":"46c8404ae73010610e7e95edb63d52f949aaa399a9061fae96b3e287732c37dd","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-810ff2ec7400bfd7fb6e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v0c\n- v0a --refutes--> v10\n- v0a --refutes--> v02\n- v0b --refutes--> v07\n- v10 --supports--> v04\n- v06 --refutes--> v0f\n- v12 --supports--> v11\n- v09 --refutes--> v05\n- v00 --supports--> v12\n- v04 --supports--> v00\n- v11 --refutes--> v08\n- v03 --refutes--> v09\n- v09 --supports--> v0a\n- v07 --supports--> v06\n- v09 --refutes--> v01\n- v08 --supports--> v0b\n- v07 --refutes--> v0e\n- v00 --supports--> v0d\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"808ea125c3f56f1da9a781b82a840a8e7811918c9f25a969693ef9c23fc1da75","schema":"livegraph.boolean-propagation.v1","spec_hash":"810ff2ec7400bfd7fb6e696629c1699926a7dfa45a20e7ec4744144cf5b0a4a2","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-1d8d0cdddc3ef295b3b9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v0f\n- v02 --refutes--> v00\n- v02 --refutes--> v0b\n- v00 --supports--> v04\n- v07 --refutes--> v05\n- v0c --supports--> v03\n- v02 --refutes--> v08\n- v07 --supports--> v06\n- v03 --supports--> v0a\n- v0a --supports--> v0e\n- v04 --refutes--> v07\n- v0d --refutes--> v0c\n- v01 --refutes--> v10\n- v12 --refutes--> v01\n- v0a --refutes--> v12\n- v05 --supports--> v0d\n- v12 --supports--> v11\n- v00 --refutes--> v09\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a4d8336547e8a5ad0321b0a4c06f5d99b5ff1f8440031ddb5c1e09e57298d116","schema":"livegraph.boolean-propagation.v1","spec_hash":"1d8d0cdddc3ef295b3b9978dbb61240c1553743f5ffb815d1314219b44bab27f","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-b074142b3275b01a023d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --supports--> v00\n- v12 --supports--> v08\n- v01 --supports--> v02\n- v0e --supports--> v11\n- v01 --refutes--> v0d\n- v04 --refutes--> v01\n- v10 --supports--> v0e\n- v03 --supports--> v0f\n- v05 --supports--> v04\n- v0b --refutes--> v07\n- v0f --supports--> v0c\n- v01 --refutes--> v09\n- v10 --supports--> v06\n- v0c --supports--> v12\n- v01 --supports--> v0b\n- v0c --refutes--> v0a\n- v07 --supports--> v10\n- v11 --supports--> v03\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5863385734d03acdb4f4965ed3ab9336d41053d42160b19d5aa7137e40f0e624","schema":"livegraph.boolean-propagation.v1","spec_hash":"b074142b3275b01a023d8e81989d0972a1a049055521b124355d975a9dd4dc24","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-4942dd6dc9f6d4165694","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v02\n- v02 --refutes--> v0e\n- v02 --refutes--> v0a\n- v05 --supports--> v07\n- v0a --supports--> v0d\n- v05 --refutes--> v0c\n- v06 --refutes--> v09\n- v06 --supports--> v11\n- v08 --refutes--> v03\n- v00 --refutes--> v0f\n- v12 --supports--> v08\n- v07 --refutes--> v12\n- v08 --supports--> v10\n- v00 --refutes--> v01\n- v0d --refutes--> v04\n- v0d --supports--> v0b\n- v0f --refutes--> v06\n- v03 --supports--> v00\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3dafc169095a1fa5bbe8ee2aa5df959dca9cc7014e8f7f9d28785260a20f35bd","schema":"livegraph.boolean-propagation.v1","spec_hash":"4942dd6dc9f6d41656948fc87059f993efe957daa658a0440edc916e41489769","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-706632bf5254ee81b2fd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v10\n- v0d --supports--> v12\n- v0a --supports--> v00\n- v01 --supports--> v11\n- v0c --supports--> v08\n- v08 --refutes--> v0f\n- v06 --supports--> v05\n- v00 --refutes--> v02\n- v0e --supports--> v01\n- v02 --supports--> v0b\n- v06 --refutes--> v09\n- v0f --supports--> v0e\n- v0d --supports--> v06\n- v0e --supports--> v07\n- v01 --supports--> v0d\n- v06 --supports--> v03\n- v03 --refutes--> v04\n- v04 --supports--> v0a\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"018ec8622408c0f192672fb16c4f3255bbb8342d9a12f71f38ca008db6a1805f","schema":"livegraph.boolean-propagation.v1","spec_hash":"706632bf5254ee81b2fd49e66c9b2a657b86d219d9f2558344182f57f57e7004","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":12,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-ccdfa14be824cf634222","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v0b\n- v11 --supports--> v01\n- v07 --refutes--> v10\n- v02 --refutes--> v00\n- v00 --refutes--> v03\n- v0b --supports--> v12\n- v00 --supports--> v06\n- v10 --supports--> v0a\n- v05 --refutes--> v0c\n- v06 --refutes--> v05\n- v11 --supports--> v09\n- v0c --refutes--> v11\n- v06 --supports--> v0f\n- v08 --refutes--> v02\n- v11 --supports--> v07\n- v00 --supports--> v0e\n- v08 --supports--> v0d\n- v04 --refutes--> v08\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e784dc4547b5e37a0c5e92c1dd6b5d668d50eafcd2f3a9c30faf7d0200a89d0f","schema":"livegraph.boolean-propagation.v1","spec_hash":"ccdfa14be824cf634222b35995cea77d2b186f5002573d0a607b3df38cc3e108","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-0ef46b146974973ff7a7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v11\n- v02 --supports--> v10\n- v00 --refutes--> v12\n- v12 --supports--> v06\n- v03 --refutes--> v0a\n- v0a --supports--> v0c\n- v0f --supports--> v09\n- v01 --supports--> v03\n- v12 --refutes--> v07\n- v0f --supports--> v0d\n- v02 --refutes--> v08\n- v02 --supports--> v0b\n- v0e --supports--> v0f\n- v11 --supports--> v0e\n- v0f --refutes--> v02\n- v10 --supports--> v05\n- v05 --supports--> v01\n- v0a --refutes--> v04\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"541ddb68343b515f8b6b4da8b73a30870e911b78b1a46f15c9e95ce7d8e809ca","schema":"livegraph.boolean-propagation.v1","spec_hash":"0ef46b146974973ff7a7c2bfbc86a423082517b8128a5996d5cbb413898df9d3","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-2bfbf2ede9eaafa69895","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v10\n- v0d --refutes--> v0f\n- v0e --refutes--> v06\n- v05 --refutes--> v02\n- v07 --refutes--> v11\n- v05 --supports--> v01\n- v08 --refutes--> v0b\n- v02 --refutes--> v12\n- v00 --refutes--> v0c\n- v10 --refutes--> v00\n- v06 --supports--> v05\n- v0e --refutes--> v04\n- v02 --refutes--> v09\n- v03 --supports--> v0d\n- v02 --refutes--> v0a\n- v0f --refutes--> v0e\n- v07 --supports--> v08\n- v08 --supports--> v03\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b18f4b45f9dd8a2cec1388434943ff24c3e5bb9ec83017958acd2dfd2cc6123d","schema":"livegraph.boolean-propagation.v1","spec_hash":"2bfbf2ede9eaafa69895c95b358036b6ff880cd61624efd8454ab69c1c8d7315","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-67d03aa439f900fb3815","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v10\n- v0f --supports--> v05\n- v07 --refutes--> v06\n- v04 --refutes--> v0f\n- v0f --refutes--> v0a\n- v01 --supports--> v02\n- v0f --supports--> v11\n- v0e --supports--> v0d\n- v0e --refutes--> v12\n- v0c --refutes--> v03\n- v0a --supports--> v0b\n- v0e --supports--> v07\n- v09 --supports--> v0e\n- v01 --supports--> v04\n- v00 --refutes--> v01\n- v08 --refutes--> v09\n- v0b --refutes--> v0c\n- v03 --supports--> v08\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"712e646d727dfb75e9db166cadb6a61b2100a1f1ea8e0dfb60af09b5b1da3abf","schema":"livegraph.boolean-propagation.v1","spec_hash":"67d03aa439f900fb3815ead82c096c85b3f7403ca5d395fc112f50602727a4c4","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":12,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-66bb7d13c08e4fa347f3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v10\n- v0c --refutes--> v09\n- v09 --refutes--> v11\n- v00 --supports--> v0b\n- v01 --supports--> v08\n- v07 --refutes--> v00\n- v05 --refutes--> v0c\n- v00 --supports--> v0d\n- v0a --supports--> v06\n- v10 --refutes--> v0a\n- v0d --supports--> v03\n- v07 --supports--> v02\n- v08 --refutes--> v0e\n- v05 --refutes--> v0f\n- v0e --refutes--> v07\n- v03 --supports--> v12\n- v09 --supports--> v01\n- v07 --refutes--> v04\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ae241289b3c2dffbfbb2a7503383ae92bd9def85bd3d60026b86a290ac295a31","schema":"livegraph.boolean-propagation.v1","spec_hash":"66bb7d13c08e4fa347f34b19cad47437874f8fc8c45871e78fcc871033403e9b","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-2ae9d3797e20c73798b1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v06\n- v0c --refutes--> v05\n- v12 --refutes--> v0e\n- v02 --supports--> v0f\n- v10 --refutes--> v00\n- v06 --refutes--> v04\n- v0e --refutes--> v0c\n- v12 --refutes--> v03\n- v06 --refutes--> v0d\n- v0e --refutes--> v08\n- v05 --refutes--> v01\n- v0d --supports--> v09\n- v10 --supports--> v11\n- v0f --supports--> v0b\n- v10 --refutes--> v12\n- v05 --supports--> v07\n- v0a --refutes--> v02\n- v09 --supports--> v10\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5ef842bc703a7594132878bbeecee1ba8c9acadd354d35c29adab426c8c50d19","schema":"livegraph.boolean-propagation.v1","spec_hash":"2ae9d3797e20c73798b187eee981ed0842a2c3a8e18d572de076964912ca034d","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-f5e3be79f0eb6c3cbc6f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v08\n- v0d --supports--> v03\n- v01 --supports--> v0e\n- v02 --refutes--> v0f\n- v07 --refutes--> v11\n- v08 --supports--> v0d\n- v03 --refutes--> v0a\n- v0f --supports--> v0c\n- v03 --supports--> v0b\n- v0a --supports--> v01\n- v02 --refutes--> v06\n- v10 --supports--> v04\n- v11 --refutes--> v02\n- v08 --refutes--> v00\n- v01 --refutes--> v05\n- v01 --supports--> v10\n- v04 --refutes--> v12\n- v09 --supports--> v07\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0f196a91c5302a1d5bb3be021859168b72f2091fbee1cdd51c1c811a5188f1be","schema":"livegraph.boolean-propagation.v1","spec_hash":"f5e3be79f0eb6c3cbc6f90fd27057db121a28b951e97698e4b2a8a89c157ea15","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-01f0f5a90e9b2fb053c2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v0f\n- v0f --refutes--> v06\n- v04 --supports--> v0c\n- v09 --supports--> v0b\n- v09 --supports--> v08\n- v02 --refutes--> v04\n- v0c --supports--> v09\n- v0a --supports--> v12\n- v12 --supports--> v00\n- v0a --refutes--> v05\n- v01 --supports--> v11\n- v03 --refutes--> v01\n- v12 --refutes--> v10\n- v0d --refutes--> v0a\n- v12 --supports--> v02\n- v12 --supports--> v0e\n- v0f --supports--> v03\n- v07 --refutes--> v0d\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8c474a450e6d1f80ce11cf68842afb59fdfe3a9102c2fff64ba8b0e22f58155a","schema":"livegraph.boolean-propagation.v1","spec_hash":"01f0f5a90e9b2fb053c2babcaba0bad6ff01c91413797b74c62c79226cf3bc0a","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-e43b920da3d9919fcaf6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v0f\n- v0b --supports--> v10\n- v08 --refutes--> v06\n- v06 --supports--> v0e\n- v11 --refutes--> v03\n- v02 --refutes--> v0b\n- v0b --supports--> v07\n- v01 --refutes--> v12\n- v05 --supports--> v11\n- v04 --supports--> v05\n- v10 --supports--> v04\n- v06 --supports--> v09\n- v0e --supports--> v00\n- v01 --refutes--> v0a\n- v12 --supports--> v0d\n- v03 --refutes--> v08\n- v04 --supports--> v0c\n- v0e --supports--> v01\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"282134201cb9e48b04065f10fda2e03958d1e611539017a9495f4a0d12baf90e","schema":"livegraph.boolean-propagation.v1","spec_hash":"e43b920da3d9919fcaf65d85e748b7d65192af457c332a708a8de693dc35b39e","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-a59429b09bb7ac3ab607","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v0a\n- v0d --refutes--> v03\n- v10 --supports--> v08\n- v0d --refutes--> v02\n- v06 --refutes--> v07\n- v11 --supports--> v04\n- v09 --supports--> v0c\n- v04 --supports--> v12\n- v00 --supports--> v06\n- v09 --refutes--> v01\n- v05 --supports--> v09\n- v12 --supports--> v10\n- v08 --supports--> v05\n- v0a --refutes--> v00\n- v07 --refutes--> v11\n- v07 --refutes--> v0f\n- v10 --refutes--> v0b\n- v06 --refutes--> v0e\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"17545b58628c9aa00e36141c509c18c56513a3f3b5ddf77864a4a9c584d4328b","schema":"livegraph.boolean-propagation.v1","spec_hash":"a59429b09bb7ac3ab607db29b6d8ec1691483066c0f94ad37694fa5744c70770","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-515c1021fa0799256ef1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v02\n- v0d --refutes--> v0e\n- v04 --refutes--> v11\n- v0b --supports--> v10\n- v08 --refutes--> v0c\n- v11 --supports--> v07\n- v0a --supports--> v0d\n- v07 --refutes--> v03\n- v0c --refutes--> v05\n- v0b --refutes--> v04\n- v06 --refutes--> v0b\n- v12 --supports--> v0a\n- v11 --supports--> v01\n- v07 --supports--> v00\n- v05 --supports--> v06\n- v05 --supports--> v09\n- v0d --supports--> v08\n- v0f --supports--> v12\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9caaec62f1f42ef93d5d9358052ea5fcca90d67823fe92b90c8fbe5e559f58e1","schema":"livegraph.boolean-propagation.v1","spec_hash":"515c1021fa0799256ef1c51372f47d2f90b61ab6a7909bff6de14daa15f60a7c","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-f678cbd41aca531f6334","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v10\n- v08 --supports--> v06\n- v04 --supports--> v0d\n- v10 --refutes--> v0c\n- v01 --refutes--> v0f\n- v0c --refutes--> v07\n- v0c --refutes--> v12\n- v0c --supports--> v0e\n- v10 --refutes--> v00\n- v06 --supports--> v0a\n- v09 --refutes--> v02\n- v0d --supports--> v0b\n- v09 --supports--> v08\n- v05 --refutes--> v04\n- v0b --refutes--> v09\n- v08 --refutes--> v03\n- v07 --supports--> v01\n- v0b --refutes--> v11\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4c51b903957f6ce7f3be4788a5b6cce168a84e243cb5d0e31e57320ff0355292","schema":"livegraph.boolean-propagation.v1","spec_hash":"f678cbd41aca531f6334d3ca38d7bc309ba4fa02ab5af5cf3ab5ddec179a3e0c","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-256162a838343a96fa7b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v0b\n- v0e --supports--> v03\n- v0e --refutes--> v0d\n- v0e --refutes--> v08\n- v0c --refutes--> v02\n- v00 --supports--> v04\n- v0b --refutes--> v0c\n- v12 --supports--> v11\n- v11 --supports--> v06\n- v06 --refutes--> v0e\n- v07 --supports--> v05\n- v04 --supports--> v01\n- v04 --supports--> v0f\n- v02 --refutes--> v10\n- v0d --supports--> v09\n- v07 --refutes--> v0a\n- v10 --supports--> v00\n- v04 --refutes--> v07\n\n[VERIFIED ROOT]\nv12 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1077ff3c455f8ba4f157ce10826792ef2a1825b44c89618bf522e4d1c83be708","schema":"livegraph.boolean-propagation.v1","spec_hash":"256162a838343a96fa7bb348eac36a9dabec50ffa7c07e6a0bae26d67e2f258e","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v12\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-ae0819af7b64f063fb54","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v0c\n- v0b --refutes--> v01\n- v0a --refutes--> v00\n- v07 --refutes--> v0d\n- v0e --supports--> v02\n- v0f --refutes--> v10\n- v0b --supports--> v05\n- v07 --refutes--> v09\n- v05 --refutes--> v0e\n- v04 --refutes--> v11\n- v04 --refutes--> v03\n- v09 --supports--> v0f\n- v00 --supports--> v0b\n- v03 --supports--> v07\n- v04 --refutes--> v06\n- v02 --supports--> v12\n- v10 --refutes--> v08\n- v02 --supports--> v04\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9211578b2478aff628772ced74b4cc59365ebe363d8fcf82f9e3586f9d3c082d","schema":"livegraph.boolean-propagation.v1","spec_hash":"ae0819af7b64f063fb54173ce45166d0901ae79e56e2722d7ab811a47d7cc656","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-b98bd8868696d2e80fe4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v07\n- v11 --supports--> v0e\n- v01 --supports--> v0c\n- v00 --refutes--> v12\n- v10 --supports--> v0d\n- v0a --supports--> v11\n- v0f --refutes--> v00\n- v02 --refutes--> v0a\n- v0d --supports--> v0b\n- v0f --supports--> v05\n- v01 --refutes--> v06\n- v07 --refutes--> v02\n- v0a --supports--> v04\n- v01 --supports--> v09\n- v02 --refutes--> v03\n- v0f --refutes--> v08\n- v0c --supports--> v10\n- v0e --refutes--> v01\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e78d74491f5c46df3c433d0a99e75097bb52739458d0600fef12c1cd85ec35ad","schema":"livegraph.boolean-propagation.v1","spec_hash":"b98bd8868696d2e80fe4193bc899828487d2c017425df35511145f2fba9bfb79","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-677018bf5c145687d614","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v11\n- v0f --refutes--> v04\n- v01 --refutes--> v0c\n- v10 --refutes--> v0f\n- v05 --refutes--> v02\n- v02 --supports--> v08\n- v11 --refutes--> v10\n- v0b --refutes--> v05\n- v0c --refutes--> v00\n- v08 --refutes--> v0d\n- v02 --supports--> v0e\n- v09 --supports--> v0b\n- v10 --refutes--> v12\n- v0d --supports--> v07\n- v01 --supports--> v03\n- v0f --supports--> v06\n- v0d --refutes--> v01\n- v09 --supports--> v0a\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5e2d21028cc7252fbf1acccc76e176dc3138dc19f6c06eac6c3331804ce3ad25","schema":"livegraph.boolean-propagation.v1","spec_hash":"677018bf5c145687d61480938126041a5986b18c9cb3514472b04324c4a3cdc9","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-ad963feb7f852259f994","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v0d\n- v03 --refutes--> v12\n- v10 --supports--> v00\n- v09 --refutes--> v11\n- v12 --supports--> v0b\n- v05 --supports--> v02\n- v0d --refutes--> v0a\n- v12 --refutes--> v0e\n- v09 --refutes--> v0c\n- v07 --supports--> v08\n- v08 --refutes--> v03\n- v0a --supports--> v07\n- v0b --supports--> v10\n- v07 --supports--> v0f\n- v12 --refutes--> v01\n- v10 --supports--> v06\n- v0a --refutes--> v04\n- v02 --supports--> v09\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2a4e41ef5107ec40fd65db2766a348e064f6e24fa2aab26117e77f0d71aaadae","schema":"livegraph.boolean-propagation.v1","spec_hash":"ad963feb7f852259f99467ee0b0f08898b4dd644286c856d814ead38fef69f20","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-a9abf123116c9d4d555f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v0f\n- v0d --supports--> v0c\n- v11 --supports--> v02\n- v09 --refutes--> v0b\n- v12 --supports--> v04\n- v09 --refutes--> v05\n- v03 --supports--> v07\n- v12 --supports--> v08\n- v06 --supports--> v11\n- v11 --refutes--> v03\n- v01 --supports--> v0e\n- v0c --supports--> v01\n- v09 --supports--> v00\n- v06 --supports--> v10\n- v0e --refutes--> v09\n- v03 --refutes--> v0a\n- v04 --refutes--> v0d\n- v0b --supports--> v06\n\n[VERIFIED ROOT]\nv12 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"38e2b1a5ff67be043c4b203a076ba63e06c2b6ef88e848fc1f616be735bd944a","schema":"livegraph.boolean-propagation.v1","spec_hash":"a9abf123116c9d4d555f0b2ed4945159f8e5156669e529924bc2dd4510c2b094","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v12\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-338535f0271b769fc147","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v07\n- v02 --refutes--> v00\n- v07 --supports--> v11\n- v09 --refutes--> v01\n- v0d --refutes--> v0a\n- v0d --refutes--> v0f\n- v11 --supports--> v08\n- v0b --refutes--> v03\n- v11 --refutes--> v0b\n- v02 --supports--> v0d\n- v0e --supports--> v05\n- v0b --refutes--> v06\n- v0f --refutes--> v09\n- v0e --supports--> v0c\n- v01 --supports--> v0e\n- v04 --supports--> v02\n- v0f --refutes--> v10\n- v03 --refutes--> v12\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e4043bad3a2327ec83808947b1e42614ed9c9cbc6495bf2062d4ea6c23914aa6","schema":"livegraph.boolean-propagation.v1","spec_hash":"338535f0271b769fc147bdedc63f4ce7eb6d93846a760724e8617ab025f88142","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-75eb5ae71be084581d2c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --supports--> v06\n- v01 --supports--> v0b\n- v10 --supports--> v04\n- v02 --supports--> v09\n- v11 --refutes--> v10\n- v04 --refutes--> v02\n- v00 --refutes--> v12\n- v01 --refutes--> v0e\n- v03 --refutes--> v05\n- v07 --supports--> v0f\n- v0e --supports--> v0a\n- v0e --refutes--> v0d\n- v0f --supports--> v11\n- v09 --refutes--> v03\n- v0e --refutes--> v08\n- v05 --refutes--> v01\n- v0e --refutes--> v0c\n- v00 --refutes--> v07\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4586e70630d6e8ee159b81c3d578138f295da9dac88957e73c6b8d4f0b3ced02","schema":"livegraph.boolean-propagation.v1","spec_hash":"75eb5ae71be084581d2ce11b04ff6d52771f3f6111fd8847a3fd5e8048a424d4","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-3c002311bffbd4967b7b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v06\n- v06 --supports--> v03\n- v0c --refutes--> v04\n- v12 --refutes--> v00\n- v02 --refutes--> v09\n- v09 --supports--> v0c\n- v00 --supports--> v0b\n- v11 --refutes--> v07\n- v07 --supports--> v0f\n- v0d --refutes--> v11\n- v0c --supports--> v12\n- v08 --supports--> v0e\n- v0d --supports--> v05\n- v06 --supports--> v10\n- v0b --refutes--> v0d\n- v0f --supports--> v08\n- v09 --supports--> v01\n- v08 --refutes--> v0a\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3a8066c5ab8ba032c8a5f3df90c3a45ea3aff0db214521b6ded395f9195500e0","schema":"livegraph.boolean-propagation.v1","spec_hash":"3c002311bffbd4967b7b504e5829ed2110fc9dafeb85fba1230467b61b2e5cb7","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-1de2422c5edb9810b28c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v02\n- v0e --refutes--> v01\n- v05 --refutes--> v0f\n- v03 --supports--> v05\n- v10 --refutes--> v04\n- v0b --refutes--> v0a\n- v04 --supports--> v0b\n- v01 --supports--> v0d\n- v0f --refutes--> v0c\n- v12 --supports--> v03\n- v07 --refutes--> v06\n- v07 --supports--> v11\n- v06 --supports--> v12\n- v07 --refutes--> v09\n- v01 --supports--> v00\n- v0f --supports--> v08\n- v01 --supports--> v07\n- v0a --supports--> v0e\n\n[VERIFIED ROOT]\nv10 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5f5f38288ab3000ef9acd19d602e9ffdf2b159a949047b479c9e9f5481d82347","schema":"livegraph.boolean-propagation.v1","spec_hash":"1de2422c5edb9810b28ce9ce7fbcf31adc55639f56d9c2194b86f1265630f71c","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-59ba77c375d258097360","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v0f\n- v0f --refutes--> v0c\n- v0f --refutes--> v09\n- v0d --refutes--> v0e\n- v0e --refutes--> v04\n- v07 --refutes--> v03\n- v02 --supports--> v0a\n- v0b --supports--> v0d\n- v0e --refutes--> v12\n- v0b --supports--> v08\n- v05 --supports--> v02\n- v04 --supports--> v10\n- v01 --supports--> v00\n- v0a --supports--> v01\n- v02 --refutes--> v11\n- v00 --refutes--> v07\n- v04 --refutes--> v06\n- v0f --refutes--> v0b\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f1c9d50a4aeb13d0010428fa08ab3f6c2d03aa7d30e51f36d516df3cb8ba80e5","schema":"livegraph.boolean-propagation.v1","spec_hash":"59ba77c375d2580973600fa7e2d10fd339ba16284f60fe358ef17d90cea0e259","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":12,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-a36b682f4aa8ca35ebda","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --supports--> v0e\n- v06 --supports--> v01\n- v04 --supports--> v00\n- v0e --supports--> v11\n- v05 --refutes--> v07\n- v01 --refutes--> v10\n- v04 --refutes--> v12\n- v11 --refutes--> v09\n- v0b --refutes--> v02\n- v02 --supports--> v0a\n- v02 --refutes--> v05\n- v07 --supports--> v03\n- v0f --supports--> v08\n- v07 --supports--> v04\n- v04 --refutes--> v0c\n- v08 --refutes--> v0b\n- v11 --refutes--> v0d\n- v01 --refutes--> v0f\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cd95aa84144e005c3a61a3999f23c8215e3f2aeb92f0f4a09480c583b2c69de3","schema":"livegraph.boolean-propagation.v1","spec_hash":"a36b682f4aa8ca35ebda89a08a2a58cec223265b7bac4c9aa789e9a57cf9f26f","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-48825d4176bd09f72748","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v0a\n- v04 --supports--> v0d\n- v08 --supports--> v12\n- v12 --supports--> v0c\n- v0d --supports--> v11\n- v01 --refutes--> v09\n- v05 --supports--> v07\n- v10 --supports--> v03\n- v04 --supports--> v0e\n- v0c --refutes--> v01\n- v09 --supports--> v02\n- v02 --supports--> v0b\n- v0c --supports--> v0f\n- v04 --supports--> v00\n- v07 --supports--> v10\n- v06 --supports--> v05\n- v10 --supports--> v04\n- v0d --refutes--> v08\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e3058e7e6d99de377e2dde1917a916b5f192754b974e7bc026c78989f2e97cb2","schema":"livegraph.boolean-propagation.v1","spec_hash":"48825d4176bd09f7274871b1078908993e08dc89543db30587b504e14ee3e1f3","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-03a495220e0d77053aab","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v05\n- v0c --refutes--> v08\n- v04 --supports--> v0f\n- v0a --supports--> v0e\n- v12 --supports--> v10\n- v05 --supports--> v0c\n- v00 --supports--> v12\n- v0f --supports--> v0b\n- v0b --refutes--> v02\n- v06 --supports--> v0d\n- v09 --refutes--> v11\n- v06 --refutes--> v04\n- v0e --supports--> v09\n- v06 --refutes--> v01\n- v09 --supports--> v06\n- v06 --supports--> v07\n- v12 --refutes--> v0a\n- v04 --refutes--> v03\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6912f44f7bb10a3954cffcd0ff6d62db0c250f40531dfda03b4a3c328e3374fb","schema":"livegraph.boolean-propagation.v1","spec_hash":"03a495220e0d77053aab120b9d33ddd5cdd86d6d5e3111606703ca158b836820","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-9c8af89f0a26382aefbf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v0e\n- v0a --refutes--> v0b\n- v00 --refutes--> v05\n- v04 --supports--> v09\n- v01 --supports--> v04\n- v0f --refutes--> v00\n- v0a --refutes--> v12\n- v05 --refutes--> v06\n- v03 --refutes--> v01\n- v10 --supports--> v03\n- v00 --supports--> v11\n- v07 --refutes--> v02\n- v0a --supports--> v08\n- v09 --refutes--> v0a\n- v01 --supports--> v0c\n- v08 --refutes--> v07\n- v06 --refutes--> v10\n- v0a --refutes--> v0d\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6b0c48d6772b9742955c4c65d36a6efcfe417efd3c86f2b2339bea03c42f3a5e","schema":"livegraph.boolean-propagation.v1","spec_hash":"9c8af89f0a26382aefbfad3267f02c71b21422158b0fd632375d717748f7e0f7","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-260d62c1cb01d67f32d1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v04\n- v04 --supports--> v03\n- v03 --refutes--> v0d\n- v0d --refutes--> v05\n- v08 --refutes--> v09\n- v04 --supports--> v0e\n- v05 --refutes--> v0b\n- v09 --supports--> v0a\n- v01 --refutes--> v10\n- v0b --supports--> v11\n- v00 --refutes--> v06\n- v09 --supports--> v01\n- v12 --refutes--> v0f\n- v12 --supports--> v00\n- v09 --supports--> v0c\n- v07 --supports--> v08\n- v06 --supports--> v02\n- v06 --refutes--> v07\n\n[VERIFIED ROOT]\nv12 = FALSE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a5762dc2c0cc829c934b0a13bfc30c485d245784bddd3228090e34c71166091b","schema":"livegraph.boolean-propagation.v1","spec_hash":"260d62c1cb01d67f32d1b4076cd900965f2a56b369bea3cf2447293acdb09adc","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v12\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-d333f98d6eea25b52c4e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v07\n- v10 --refutes--> v0a\n- v0c --refutes--> v11\n- v07 --supports--> v05\n- v0c --supports--> v09\n- v12 --supports--> v00\n- v00 --refutes--> v06\n- v12 --supports--> v0e\n- v10 --supports--> v0f\n- v0b --supports--> v02\n- v0a --supports--> v0b\n- v04 --supports--> v08\n- v02 --supports--> v12\n- v04 --refutes--> v01\n- v00 --refutes--> v03\n- v03 --supports--> v0c\n- v09 --refutes--> v04\n- v10 --refutes--> v0d\n\n[VERIFIED ROOT]\nv10 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"414ed844919a8b0689485ff4096dae98434bb18136e22cc96034c1399b6ebc89","schema":"livegraph.boolean-propagation.v1","spec_hash":"d333f98d6eea25b52c4e855e66f3660c23c482c7df231b1bde1c957dfc4f1eb5","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-4c7d1c18ce8904755a8c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v05\n- v00 --supports--> v11\n- v08 --refutes--> v10\n- v11 --refutes--> v0d\n- v08 --refutes--> v04\n- v01 --supports--> v07\n- v08 --supports--> v0c\n- v03 --refutes--> v02\n- v12 --supports--> v06\n- v09 --refutes--> v00\n- v0e --refutes--> v0a\n- v05 --refutes--> v0f\n- v01 --refutes--> v0b\n- v05 --refutes--> v08\n- v0c --refutes--> v09\n- v11 --supports--> v01\n- v0e --refutes--> v03\n- v07 --refutes--> v12\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"78649b857dd84a0b7a2bd80bb72a6ac57c70dbe0493eef1cba4668b473713bd7","schema":"livegraph.boolean-propagation.v1","spec_hash":"4c7d1c18ce8904755a8cb3b5a5ae40b3dd7766154d4b2d6f0a7fd3c30e58f1d8","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-39fe8ec148eb8673fc5e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v01\n- v07 --supports--> v12\n- v12 --refutes--> v10\n- v0b --supports--> v00\n- v05 --supports--> v02\n- v0b --refutes--> v05\n- v04 --refutes--> v0a\n- v0d --refutes--> v07\n- v08 --refutes--> v0d\n- v0d --supports--> v09\n- v05 --refutes--> v11\n- v10 --refutes--> v04\n- v05 --supports--> v0c\n- v05 --supports--> v06\n- v0a --supports--> v0f\n- v12 --supports--> v03\n- v0e --supports--> v08\n- v01 --supports--> v0b\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ffdec4297a655714dccab760f80eb9665367d57193c334964f79f288119b5a98","schema":"livegraph.boolean-propagation.v1","spec_hash":"39fe8ec148eb8673fc5eabb9b8c37ec5de503215b76c402cbaadaa5ac5c97114","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-931e6500e879a843c180","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v09\n- v0c --refutes--> v0f\n- v09 --supports--> v10\n- v11 --supports--> v0a\n- v0d --supports--> v12\n- v03 --supports--> v01\n- v0f --supports--> v00\n- v0e --supports--> v0d\n- v0e --refutes--> v0b\n- v08 --supports--> v02\n- v0d --supports--> v08\n- v05 --refutes--> v03\n- v09 --refutes--> v04\n- v00 --supports--> v0e\n- v08 --supports--> v11\n- v09 --refutes--> v06\n- v10 --supports--> v0c\n- v11 --supports--> v07\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"37f9859d92e76fb9e813c45f60e6cb5392a69a109ae46a072811013198cc7f86","schema":"livegraph.boolean-propagation.v1","spec_hash":"931e6500e879a843c1807de0b493fddb2bf6b8c9928a2eb0aad0b36f43cc9fbe","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-e98113d703fa5746ec00","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v08\n- v10 --supports--> v05\n- v09 --supports--> v12\n- v00 --refutes--> v0e\n- v0b --supports--> v07\n- v01 --supports--> v02\n- v03 --refutes--> v04\n- v09 --supports--> v0f\n- v00 --refutes--> v10\n- v05 --refutes--> v06\n- v0d --supports--> v01\n- v07 --supports--> v03\n- v09 --refutes--> v00\n- v02 --refutes--> v0a\n- v06 --supports--> v0d\n- v0b --supports--> v11\n- v04 --supports--> v09\n- v04 --supports--> v0c\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9fea07e45872106b7c5eb114103bd863dd8f62189637a221ebbd3d07570c3e81","schema":"livegraph.boolean-propagation.v1","spec_hash":"e98113d703fa5746ec00acaa6a6685e10bc471b9afefd5e1a94e74723f413c30","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-e1a413b39879bd1ea427","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v06\n- v0c --supports--> v0e\n- v09 --supports--> v01\n- v0e --supports--> v07\n- v0f --supports--> v12\n- v0d --refutes--> v05\n- v04 --refutes--> v0f\n- v07 --supports--> v08\n- v0b --refutes--> v02\n- v03 --supports--> v0c\n- v10 --refutes--> v04\n- v05 --supports--> v10\n- v10 --supports--> v00\n- v0f --supports--> v03\n- v0e --refutes--> v11\n- v02 --supports--> v0d\n- v03 --refutes--> v0a\n- v07 --refutes--> v09\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"241c9f40f3353590db869d52df6a4d2630c9437768d08fe36d31e851fa4f3f89","schema":"livegraph.boolean-propagation.v1","spec_hash":"e1a413b39879bd1ea4276e99dcac81bf7011a98f57fb066588f74511c79080c9","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-97fd3aa77776447b40c0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v03\n- v0c --supports--> v0d\n- v06 --supports--> v00\n- v03 --supports--> v0e\n- v00 --supports--> v09\n- v08 --supports--> v04\n- v0b --supports--> v10\n- v08 --refutes--> v11\n- v11 --refutes--> v05\n- v00 --supports--> v0a\n- v00 --supports--> v0f\n- v0c --supports--> v0b\n- v0c --supports--> v01\n- v0f --refutes--> v02\n- v02 --supports--> v0c\n- v07 --refutes--> v06\n- v0b --supports--> v12\n- v0e --refutes--> v07\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fb1c8bd82efe89cf4114862e2b12796fcbecf75cc743dd0217a8288efaeeb09d","schema":"livegraph.boolean-propagation.v1","spec_hash":"97fd3aa77776447b40c08f6f1d41867609dc9996f2207f475d25f1e1bd4e5980","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-55b3d9cc1ea6650edb28","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v02\n- v0b --refutes--> v04\n- v0e --refutes--> v0d\n- v0d --supports--> v0a\n- v06 --supports--> v00\n- v0b --supports--> v08\n- v03 --refutes--> v10\n- v12 --refutes--> v01\n- v05 --refutes--> v0f\n- v0c --supports--> v0b\n- v02 --supports--> v12\n- v11 --refutes--> v0e\n- v09 --supports--> v07\n- v05 --supports--> v09\n- v06 --refutes--> v05\n- v12 --refutes--> v0c\n- v08 --supports--> v11\n- v03 --supports--> v06\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a75c96eabd39354898692264af6b3dff88c96974daf989abf4f79ac7129cc26a","schema":"livegraph.boolean-propagation.v1","spec_hash":"55b3d9cc1ea6650edb281d4a26b745cad220c05724b6a7403c554f45dab6f315","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-aee43edc03f75123da4c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v04\n- v04 --refutes--> v03\n- v0c --supports--> v0a\n- v11 --supports--> v00\n- v11 --supports--> v09\n- v0e --refutes--> v02\n- v06 --supports--> v0e\n- v02 --refutes--> v01\n- v00 --refutes--> v0b\n- v00 --refutes--> v05\n- v0c --refutes--> v12\n- v0e --supports--> v08\n- v01 --refutes--> v0c\n- v03 --supports--> v0f\n- v0b --supports--> v10\n- v11 --supports--> v07\n- v0a --refutes--> v11\n- v0d --refutes--> v06\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c1f3faf587ed2d48c12a27dbe29699ba8eba88a5f10a87f02f48bb02f94f7eec","schema":"livegraph.boolean-propagation.v1","spec_hash":"aee43edc03f75123da4c3b303fe5cbaaaf0fe4f24c67a8007f7dd30b2409308d","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-73c3e9b8fabab582d700","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v04\n- v08 --refutes--> v0d\n- v00 --refutes--> v03\n- v0c --supports--> v08\n- v07 --refutes--> v06\n- v03 --supports--> v02\n- v11 --supports--> v09\n- v11 --refutes--> v0f\n- v0a --supports--> v11\n- v02 --refutes--> v07\n- v0b --refutes--> v0a\n- v0b --supports--> v12\n- v0a --refutes--> v05\n- v0e --supports--> v0c\n- v06 --supports--> v10\n- v0c --refutes--> v01\n- v0f --supports--> v0e\n- v08 --supports--> v00\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2da411e2d36afff53e45296fe06c57795bae504d6b903bf4f72b036e4609702b","schema":"livegraph.boolean-propagation.v1","spec_hash":"73c3e9b8fabab582d700ee238da0ef9e622d36c8278ce2a0e3e868e39108ada9","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-66d41b3c0be1e305b650","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v04\n- v06 --refutes--> v10\n- v00 --supports--> v12\n- v0b --supports--> v09\n- v03 --refutes--> v0a\n- v09 --supports--> v0c\n- v07 --refutes--> v0b\n- v05 --supports--> v02\n- v04 --supports--> v0d\n- v0f --refutes--> v03\n- v10 --refutes--> v08\n- v03 --supports--> v0e\n- v07 --supports--> v11\n- v12 --refutes--> v01\n- v05 --refutes--> v07\n- v09 --supports--> v00\n- v12 --supports--> v0f\n- v04 --supports--> v05\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a7cf17460c0d4319fff0d9e65fa95ff450255ffceccea2e44f2914c796b266dd","schema":"livegraph.boolean-propagation.v1","spec_hash":"66d41b3c0be1e305b65056ba636ee75ba047ff1436838d1bf5ade1bee42a52e9","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-9d347ebb1362b3121a45","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --supports--> v12\n- v09 --refutes--> v0e\n- v0f --supports--> v05\n- v11 --refutes--> v0a\n- v05 --refutes--> v04\n- v0b --refutes--> v0f\n- v0a --refutes--> v06\n- v0e --supports--> v00\n- v05 --refutes--> v07\n- v06 --refutes--> v02\n- v10 --supports--> v09\n- v0f --supports--> v03\n- v05 --refutes--> v0c\n- v00 --supports--> v11\n- v01 --supports--> v10\n- v0b --refutes--> v0d\n- v11 --supports--> v08\n- v02 --supports--> v0b\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ad6eeb587d84f68dcb2b3a5a1960da24e0c5c3ddbf6c3a965d4f8da867aa2841","schema":"livegraph.boolean-propagation.v1","spec_hash":"9d347ebb1362b3121a4548574b4b58a9421cc4778fe94febc47a555b86537bdd","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-44989cda58482266f69b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v08\n- v01 --supports--> v0d\n- v06 --refutes--> v12\n- v0b --supports--> v11\n- v06 --refutes--> v0f\n- v08 --refutes--> v0b\n- v0d --supports--> v0c\n- v09 --supports--> v0e\n- v11 --supports--> v01\n- v04 --refutes--> v09\n- v0e --supports--> v03\n- v0c --refutes--> v04\n- v11 --supports--> v05\n- v01 --refutes--> v07\n- v0e --refutes--> v10\n- v12 --supports--> v0a\n- v0e --refutes--> v00\n- v06 --supports--> v02\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ec7283b42f1aee38df8c0ad9d42de1fb42f14efb9c35908150a0d987244cc5d8","schema":"livegraph.boolean-propagation.v1","spec_hash":"44989cda58482266f69b04509975eae9b70533f34ca0845b99bd5328e7a32a7b","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-46c596630fa405f98555","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v12\n- v07 --refutes--> v06\n- v06 --supports--> v04\n- v0d --refutes--> v0e\n- v0a --supports--> v00\n- v06 --supports--> v0c\n- v05 --supports--> v01\n- v08 --supports--> v07\n- v02 --refutes--> v09\n- v0f --supports--> v05\n- v01 --supports--> v03\n- v02 --supports--> v0b\n- v01 --supports--> v10\n- v04 --supports--> v0f\n- v00 --supports--> v02\n- v0d --supports--> v0a\n- v0b --refutes--> v08\n- v01 --supports--> v11\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"55babec42d5e00eb1512cf5a4080395677cbc13af02aeebfa225ad05e9162865","schema":"livegraph.boolean-propagation.v1","spec_hash":"46c596630fa405f9855524a6c0f06641787a5fd020caaf48656ec3ed0ba5b688","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":12,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-aa6a2c73cb1ca1251593","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v05\n- v0e --refutes--> v09\n- v0f --refutes--> v02\n- v03 --refutes--> v12\n- v03 --refutes--> v0d\n- v0b --supports--> v01\n- v09 --refutes--> v11\n- v05 --supports--> v03\n- v10 --refutes--> v08\n- v0e --refutes--> v0a\n- v07 --supports--> v0e\n- v0d --refutes--> v06\n- v0f --supports--> v0b\n- v08 --supports--> v0f\n- v05 --supports--> v04\n- v11 --refutes--> v0c\n- v0d --supports--> v10\n- v09 --supports--> v00\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a98a24b921bf1b8c7519b1906d2ba41c148a74b33d4d04480e284aa6aeac803e","schema":"livegraph.boolean-propagation.v1","spec_hash":"aa6a2c73cb1ca12515932068be86f882878c7305148a644ab9633fdf20089338","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-92a6ff2e95d5ce8aafdb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v05\n- v10 --refutes--> v0c\n- v0e --supports--> v00\n- v08 --supports--> v12\n- v0c --refutes--> v04\n- v05 --refutes--> v08\n- v0d --supports--> v0e\n- v09 --refutes--> v0b\n- v08 --refutes--> v11\n- v09 --refutes--> v06\n- v00 --supports--> v0a\n- v0f --refutes--> v03\n- v0e --supports--> v01\n- v0c --supports--> v09\n- v08 --supports--> v10\n- v02 --refutes--> v0f\n- v0b --supports--> v0d\n- v00 --supports--> v02\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"444793d7ffb353185c1c2da6b7dc402d5b24969132c3a9305ea7d145fd0fb267","schema":"livegraph.boolean-propagation.v1","spec_hash":"92a6ff2e95d5ce8aafdbe2f1eef78467a8efd2722cba929760bbd5ce33f5f2a2","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-0a76ec5c08a97f22b324","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v09\n- v0d --supports--> v0b\n- v06 --refutes--> v0f\n- v09 --supports--> v0c\n- v05 --supports--> v11\n- v08 --supports--> v02\n- v02 --supports--> v10\n- v02 --refutes--> v00\n- v12 --supports--> v04\n- v03 --refutes--> v07\n- v0b --supports--> v0e\n- v0d --refutes--> v01\n- v10 --refutes--> v06\n- v12 --supports--> v0a\n- v04 --supports--> v03\n- v0e --refutes--> v05\n- v06 --supports--> v0d\n- v03 --refutes--> v08\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9a8f2a573d7f7996d689443d631223d57580d28d2fe7439daf09b075995a9dd3","schema":"livegraph.boolean-propagation.v1","spec_hash":"0a76ec5c08a97f22b3242daf638bfe36f8251a206d3ed1554d12f9318c21f7aa","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-08f5c59e4a194e74ad40","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v06\n- v0c --supports--> v02\n- v01 --supports--> v0d\n- v03 --refutes--> v0e\n- v02 --refutes--> v00\n- v0e --refutes--> v04\n- v09 --refutes--> v0f\n- v0f --refutes--> v0a\n- v04 --supports--> v09\n- v0d --supports--> v0b\n- v00 --refutes--> v11\n- v12 --supports--> v01\n- v0d --refutes--> v03\n- v12 --supports--> v07\n- v09 --refutes--> v08\n- v00 --refutes--> v05\n- v0a --supports--> v0c\n- v02 --supports--> v10\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8e11e56431fe9bd220d012192c88ddd429dab0f042b8e13a274c42aee50b0d37","schema":"livegraph.boolean-propagation.v1","spec_hash":"08f5c59e4a194e74ad402141dd12bf222622d5cb5aea8b1340b76b288da01db9","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-fddc9ec460772980aa4b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v0c\n- v0e --supports--> v0d\n- v0f --supports--> v12\n- v0b --refutes--> v04\n- v08 --refutes--> v07\n- v11 --supports--> v0a\n- v08 --refutes--> v0e\n- v01 --refutes--> v00\n- v0d --refutes--> v0b\n- v0a --supports--> v08\n- v00 --supports--> v0f\n- v0b --refutes--> v10\n- v12 --refutes--> v06\n- v10 --refutes--> v03\n- v10 --refutes--> v09\n- v11 --refutes--> v02\n- v03 --supports--> v01\n- v0b --supports--> v05\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8988628f2019dc00a372b3a162152f8b4c0c7fec694291dfc53836967eaf5d5d","schema":"livegraph.boolean-propagation.v1","spec_hash":"fddc9ec460772980aa4b0e768c99df9a2729c3538cef0a93852cf5e88f2236fb","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-c4817981f45b2a0eb519","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v12\n- v11 --refutes--> v0b\n- v0b --supports--> v09\n- v05 --refutes--> v06\n- v04 --refutes--> v02\n- v11 --refutes--> v0c\n- v09 --refutes--> v01\n- v0d --refutes--> v08\n- v0f --supports--> v0e\n- v0d --refutes--> v10\n- v04 --refutes--> v03\n- v07 --supports--> v0a\n- v03 --supports--> v07\n- v00 --supports--> v0d\n- v09 --supports--> v05\n- v06 --supports--> v0f\n- v0a --refutes--> v11\n- v0f --refutes--> v00\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d72496757e2e6e07fcfa6a185fef2e3a227ba9843e30d89ff136081ea2029262","schema":"livegraph.boolean-propagation.v1","spec_hash":"c4817981f45b2a0eb51915e8637728110d6567f4fd518057e77dafcf29a50817","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-5f7a78f500c7a47c841d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v11\n- v0d --refutes--> v04\n- v0a --supports--> v07\n- v00 --supports--> v09\n- v09 --refutes--> v08\n- v0f --refutes--> v0e\n- v10 --supports--> v0d\n- v08 --supports--> v06\n- v08 --supports--> v0b\n- v04 --supports--> v0a\n- v08 --refutes--> v01\n- v05 --refutes--> v10\n- v0a --supports--> v0f\n- v00 --refutes--> v12\n- v06 --supports--> v02\n- v0f --supports--> v03\n- v03 --supports--> v00\n- v0c --refutes--> v05\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3c6ad1a24bb466f33e30c3c6326ac432889ed0d07af66b2b81b3cb439cd743c1","schema":"livegraph.boolean-propagation.v1","spec_hash":"5f7a78f500c7a47c841dc6c08df1c55087097815a4204f4a31a1d8ef6a8123c2","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-f81d511394841415e707","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v00\n- v0e --refutes--> v05\n- v02 --supports--> v10\n- v0b --refutes--> v0d\n- v0e --supports--> v12\n- v10 --supports--> v0c\n- v03 --supports--> v04\n- v06 --supports--> v08\n- v04 --refutes--> v02\n- v11 --supports--> v07\n- v04 --supports--> v0f\n- v0c --supports--> v06\n- v07 --supports--> v03\n- v02 --supports--> v09\n- v02 --supports--> v01\n- v08 --refutes--> v0b\n- v0d --refutes--> v0e\n- v06 --supports--> v0a\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"87eaf4a9991c45a525f60324fd0acda79dcf61bd21bcf37fd2a535816d749248","schema":"livegraph.boolean-propagation.v1","spec_hash":"f81d511394841415e707c65da4d99d33dc665fac10d8d259748aa37c200b1fdb","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-b2806fb1144dfcab5fe2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v03\n- v06 --refutes--> v02\n- v12 --refutes--> v00\n- v02 --supports--> v12\n- v10 --supports--> v0c\n- v04 --supports--> v05\n- v10 --supports--> v0f\n- v04 --refutes--> v06\n- v0f --supports--> v09\n- v0f --supports--> v0b\n- v01 --refutes--> v07\n- v0d --supports--> v01\n- v08 --refutes--> v10\n- v07 --supports--> v04\n- v02 --refutes--> v0e\n- v03 --refutes--> v0d\n- v0a --refutes--> v08\n- v04 --supports--> v11\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9bf610e03d49664bd8ee4641ea638b94238af5bba2085257109f12dd92f88215","schema":"livegraph.boolean-propagation.v1","spec_hash":"b2806fb1144dfcab5fe21ab09e998ed9a23b29aba733c99ce3cbdfae4ce52387","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-2781042207722bc49f07","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v06\n- v07 --supports--> v02\n- v0c --refutes--> v0d\n- v03 --supports--> v05\n- v06 --supports--> v0b\n- v0f --supports--> v12\n- v05 --supports--> v07\n- v07 --supports--> v09\n- v0b --supports--> v04\n- v07 --refutes--> v11\n- v0f --supports--> v0e\n- v08 --refutes--> v0c\n- v07 --supports--> v0a\n- v08 --supports--> v01\n- v00 --supports--> v0f\n- v0c --supports--> v10\n- v09 --refutes--> v08\n- v12 --refutes--> v03\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"71172aadff9f8a10e10b233e795b241d5b491854269aaa7554e9a3c7bb70bd80","schema":"livegraph.boolean-propagation.v1","spec_hash":"2781042207722bc49f07a5f253854257aee5be2e985b95a77cbdb2d4a37ccb90","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-9f23e8978ca195fa2658","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --supports--> v0a\n- v09 --refutes--> v0c\n- v11 --supports--> v08\n- v04 --supports--> v00\n- v0e --supports--> v0b\n- v0b --refutes--> v12\n- v09 --supports--> v0f\n- v01 --supports--> v05\n- v0b --refutes--> v09\n- v05 --refutes--> v10\n- v02 --supports--> v0e\n- v11 --refutes--> v02\n- v0b --refutes--> v0d\n- v07 --refutes--> v11\n- v09 --supports--> v03\n- v04 --refutes--> v06\n- v0c --refutes--> v04\n- v00 --supports--> v01\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2c8553fb9c13914188419cd35334896ae2c86903a5d553d2018adf84141b52ac","schema":"livegraph.boolean-propagation.v1","spec_hash":"9f23e8978ca195fa2658da00ce350e600bca8c77407d81f6d661232a9d278ac9","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-5617690e421a164bf0e6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v05\n- v07 --refutes--> v02\n- v0b --supports--> v11\n- v03 --refutes--> v00\n- v0d --refutes--> v09\n- v05 --refutes--> v10\n- v10 --supports--> v08\n- v0c --supports--> v07\n- v02 --refutes--> v0e\n- v03 --refutes--> v01\n- v12 --supports--> v0c\n- v0b --supports--> v0f\n- v09 --supports--> v12\n- v0a --supports--> v04\n- v10 --refutes--> v0b\n- v11 --refutes--> v0d\n- v02 --refutes--> v06\n- v0a --supports--> v03\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"617b9b2a67f728f0263582b69f9958a6dea59a638719caf9b93d04ed16642e2b","schema":"livegraph.boolean-propagation.v1","spec_hash":"5617690e421a164bf0e637a72a84693ed09e17673c6fa73917726b75b17f9138","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-c727d1231a5f3cc6fbae","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v00\n- v0c --refutes--> v11\n- v09 --refutes--> v02\n- v0a --refutes--> v0f\n- v00 --supports--> v12\n- v07 --refutes--> v03\n- v0c --supports--> v10\n- v03 --refutes--> v0a\n- v00 --supports--> v08\n- v0c --supports--> v06\n- v02 --refutes--> v0c\n- v0f --supports--> v09\n- v10 --refutes--> v05\n- v0e --supports--> v07\n- v0c --refutes--> v0b\n- v01 --supports--> v0d\n- v01 --supports--> v0e\n- v0c --supports--> v04\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6035b6c17b252776c29c81a7365dc01a21ac95f1cf1a14acc4019d6b5e96fa8a","schema":"livegraph.boolean-propagation.v1","spec_hash":"c727d1231a5f3cc6fbae4a770907fa89c3002a208a3602aaf391ba03df16168e","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-4fe44647c77791e8357d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v00\n- v0b --refutes--> v0f\n- v04 --refutes--> v11\n- v05 --supports--> v0b\n- v02 --refutes--> v08\n- v07 --supports--> v02\n- v04 --supports--> v05\n- v0f --refutes--> v06\n- v05 --refutes--> v03\n- v0a --supports--> v04\n- v01 --refutes--> v0d\n- v0d --supports--> v0a\n- v06 --supports--> v12\n- v09 --refutes--> v07\n- v08 --refutes--> v01\n- v08 --supports--> v0c\n- v04 --refutes--> v0e\n- v07 --supports--> v10\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bc2539c1765a7b3d9b58dc53f18fa9bdd13d4e23b82137e323683ceb71a1b80e","schema":"livegraph.boolean-propagation.v1","spec_hash":"4fe44647c77791e8357dce3b08ccc00af09bc2ed0ccfef688c9268e60c3d70c3","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":12,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-e79e1eab530e81367128","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v06\n- v09 --supports--> v04\n- v03 --refutes--> v11\n- v05 --refutes--> v00\n- v03 --refutes--> v0e\n- v0a --refutes--> v0b\n- v01 --refutes--> v0c\n- v06 --supports--> v03\n- v01 --refutes--> v0d\n- v0f --refutes--> v0a\n- v0e --refutes--> v09\n- v03 --refutes--> v08\n- v05 --supports--> v07\n- v10 --supports--> v01\n- v10 --refutes--> v02\n- v03 --refutes--> v12\n- v0b --supports--> v10\n- v07 --refutes--> v0f\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"96b969e35fc9f3620bd0c3f75ed6f246420477aa919a82fdcb014a15e150fb85","schema":"livegraph.boolean-propagation.v1","spec_hash":"e79e1eab530e8136712898db046ad06c30663cf73c05fa4e6a384d53aa3c1c45","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-89181c4e3c3d94d78e07","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v08\n- v08 --supports--> v01\n- v06 --supports--> v0a\n- v0a --refutes--> v09\n- v05 --refutes--> v10\n- v0d --refutes--> v0f\n- v09 --refutes--> v11\n- v04 --refutes--> v0e\n- v0d --refutes--> v07\n- v0a --supports--> v0b\n- v12 --supports--> v05\n- v07 --supports--> v00\n- v11 --refutes--> v12\n- v0d --supports--> v02\n- v08 --supports--> v03\n- v0c --supports--> v04\n- v08 --refutes--> v06\n- v04 --supports--> v0d\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9053af1e535ada69533c0a79edbe5863ad63d1e6d8aee4f5e97ef5c7a5a1b2dd","schema":"livegraph.boolean-propagation.v1","spec_hash":"89181c4e3c3d94d78e072ff24e2b14172ae3853fe45fc7bef106efa56b832fde","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-44c9b0edfa944e158f7f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v0d\n- v06 --supports--> v0e\n- v0d --supports--> v02\n- v0e --refutes--> v0c\n- v03 --refutes--> v0f\n- v12 --refutes--> v06\n- v0a --refutes--> v05\n- v06 --supports--> v00\n- v08 --refutes--> v01\n- v03 --supports--> v0b\n- v07 --supports--> v08\n- v08 --supports--> v09\n- v05 --refutes--> v12\n- v0a --refutes--> v04\n- v01 --refutes--> v03\n- v0e --refutes--> v10\n- v0a --refutes--> v11\n- v0c --supports--> v07\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f96fb8c5c54790d7c0718d7e8f1e15086c85e70f079bfa38a3d25b25f352b0ab","schema":"livegraph.boolean-propagation.v1","spec_hash":"44c9b0edfa944e158f7f9f54809c6bb49590f07cc92ac4cecb29521a122cd450","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-2767b4be93af77060ad5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v01\n- v10 --supports--> v11\n- v0e --supports--> v12\n- v0c --supports--> v0d\n- v0a --refutes--> v08\n- v01 --supports--> v0c\n- v08 --refutes--> v09\n- v0d --refutes--> v04\n- v05 --refutes--> v0a\n- v01 --supports--> v0f\n- v00 --refutes--> v06\n- v08 --supports--> v02\n- v01 --supports--> v07\n- v11 --supports--> v05\n- v0b --refutes--> v0e\n- v00 --refutes--> v0b\n- v01 --supports--> v03\n- v02 --supports--> v00\n\n[VERIFIED ROOT]\nv10 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"502188321d2d4ca0ae0196de4af21af39e0a5c11ec20e33940e24eb4e1b0d8ac","schema":"livegraph.boolean-propagation.v1","spec_hash":"2767b4be93af77060ad5214a0044284c28ad2281a576a6715e68c7240931ebb4","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-9d24fc11d6993a63ff98","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v10\n- v01 --refutes--> v02\n- v11 --refutes--> v00\n- v06 --refutes--> v0b\n- v00 --refutes--> v08\n- v11 --refutes--> v09\n- v04 --supports--> v07\n- v08 --supports--> v05\n- v04 --supports--> v0d\n- v07 --refutes--> v12\n- v11 --refutes--> v0f\n- v11 --supports--> v0e\n- v11 --refutes--> v0a\n- v0b --supports--> v04\n- v05 --refutes--> v01\n- v12 --refutes--> v11\n- v03 --refutes--> v0c\n- v02 --supports--> v03\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a59ba89f1516032f6d02a292712ab8b07a379921580554374d864b01792fd262","schema":"livegraph.boolean-propagation.v1","spec_hash":"9d24fc11d6993a63ff980fa63fa548714ced31c6d4e06b29b6ce87d47608b281","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-fe86db62e874984971a6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v0a\n- v04 --refutes--> v0f\n- v0e --supports--> v0d\n- v03 --refutes--> v0c\n- v05 --supports--> v09\n- v0f --supports--> v10\n- v01 --refutes--> v0b\n- v0d --refutes--> v03\n- v0b --supports--> v12\n- v0f --supports--> v0e\n- v04 --supports--> v00\n- v0a --supports--> v06\n- v0b --refutes--> v07\n- v0c --supports--> v02\n- v06 --refutes--> v04\n- v06 --refutes--> v08\n- v0b --supports--> v11\n- v05 --refutes--> v01\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0a4eacb898f5912ba16a8a1cce0827461c1c100342969f47661b6856568f4710","schema":"livegraph.boolean-propagation.v1","spec_hash":"fe86db62e874984971a6b3cc34ddafbb5e2d0b72197d3130c8cc13478254efc6","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-fd5b7f0f4eb394ff9d1b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --supports--> v08\n- v07 --refutes--> v00\n- v02 --supports--> v05\n- v04 --refutes--> v02\n- v10 --supports--> v12\n- v0c --refutes--> v10\n- v00 --refutes--> v01\n- v0b --refutes--> v0e\n- v01 --refutes--> v03\n- v06 --refutes--> v0b\n- v0b --supports--> v09\n- v0e --supports--> v0a\n- v12 --refutes--> v04\n- v0c --supports--> v0f\n- v00 --supports--> v0d\n- v03 --refutes--> v0c\n- v06 --supports--> v11\n- v02 --supports--> v06\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"884632e154cca01c646cdbfa24e3c1502bb6fac3bb9be9d7f57480c20bd0f4be","schema":"livegraph.boolean-propagation.v1","spec_hash":"fd5b7f0f4eb394ff9d1b55f100de5f42b69f3cd6fec2962bdc0d8f3bf8a5bc18","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-09c670aff0bb70aac1d1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v11\n- v00 --refutes--> v0d\n- v0d --refutes--> v03\n- v09 --supports--> v12\n- v0e --supports--> v08\n- v03 --refutes--> v07\n- v0c --supports--> v09\n- v05 --supports--> v01\n- v10 --supports--> v02\n- v03 --refutes--> v10\n- v05 --refutes--> v0a\n- v09 --refutes--> v0e\n- v04 --refutes--> v00\n- v04 --refutes--> v06\n- v08 --refutes--> v0f\n- v02 --refutes--> v0c\n- v09 --refutes--> v0b\n- v05 --refutes--> v04\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ba0c64049478ab0a34b8b3bb3c46574f0437eebf89bb2132abec42091bb133c8","schema":"livegraph.boolean-propagation.v1","spec_hash":"09c670aff0bb70aac1d19b36df9b443ca6dc745ad7055579bb1d4d4e2128db2e","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-e8a6b8da704fcaa71e62","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v06\n- v01 --supports--> v0b\n- v0e --refutes--> v05\n- v02 --supports--> v12\n- v0f --supports--> v09\n- v03 --supports--> v0f\n- v0b --refutes--> v04\n- v0c --supports--> v03\n- v00 --supports--> v0e\n- v03 --refutes--> v11\n- v09 --refutes--> v02\n- v12 --refutes--> v00\n- v0f --refutes--> v10\n- v06 --refutes--> v0c\n- v06 --refutes--> v07\n- v0f --supports--> v0a\n- v0e --refutes--> v01\n- v0f --refutes--> v0d\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c7132cc48488ca1e7f9b91b60e0675584f31a0e3cbb586d12a2d8cf9fa5781ae","schema":"livegraph.boolean-propagation.v1","spec_hash":"e8a6b8da704fcaa71e628aa43f5aecc56247dff4f34da77b59a44b5c779b3f82","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-05a5c3ad2834d8fbd6ab","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --supports--> v0d\n- v03 --supports--> v06\n- v0f --supports--> v11\n- v07 --supports--> v02\n- v03 --supports--> v0f\n- v08 --refutes--> v0b\n- v0a --refutes--> v07\n- v0c --supports--> v0a\n- v00 --refutes--> v08\n- v11 --refutes--> v0e\n- v0e --supports--> v12\n- v07 --refutes--> v09\n- v0a --supports--> v01\n- v12 --refutes--> v0c\n- v0b --refutes--> v05\n- v0f --refutes--> v04\n- v05 --refutes--> v03\n- v11 --refutes--> v10\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c5e6116d41e0222234b2c21035c135d2a2df0141aba1bdb8112842543d6dd283","schema":"livegraph.boolean-propagation.v1","spec_hash":"05a5c3ad2834d8fbd6ab286842b87b7e804e9c172bb1b7899d065c963072da85","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-7188677e4a8696c117fb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v11\n- v0e --refutes--> v10\n- v11 --supports--> v0c\n- v08 --supports--> v0e\n- v07 --refutes--> v05\n- v11 --supports--> v0f\n- v01 --refutes--> v09\n- v06 --refutes--> v04\n- v02 --supports--> v00\n- v12 --supports--> v08\n- v0e --supports--> v02\n- v0e --refutes--> v0d\n- v12 --supports--> v03\n- v05 --refutes--> v01\n- v00 --refutes--> v06\n- v09 --refutes--> v12\n- v0e --refutes--> v0a\n- v07 --refutes--> v0b\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"884e2e8eddfe52a6c8a7fb77d390850134405dc0506002c21fced2db9de06b05","schema":"livegraph.boolean-propagation.v1","spec_hash":"7188677e4a8696c117fbdb32ebfb80f922a5270e6aa20e67c6bd4541620fe16e","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-ddd58c5ebf50eaaf0c97","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v0a\n- v12 --refutes--> v07\n- v0e --supports--> v10\n- v06 --refutes--> v03\n- v0c --refutes--> v02\n- v11 --refutes--> v12\n- v12 --refutes--> v04\n- v0b --refutes--> v05\n- v08 --refutes--> v0b\n- v0d --supports--> v0f\n- v0e --refutes--> v06\n- v02 --refutes--> v09\n- v01 --refutes--> v08\n- v0b --supports--> v0d\n- v06 --refutes--> v01\n- v00 --refutes--> v11\n- v12 --refutes--> v0e\n- v02 --refutes--> v00\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"55d9f4d2643759a486b571d729887d0360886cbea08b7715e8bb7db975408f93","schema":"livegraph.boolean-propagation.v1","spec_hash":"ddd58c5ebf50eaaf0c976509cdfa2028f0204ac7f25790a242fee026d1428940","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-e6d10fd3767402d35910","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v00\n- v12 --supports--> v03\n- v0c --supports--> v0f\n- v10 --refutes--> v07\n- v0f --supports--> v0d\n- v0f --supports--> v05\n- v04 --refutes--> v06\n- v0b --supports--> v04\n- v0e --supports--> v01\n- v07 --refutes--> v0a\n- v11 --refutes--> v0e\n- v12 --supports--> v0c\n- v0d --supports--> v10\n- v0e --refutes--> v09\n- v0a --refutes--> v11\n- v02 --supports--> v08\n- v04 --supports--> v02\n- v02 --refutes--> v12\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7da34121363183be88ec5214e8d6337f68a94dc12b48f7e3371763fdeecfb4f8","schema":"livegraph.boolean-propagation.v1","spec_hash":"e6d10fd3767402d3591011c57a3af9db70e9724a1737159dd9ff7ee24523f93b","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-d12ef9ce8a197b01247b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v0d\n- v05 --refutes--> v00\n- v07 --refutes--> v03\n- v09 --refutes--> v01\n- v0c --supports--> v0b\n- v02 --refutes--> v10\n- v12 --supports--> v0c\n- v02 --supports--> v11\n- v07 --supports--> v12\n- v11 --refutes--> v05\n- v0c --refutes--> v08\n- v0a --supports--> v0f\n- v08 --refutes--> v0a\n- v07 --refutes--> v06\n- v00 --refutes--> v07\n- v0e --refutes--> v02\n- v00 --refutes--> v04\n- v01 --supports--> v0e\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"27ee631022523a8968bda623442b8e7f37836e1a0d962f7db883c4e37087dd01","schema":"livegraph.boolean-propagation.v1","spec_hash":"d12ef9ce8a197b01247b5475c18f78626b326818a08df405db50d103ccbdf636","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-3db6799955ab3e886f77","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v05\n- v0e --refutes--> v07\n- v0e --supports--> v12\n- v0a --supports--> v0e\n- v00 --refutes--> v0b\n- v07 --refutes--> v0c\n- v00 --supports--> v09\n- v0d --supports--> v11\n- v03 --supports--> v02\n- v02 --refutes--> v00\n- v0f --refutes--> v0a\n- v05 --supports--> v04\n- v04 --refutes--> v0d\n- v00 --supports--> v06\n- v02 --supports--> v10\n- v06 --supports--> v08\n- v0d --refutes--> v01\n- v0d --refutes--> v0f\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d280a86ef00bb988f6af3e15aaf401a05bf155a19b8c8ea686e5295a79095122","schema":"livegraph.boolean-propagation.v1","spec_hash":"3db6799955ab3e886f77a7716cbe4636e68ca52535d81639464bdc61f8f09c0a","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":12,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-bfd671551433f76355b0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v05\n- v00 --supports--> v0a\n- v00 --supports--> v12\n- v0f --supports--> v01\n- v10 --supports--> v0b\n- v00 --supports--> v03\n- v04 --refutes--> v0e\n- v10 --refutes--> v02\n- v04 --supports--> v0f\n- v0a --supports--> v10\n- v02 --supports--> v0c\n- v0f --supports--> v06\n- v08 --refutes--> v11\n- v07 --supports--> v09\n- v0c --supports--> v0d\n- v11 --refutes--> v07\n- v06 --refutes--> v00\n- v09 --refutes--> v04\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"80a1f6c39823a83d89563fce87d685f56bd4db334f6b7c2e23108d0b670a7aea","schema":"livegraph.boolean-propagation.v1","spec_hash":"bfd671551433f76355b050933e2fefc187f0d209cfbff5eab8b0e7a64626b0ea","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-330debfe342b10a18017","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v12\n- v0b --supports--> v00\n- v01 --refutes--> v04\n- v00 --supports--> v06\n- v0e --supports--> v08\n- v0c --refutes--> v0e\n- v08 --refutes--> v10\n- v02 --refutes--> v0c\n- v05 --refutes--> v0f\n- v05 --refutes--> v07\n- v09 --supports--> v05\n- v01 --supports--> v03\n- v03 --refutes--> v09\n- v0a --supports--> v0b\n- v05 --supports--> v02\n- v00 --refutes--> v11\n- v00 --refutes--> v0d\n- v10 --refutes--> v0a\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"00e909c6056c5b687cb27a45605d98b908a58eefa4a12c4794465e72939b3756","schema":"livegraph.boolean-propagation.v1","spec_hash":"330debfe342b10a1801720f332c5c13ba9900e281855363e366937f64260e851","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-272f57a1643e2f5a2471","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v11\n- v03 --refutes--> v0b\n- v0a --refutes--> v09\n- v11 --refutes--> v01\n- v0d --supports--> v08\n- v05 --refutes--> v06\n- v0c --refutes--> v0d\n- v0f --refutes--> v0c\n- v11 --supports--> v04\n- v11 --supports--> v0f\n- v0c --supports--> v10\n- v00 --supports--> v0e\n- v0e --refutes--> v0a\n- v12 --supports--> v05\n- v0a --refutes--> v12\n- v0a --refutes--> v02\n- v07 --refutes--> v03\n- v0d --supports--> v07\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f1ae5202ff09540b1fd2f8960a7405fb485f09a39187fe83fa58ce10fc700534","schema":"livegraph.boolean-propagation.v1","spec_hash":"272f57a1643e2f5a247119e5d90a05532af587b4b1ba4c93a7cfbc8386953c83","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-65c4361ab1d59fd05d90","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v0f\n- v0b --supports--> v02\n- v0d --refutes--> v10\n- v0b --refutes--> v05\n- v00 --refutes--> v01\n- v03 --refutes--> v06\n- v10 --supports--> v11\n- v05 --refutes--> v12\n- v09 --supports--> v03\n- v01 --supports--> v09\n- v0f --supports--> v0b\n- v09 --refutes--> v0a\n- v10 --supports--> v0c\n- v06 --refutes--> v07\n- v01 --supports--> v0e\n- v11 --supports--> v04\n- v04 --supports--> v08\n- v04 --supports--> v00\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7ab8c91224f357f957835b22165b3cd3b1c304d75b9083c28cadd71c48ff2098","schema":"livegraph.boolean-propagation.v1","spec_hash":"65c4361ab1d59fd05d90994365e2570a56c20949b90235df0f82426538725101","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-b8ef49ae8337f7dde21c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v02\n- v03 --supports--> v11\n- v09 --supports--> v06\n- v00 --supports--> v10\n- v06 --refutes--> v03\n- v11 --refutes--> v0a\n- v0b --supports--> v0e\n- v00 --supports--> v0f\n- v0d --refutes--> v00\n- v09 --supports--> v12\n- v0f --refutes--> v04\n- v0e --supports--> v01\n- v06 --refutes--> v0c\n- v08 --refutes--> v07\n- v11 --supports--> v08\n- v02 --supports--> v0d\n- v07 --supports--> v0b\n- v08 --supports--> v05\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"453eb8d19ea47afa5e99429d09da578fdb425db17ac317ea1dfefd36dc8dcda0","schema":"livegraph.boolean-propagation.v1","spec_hash":"b8ef49ae8337f7dde21ce0f7cfc15b4e5acd7d735598615a7214605794905305","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-188b2280a06fedc746a7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v02\n- v09 --supports--> v03\n- v12 --supports--> v0f\n- v06 --refutes--> v07\n- v12 --refutes--> v00\n- v0b --refutes--> v0d\n- v09 --refutes--> v01\n- v0f --supports--> v09\n- v11 --refutes--> v0e\n- v0a --supports--> v0c\n- v11 --refutes--> v10\n- v05 --refutes--> v0b\n- v10 --supports--> v08\n- v03 --refutes--> v04\n- v04 --refutes--> v11\n- v10 --supports--> v05\n- v0d --refutes--> v06\n- v06 --refutes--> v0a\n\n[VERIFIED ROOT]\nv12 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ebddc35ea7fba887bd9ff65db4c03ccf6c1584c20d6895eb0a23ece743c21a38","schema":"livegraph.boolean-propagation.v1","spec_hash":"188b2280a06fedc746a7cb9f0d34fe13e45f71c37763f487d7bd9b1680295d1f","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v12\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-c51cae4f38540984cee1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v01\n- v05 --supports--> v03\n- v02 --refutes--> v07\n- v03 --supports--> v0b\n- v12 --refutes--> v00\n- v12 --supports--> v06\n- v04 --refutes--> v10\n- v0f --refutes--> v08\n- v02 --refutes--> v0e\n- v0d --supports--> v04\n- v11 --refutes--> v0f\n- v12 --refutes--> v0a\n- v0c --refutes--> v0d\n- v0b --supports--> v02\n- v04 --supports--> v05\n- v02 --supports--> v12\n- v08 --refutes--> v0c\n- v06 --refutes--> v09\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"884e9a5895bd9e912a96aa773657ac73f5e2d8899b06afb7b7fa528a48201132","schema":"livegraph.boolean-propagation.v1","spec_hash":"c51cae4f38540984cee19df74769404b90bc7e4993f33d5294845987641eafa6","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-2030444ff6f4a3698333","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v0b\n- v12 --supports--> v03\n- v0b --refutes--> v02\n- v00 --supports--> v0d\n- v0a --supports--> v01\n- v00 --supports--> v0e\n- v0f --refutes--> v11\n- v08 --supports--> v0f\n- v03 --supports--> v10\n- v0a --refutes--> v0c\n- v11 --supports--> v0a\n- v03 --refutes--> v07\n- v0f --refutes--> v05\n- v06 --supports--> v09\n- v01 --supports--> v12\n- v02 --supports--> v06\n- v02 --refutes--> v04\n- v09 --refutes--> v00\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d09bcbea6eca35765a1dad6f9af9750a78dfc93dcafa5c51911c99f84d832a04","schema":"livegraph.boolean-propagation.v1","spec_hash":"2030444ff6f4a36983333ebaf7eda78f4ca1a017299b2100fccc8d6fff839b20","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-859fd40c7d912cf60de3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v0f\n- v06 --supports--> v0b\n- v06 --refutes--> v05\n- v00 --refutes--> v02\n- v0c --refutes--> v12\n- v07 --supports--> v04\n- v0a --supports--> v0c\n- v10 --supports--> v01\n- v08 --refutes--> v0e\n- v00 --refutes--> v0a\n- v07 --refutes--> v08\n- v0c --supports--> v03\n- v0e --supports--> v06\n- v05 --supports--> v00\n- v0d --refutes--> v10\n- v08 --supports--> v11\n- v0f --refutes--> v09\n- v10 --supports--> v07\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d0d987d16a35805961bdff3c08c31ba587488d119f5062a83c129f73e1e2c971","schema":"livegraph.boolean-propagation.v1","spec_hash":"859fd40c7d912cf60de384ed38585f8a1c3d75c02e7a503914f3086febc3d9b5","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-08075c4e9a8072fca877","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v08\n- v0d --supports--> v0b\n- v00 --supports--> v0c\n- v05 --refutes--> v00\n- v01 --refutes--> v0a\n- v0a --supports--> v0e\n- v04 --supports--> v0d\n- v01 --supports--> v0f\n- v12 --refutes--> v05\n- v06 --supports--> v03\n- v03 --supports--> v04\n- v0d --supports--> v11\n- v06 --refutes--> v07\n- v0b --supports--> v12\n- v00 --refutes--> v01\n- v05 --refutes--> v10\n- v0e --refutes--> v09\n- v09 --refutes--> v02\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"aa89dc244e3feb499086cea1988487a5ad18da643ce611cf349b22c6a1bb0c7b","schema":"livegraph.boolean-propagation.v1","spec_hash":"08075c4e9a8072fca877f796d12e21d5959eefb0a3c969cd4e14a86a6ab43b48","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-0a6a931a6504a8792192","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v04\n- v08 --refutes--> v03\n- v0a --refutes--> v05\n- v0a --supports--> v07\n- v05 --supports--> v0b\n- v0b --refutes--> v11\n- v02 --supports--> v09\n- v09 --supports--> v0d\n- v06 --refutes--> v10\n- v12 --supports--> v0c\n- v00 --supports--> v06\n- v11 --refutes--> v01\n- v0d --refutes--> v0a\n- v0b --refutes--> v0e\n- v10 --refutes--> v0f\n- v12 --refutes--> v08\n- v10 --supports--> v02\n- v11 --refutes--> v12\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3cb55e7f029ba0f1910d433f49f9b7e7b896a647f37a4192690f147800105396","schema":"livegraph.boolean-propagation.v1","spec_hash":"0a6a931a6504a87921921b596271c6748d25a42d330cfcfad1eac3025a02562c","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-af2ccdcc5194a7a5369a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v11\n- v06 --refutes--> v04\n- v12 --refutes--> v0e\n- v09 --supports--> v10\n- v00 --supports--> v06\n- v04 --supports--> v0f\n- v12 --supports--> v0c\n- v0b --refutes--> v0d\n- v09 --refutes--> v07\n- v09 --supports--> v0a\n- v0f --supports--> v02\n- v02 --refutes--> v01\n- v09 --supports--> v05\n- v07 --refutes--> v00\n- v09 --refutes--> v03\n- v0f --supports--> v08\n- v01 --supports--> v0b\n- v0d --supports--> v12\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"96b76f62df0a031ee5c2b6e4b951eac4541e87136e442b2ed9a6f1cae3bbdd8e","schema":"livegraph.boolean-propagation.v1","spec_hash":"af2ccdcc5194a7a5369abc4fae969e572380199b7f2151ed44feae7a85ad8ca9","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-2e570416a834becdaa92","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v08\n- v06 --supports--> v11\n- v01 --supports--> v06\n- v09 --refutes--> v0b\n- v02 --refutes--> v12\n- v0e --refutes--> v09\n- v03 --supports--> v00\n- v02 --refutes--> v10\n- v0a --refutes--> v07\n- v07 --refutes--> v02\n- v0d --refutes--> v0f\n- v09 --refutes--> v0d\n- v0e --refutes--> v0c\n- v00 --refutes--> v0a\n- v04 --supports--> v01\n- v05 --supports--> v03\n- v06 --supports--> v05\n- v02 --supports--> v0e\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"483a2421cd9c23011f3483a0343ea5c31be604f9bc4c4182359bbbe3f7cb3d77","schema":"livegraph.boolean-propagation.v1","spec_hash":"2e570416a834becdaa92a0f8edd0d192c2d37fb5f2034dbe3ce4eb5fe3ad7f32","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"}],\"hop\":12,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-591e0996582779df0bb4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v09\n- v02 --supports--> v0c\n- v02 --supports--> v12\n- v0e --supports--> v02\n- v0f --refutes--> v11\n- v0b --supports--> v01\n- v0f --refutes--> v10\n- v11 --refutes--> v06\n- v05 --supports--> v08\n- v05 --supports--> v0d\n- v03 --refutes--> v07\n- v00 --refutes--> v0f\n- v0b --supports--> v03\n- v0d --supports--> v00\n- v0c --refutes--> v05\n- v0a --refutes--> v0e\n- v07 --refutes--> v0a\n- v0e --refutes--> v04\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"041edd433beffe461e3656726c2adaea12417d0e859d8067c143da6bb3c40bab","schema":"livegraph.boolean-propagation.v1","spec_hash":"591e0996582779df0bb484e6e1ab1a6e5e0c126ea5fd8d3f42419b198dd572d4","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-f05ca8f8509d72f3b8b9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v06\n- v0a --refutes--> v0f\n- v0d --supports--> v0a\n- v08 --refutes--> v0d\n- v11 --supports--> v10\n- v0f --supports--> v0e\n- v02 --supports--> v04\n- v05 --supports--> v03\n- v08 --refutes--> v09\n- v0e --refutes--> v07\n- v0c --supports--> v01\n- v03 --supports--> v08\n- v01 --refutes--> v0b\n- v07 --supports--> v11\n- v11 --refutes--> v0c\n- v02 --supports--> v05\n- v02 --refutes--> v00\n- v0c --refutes--> v12\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cb4b9ae0596da9854fa9b5a9eec0b08eb052eae4ef4e514af0adbe7cab4383b6","schema":"livegraph.boolean-propagation.v1","spec_hash":"f05ca8f8509d72f3b8b9f58a0924fac6b51b9c28f41eeb22a0d7603bb8d8f084","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-49738b904be98c52575a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v00\n- v0b --refutes--> v05\n- v06 --supports--> v0a\n- v0e --refutes--> v06\n- v0d --supports--> v04\n- v0b --supports--> v01\n- v10 --refutes--> v0e\n- v06 --supports--> v11\n- v07 --supports--> v0b\n- v0d --supports--> v12\n- v03 --refutes--> v0d\n- v06 --refutes--> v08\n- v0a --supports--> v0f\n- v0c --refutes--> v09\n- v05 --supports--> v03\n- v04 --supports--> v02\n- v04 --refutes--> v0c\n- v09 --supports--> v10\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c40cfbfaa35cccae5bcfaaad9956b5cbf26d4316ccb57a8e4fa1b524fb3197bb","schema":"livegraph.boolean-propagation.v1","spec_hash":"49738b904be98c52575a906040cf2d1a604506df8ec4f7ab12548de72fe89f39","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-c46e54585743b99b4138","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v08\n- v0d --supports--> v06\n- v03 --refutes--> v07\n- v0f --refutes--> v05\n- v01 --supports--> v0d\n- v12 --refutes--> v10\n- v06 --supports--> v0f\n- v10 --refutes--> v09\n- v03 --refutes--> v00\n- v03 --supports--> v0b\n- v06 --supports--> v02\n- v0f --refutes--> v03\n- v07 --supports--> v0c\n- v08 --supports--> v11\n- v0d --refutes--> v04\n- v11 --refutes--> v01\n- v07 --supports--> v0e\n- v0e --refutes--> v12\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c9cb7f40c40c739305561088cb128fbd87c0f0bd5e72e799290f88397b2c9739","schema":"livegraph.boolean-propagation.v1","spec_hash":"c46e54585743b99b4138e23701a47e1b31684d2c7ce2b3d17a3b31abfafe4fa1","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-51e1567343feeed87d0a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v05\n- v11 --refutes--> v10\n- v07 --refutes--> v08\n- v0c --refutes--> v0f\n- v0a --refutes--> v09\n- v0c --supports--> v06\n- v04 --supports--> v03\n- v0d --refutes--> v0b\n- v04 --refutes--> v0a\n- v00 --supports--> v04\n- v00 --refutes--> v02\n- v10 --supports--> v0c\n- v0e --supports--> v00\n- v0c --supports--> v01\n- v09 --supports--> v0d\n- v0d --supports--> v12\n- v06 --refutes--> v0e\n- v08 --refutes--> v11\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5d01439263b8d0f6fc33e6a27d452d987ad6a1ae6153687f43e2bf904120ba95","schema":"livegraph.boolean-propagation.v1","spec_hash":"51e1567343feeed87d0aa8155a8b0ccd81f293213b1e55c74102ae149da8c48a","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-ca5df80618869cdfda16","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v12\n- v02 --supports--> v08\n- v06 --supports--> v05\n- v07 --supports--> v0c\n- v10 --refutes--> v03\n- v05 --supports--> v02\n- v05 --refutes--> v11\n- v00 --supports--> v07\n- v09 --supports--> v00\n- v0c --supports--> v0d\n- v0d --refutes--> v0a\n- v08 --supports--> v09\n- v09 --refutes--> v0f\n- v03 --refutes--> v04\n- v0a --supports--> v10\n- v03 --supports--> v0e\n- v08 --supports--> v01\n- v05 --supports--> v0b\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"38655defc84b285d05985c6ee2bdcabf99a2f94a50a5dfa30bee02a083903fcb","schema":"livegraph.boolean-propagation.v1","spec_hash":"ca5df80618869cdfda168f661c3af608c2e08df5c81b177f7353149259bc63b2","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-18b0d174f46ec6faebbb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v0e\n- v12 --supports--> v0f\n- v03 --supports--> v11\n- v07 --refutes--> v0c\n- v0d --refutes--> v07\n- v03 --refutes--> v02\n- v01 --supports--> v06\n- v09 --supports--> v03\n- v04 --refutes--> v09\n- v00 --supports--> v0a\n- v04 --supports--> v08\n- v04 --refutes--> v10\n- v0b --refutes--> v00\n- v02 --refutes--> v05\n- v02 --refutes--> v01\n- v07 --supports--> v0b\n- v06 --supports--> v12\n- v0e --supports--> v04\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6c0279f139bf01d7e6f664efcf228e2e0bd2b33d31de4c31e5b3b7ed8ab615ff","schema":"livegraph.boolean-propagation.v1","spec_hash":"18b0d174f46ec6faebbb068bd52f6a34e67ac255a3c18aa98c2cbc34949866ab","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-aaed10a2d039653c3944","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v0a\n- v0b --supports--> v01\n- v0d --refutes--> v06\n- v01 --refutes--> v0f\n- v03 --refutes--> v09\n- v12 --refutes--> v0e\n- v09 --refutes--> v0d\n- v05 --supports--> v03\n- v06 --supports--> v02\n- v10 --supports--> v07\n- v08 --refutes--> v11\n- v0d --refutes--> v00\n- v11 --refutes--> v05\n- v12 --supports--> v0c\n- v02 --refutes--> v12\n- v09 --refutes--> v04\n- v10 --supports--> v08\n- v01 --supports--> v10\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"92239fc4ad7cffb4e7b5096b8d3dc7660193263a45236def6dee51148bcdc4e7","schema":"livegraph.boolean-propagation.v1","spec_hash":"aaed10a2d039653c3944f261523a16a8de6b8298ff618aad5cdc033fc6b5f895","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-d58628d40853808f7116","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --supports--> v03\n- v10 --supports--> v05\n- v02 --refutes--> v00\n- v0b --supports--> v12\n- v0f --refutes--> v0c\n- v01 --supports--> v02\n- v03 --refutes--> v11\n- v07 --refutes--> v0e\n- v02 --supports--> v10\n- v05 --refutes--> v0d\n- v05 --supports--> v06\n- v0e --refutes--> v09\n- v0e --refutes--> v0a\n- v07 --supports--> v04\n- v0e --supports--> v0b\n- v11 --refutes--> v01\n- v0d --supports--> v0f\n- v05 --refutes--> v08\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"df7342df6c4f0a881bc9840e585f0e37a2dda6d54a1ce9c7cb64600e33a69006","schema":"livegraph.boolean-propagation.v1","spec_hash":"d58628d40853808f71164c1fc0ca0fbb16b525f5b31d015ad2fa26da8876bd99","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-4cfa71ffaabbe0716c84","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v03\n- v05 --refutes--> v01\n- v02 --supports--> v07\n- v0c --refutes--> v0e\n- v00 --supports--> v05\n- v02 --refutes--> v08\n- v0f --supports--> v09\n- v0b --supports--> v00\n- v09 --supports--> v02\n- v0c --refutes--> v11\n- v0a --supports--> v0c\n- v0b --refutes--> v12\n- v0c --supports--> v0f\n- v0d --refutes--> v04\n- v0f --refutes--> v10\n- v01 --supports--> v0d\n- v03 --refutes--> v0a\n- v02 --refutes--> v06\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ae22a339b5c2db2e63c05164ecd48efc79e57a244feee843bdbcc11ed24cc926","schema":"livegraph.boolean-propagation.v1","spec_hash":"4cfa71ffaabbe0716c8404cf78deaf9faa8fb2d09e801f97739fb00a25e6b7c0","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":12,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-346a4a8258630ef5bbe9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v05\n- v06 --supports--> v00\n- v06 --supports--> v09\n- v0c --refutes--> v0a\n- v12 --supports--> v02\n- v03 --refutes--> v10\n- v10 --supports--> v01\n- v06 --supports--> v04\n- v02 --supports--> v0b\n- v0f --refutes--> v0c\n- v0d --refutes--> v07\n- v01 --refutes--> v12\n- v12 --supports--> v08\n- v0d --supports--> v03\n- v0c --refutes--> v11\n- v0b --refutes--> v0e\n- v0a --supports--> v06\n- v00 --supports--> v0d\n\n[VERIFIED ROOT]\nv0f = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c97bacb6406f66cd383c01ac01d8c5e40873317247d3067edb9656e3e6494364","schema":"livegraph.boolean-propagation.v1","spec_hash":"346a4a8258630ef5bbe95a58ee1ce521746a3b8e4f2139ba0ebdaa6b6d94e455","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-b0a32b1c4871c2565db6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v09\n- v11 --supports--> v0a\n- v0e --supports--> v0d\n- v11 --refutes--> v05\n- v0a --refutes--> v02\n- v0b --supports--> v03\n- v02 --refutes--> v0b\n- v04 --refutes--> v12\n- v0d --supports--> v00\n- v01 --supports--> v0e\n- v04 --supports--> v0c\n- v0d --supports--> v06\n- v12 --refutes--> v11\n- v03 --supports--> v10\n- v0f --supports--> v08\n- v01 --refutes--> v07\n- v0f --supports--> v01\n- v06 --supports--> v04\n\n[VERIFIED ROOT]\nv0f = FALSE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a4233a04a5022321c15623984e20fe5371e331dbdc194e643a3c89126adaa23d","schema":"livegraph.boolean-propagation.v1","spec_hash":"b0a32b1c4871c2565db6d635f41955b4bd4051e3ede49eb5d5582299fdcb3318","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-6729fa4ba42f7687bb43","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v08\n- v09 --refutes--> v06\n- v12 --supports--> v01\n- v0e --refutes--> v04\n- v01 --supports--> v00\n- v05 --refutes--> v0a\n- v07 --refutes--> v03\n- v0c --refutes--> v07\n- v07 --refutes--> v0d\n- v0d --supports--> v02\n- v00 --supports--> v0e\n- v06 --refutes--> v05\n- v04 --supports--> v0c\n- v12 --refutes--> v0f\n- v10 --refutes--> v09\n- v12 --refutes--> v11\n- v00 --supports--> v0b\n- v05 --refutes--> v12\n\n[VERIFIED ROOT]\nv10 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8188338b86445227287cf76375a163195eb357882a4d2404f03e7b37925e45c4","schema":"livegraph.boolean-propagation.v1","spec_hash":"6729fa4ba42f7687bb431ae93f0b09006fbe32ed32f4724adc68ff5f1a0bc08a","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-bbe5b5819ef15365e126","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v09\n- v03 --refutes--> v0e\n- v11 --refutes--> v07\n- v07 --supports--> v03\n- v04 --supports--> v05\n- v05 --refutes--> v11\n- v03 --supports--> v12\n- v10 --supports--> v00\n- v0a --refutes--> v02\n- v10 --refutes--> v0d\n- v0e --supports--> v08\n- v0e --supports--> v06\n- v0a --refutes--> v10\n- v06 --supports--> v0b\n- v0a --supports--> v0c\n- v0f --supports--> v0a\n- v01 --supports--> v0f\n- v09 --supports--> v01\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"78a31f8a6f2a7cbd7a18eec2e26724b7cd30457d17fcd306414c0fb0cc811622","schema":"livegraph.boolean-propagation.v1","spec_hash":"bbe5b5819ef15365e126b58b5a09c49892451e643ca43b400a855aaf3699af4e","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-b58dd174f76d977ae787","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v01\n- v09 --refutes--> v0f\n- v0a --supports--> v09\n- v05 --refutes--> v04\n- v10 --supports--> v0b\n- v03 --refutes--> v0c\n- v0f --refutes--> v07\n- v02 --supports--> v10\n- v0a --refutes--> v0e\n- v04 --supports--> v08\n- v0b --supports--> v06\n- v06 --refutes--> v05\n- v10 --refutes--> v11\n- v00 --supports--> v03\n- v04 --refutes--> v0d\n- v0c --refutes--> v12\n- v0c --supports--> v0a\n- v0f --supports--> v02\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3e26ea1906408f301b211aaa78fca136d23fe9e914d9c1c84d7887db94f99bef","schema":"livegraph.boolean-propagation.v1","spec_hash":"b58dd174f76d977ae78728461bfb7e05dba7dd50b2a0e983ec7735e5829ca616","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-1029b553f6bbe3c324ea","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v00\n- v09 --refutes--> v06\n- v05 --supports--> v04\n- v0a --refutes--> v05\n- v05 --supports--> v08\n- v04 --refutes--> v01\n- v07 --supports--> v09\n- v0f --refutes--> v03\n- v0f --refutes--> v10\n- v0a --refutes--> v0b\n- v09 --refutes--> v12\n- v02 --refutes--> v0d\n- v0e --supports--> v07\n- v11 --supports--> v0f\n- v01 --refutes--> v11\n- v0d --refutes--> v0e\n- v10 --refutes--> v02\n- v05 --refutes--> v0c\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"773daca3baf7e352de6f063eb6cdde5ea604f0fed3b4e9ff62e283bcfb81bc30","schema":"livegraph.boolean-propagation.v1","spec_hash":"1029b553f6bbe3c324ead84c821f196f00aa09ddbd79aafb98cc11a5fb142f58","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-f5c4f8d13ba4fcd3c4cf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v0d\n- v09 --supports--> v12\n- v09 --refutes--> v10\n- v09 --supports--> v06\n- v02 --refutes--> v0a\n- v06 --refutes--> v01\n- v08 --refutes--> v05\n- v0f --refutes--> v11\n- v00 --supports--> v0e\n- v0c --supports--> v08\n- v0e --supports--> v09\n- v07 --supports--> v02\n- v03 --supports--> v07\n- v0a --supports--> v0c\n- v0f --refutes--> v03\n- v06 --refutes--> v04\n- v06 --refutes--> v0f\n- v0d --supports--> v0b\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fe05682f38701f34f47002006998d4d55c69a4a5c36421aac05658ad168c626f","schema":"livegraph.boolean-propagation.v1","spec_hash":"f5c4f8d13ba4fcd3c4cf68b379363c355a52686f5a1db4e81b7e0a0263f55103","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":12,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-a08426d99dc16089db78","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v0e\n- v0b --refutes--> v00\n- v04 --supports--> v0c\n- v0a --refutes--> v02\n- v05 --refutes--> v09\n- v10 --refutes--> v03\n- v0e --refutes--> v0d\n- v05 --refutes--> v07\n- v12 --refutes--> v11\n- v02 --refutes--> v01\n- v03 --refutes--> v05\n- v04 --refutes--> v08\n- v12 --refutes--> v04\n- v07 --supports--> v0f\n- v0d --refutes--> v06\n- v0b --refutes--> v12\n- v07 --supports--> v0b\n- v0d --refutes--> v10\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"40fbaf14eccb9db80dfd59b4b954c58f5484a96cf923410d17b59d6077d84a6f","schema":"livegraph.boolean-propagation.v1","spec_hash":"a08426d99dc16089db782b6ab579d36ecd310768fce49464c89807961d3d5e5d","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-edadb1e0fbe713f1a101","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v0a\n- v03 --refutes--> v04\n- v0c --supports--> v10\n- v12 --refutes--> v0c\n- v07 --supports--> v06\n- v11 --refutes--> v0e\n- v12 --refutes--> v0f\n- v05 --supports--> v0b\n- v03 --refutes--> v01\n- v04 --refutes--> v12\n- v00 --refutes--> v02\n- v09 --supports--> v0d\n- v08 --supports--> v11\n- v0d --supports--> v00\n- v05 --supports--> v03\n- v0c --refutes--> v09\n- v0e --refutes--> v05\n- v00 --refutes--> v07\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"552c401c14245421eca9fd49579787c433e2b8339c976e6095ce8cf4fc39388b","schema":"livegraph.boolean-propagation.v1","spec_hash":"edadb1e0fbe713f1a101ddb1b5e34456b5cc404c485d01ba63b65a5e1c155144","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-7ac3fca2d76cad7a6208","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v04\n- v09 --refutes--> v00\n- v09 --refutes--> v06\n- v03 --refutes--> v0c\n- v0b --refutes--> v0a\n- v0b --refutes--> v0e\n- v04 --refutes--> v01\n- v11 --supports--> v08\n- v0e --supports--> v10\n- v00 --refutes--> v02\n- v04 --refutes--> v0d\n- v05 --supports--> v0f\n- v07 --refutes--> v11\n- v08 --refutes--> v12\n- v0e --refutes--> v07\n- v03 --refutes--> v09\n- v0d --refutes--> v03\n- v12 --supports--> v05\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"119676f5a939fd75dccedbb767e25665393a434fd94ae1fcf76587fea15e0a42","schema":"livegraph.boolean-propagation.v1","spec_hash":"7ac3fca2d76cad7a62085b44d37e0698f0b65a0c8d78c17d57e43a0039a2b92d","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-c2ced07b85b6dfe7aea8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --supports--> v0b\n- v0d --supports--> v0e\n- v10 --supports--> v02\n- v07 --supports--> v03\n- v00 --refutes--> v07\n- v10 --refutes--> v09\n- v0c --supports--> v01\n- v11 --supports--> v0d\n- v04 --refutes--> v12\n- v12 --supports--> v0a\n- v01 --refutes--> v00\n- v0f --supports--> v04\n- v12 --supports--> v11\n- v12 --supports--> v06\n- v01 --refutes--> v05\n- v0e --refutes--> v0c\n- v10 --refutes--> v0f\n- v08 --refutes--> v10\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9080267d150c0409a41257cbd6dbc7bc1e8d40b338c4d540e733335d898e67d7","schema":"livegraph.boolean-propagation.v1","spec_hash":"c2ced07b85b6dfe7aea86eb0bc812090ae5ae1d90a92c80d08e3a4bbaf8b013e","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-579fea97c23306110620","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --supports--> v07\n- v09 --supports--> v05\n- v0d --refutes--> v00\n- v00 --supports--> v02\n- v09 --supports--> v11\n- v00 --refutes--> v03\n- v0e --supports--> v0c\n- v0b --supports--> v01\n- v05 --refutes--> v0a\n- v0d --supports--> v06\n- v0a --supports--> v0f\n- v02 --refutes--> v09\n- v10 --refutes--> v0e\n- v0b --supports--> v12\n- v0b --refutes--> v08\n- v0f --refutes--> v0b\n- v0c --supports--> v0d\n- v0e --refutes--> v04\n\n[VERIFIED ROOT]\nv10 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"929042bb80345428de6ccb6f880140af9f9aed99713cd7f6764691e0c171ab3c","schema":"livegraph.boolean-propagation.v1","spec_hash":"579fea97c23306110620b947f4062e998540cce792df8bcdae23d57ae7d49bcc","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-259e60f143a36d337f7d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v05\n- v0e --refutes--> v0c\n- v07 --supports--> v01\n- v12 --supports--> v03\n- v06 --supports--> v02\n- v00 --supports--> v04\n- v0d --supports--> v12\n- v0a --refutes--> v0e\n- v01 --supports--> v00\n- v12 --supports--> v10\n- v0f --refutes--> v06\n- v0e --refutes--> v09\n- v08 --refutes--> v0a\n- v04 --supports--> v11\n- v04 --supports--> v0f\n- v0e --refutes--> v0b\n- v12 --supports--> v07\n- v02 --refutes--> v08\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b645b5a2e2f5b984b8c1c55ab8285cb9a477f66be13573406849cfe9ef2fbeb2","schema":"livegraph.boolean-propagation.v1","spec_hash":"259e60f143a36d337f7df1f8eebf35ab35d780f40f0d4d0f1d38825050d7d7a7","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-48ebe538d0037989098b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v09\n- v02 --supports--> v00\n- v03 --supports--> v05\n- v09 --supports--> v0c\n- v02 --refutes--> v0d\n- v07 --supports--> v04\n- v10 --supports--> v01\n- v06 --supports--> v0b\n- v0e --supports--> v03\n- v03 --refutes--> v0f\n- v02 --refutes--> v06\n- v11 --refutes--> v02\n- v08 --supports--> v10\n- v05 --supports--> v07\n- v04 --refutes--> v0a\n- v11 --supports--> v12\n- v06 --refutes--> v08\n- v09 --supports--> v11\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2a03e8850ca768fea31d5e9627547a09a8c993b35d871869d3e2b9a3dab130d2","schema":"livegraph.boolean-propagation.v1","spec_hash":"48ebe538d0037989098bdadfc26fac11e40f2af9018755e4a2e368055a77be83","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-2c98a22fa58fec83d877","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v05\n- v0f --supports--> v11\n- v01 --refutes--> v06\n- v07 --supports--> v03\n- v02 --supports--> v00\n- v02 --supports--> v12\n- v09 --refutes--> v10\n- v0a --refutes--> v08\n- v0c --supports--> v07\n- v07 --supports--> v01\n- v04 --supports--> v0f\n- v11 --refutes--> v02\n- v10 --supports--> v0a\n- v02 --supports--> v0e\n- v02 --supports--> v0d\n- v08 --refutes--> v0c\n- v06 --supports--> v04\n- v0a --supports--> v0b\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6d998b9f8d605dca3b62a341217fccec7b440c15b6fd9e7e8109a66770fa3df5","schema":"livegraph.boolean-propagation.v1","spec_hash":"2c98a22fa58fec83d8774909138944b139d039f6d95bf7a4349b378ab9753794","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-4865e6fc2783af5a5f7e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v05\n- v05 --supports--> v10\n- v11 --supports--> v03\n- v04 --supports--> v09\n- v0a --supports--> v11\n- v02 --supports--> v07\n- v0b --refutes--> v12\n- v08 --refutes--> v06\n- v12 --refutes--> v00\n- v07 --refutes--> v08\n- v08 --refutes--> v0f\n- v0f --supports--> v0b\n- v04 --supports--> v02\n- v03 --refutes--> v0e\n- v05 --supports--> v01\n- v01 --supports--> v04\n- v07 --refutes--> v0d\n- v07 --supports--> v0c\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ea3ba2dfe73b85b287802c37e730520148894ada60624a8831a07402f1b7ac6f","schema":"livegraph.boolean-propagation.v1","spec_hash":"4865e6fc2783af5a5f7e94677805e5ea4113fb272018bab8db73a442d512f935","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-820bc29c7c4b56418a12","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v07\n- v01 --supports--> v06\n- v04 --supports--> v05\n- v0c --refutes--> v04\n- v02 --refutes--> v0d\n- v05 --supports--> v11\n- v08 --supports--> v01\n- v0e --refutes--> v08\n- v0e --refutes--> v10\n- v0d --refutes--> v0e\n- v0a --supports--> v0b\n- v0a --refutes--> v00\n- v03 --refutes--> v09\n- v0d --supports--> v0f\n- v06 --supports--> v0c\n- v0b --supports--> v02\n- v05 --supports--> v03\n- v08 --supports--> v12\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"036175376041fbc57c6f0a25c0e6605f561d71eab5598dee7032cd90975ad43c","schema":"livegraph.boolean-propagation.v1","spec_hash":"820bc29c7c4b56418a1244801858103186cc5a0bf996af9eb0e8e9ef51831a06","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"}],\"hop\":12,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-fb49a4c8430cc1d91ea4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v12\n- v11 --supports--> v0a\n- v10 --refutes--> v0e\n- v05 --supports--> v0b\n- v02 --refutes--> v11\n- v11 --supports--> v09\n- v0e --refutes--> v03\n- v0b --refutes--> v0d\n- v0b --refutes--> v07\n- v00 --supports--> v02\n- v0a --supports--> v04\n- v01 --refutes--> v00\n- v01 --supports--> v0c\n- v06 --refutes--> v05\n- v04 --supports--> v06\n- v0b --supports--> v0f\n- v03 --refutes--> v01\n- v06 --supports--> v08\n\n[VERIFIED ROOT]\nv10 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d89ce2722d596d92866dcc02bf3aeaee02c629c0640da3e8ca6e409c79e22d1c","schema":"livegraph.boolean-propagation.v1","spec_hash":"fb49a4c8430cc1d91ea467134b6b307ca464910f3ed5896a61d597978a79a290","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-031b137caf336edcd6ca","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v07\n- v07 --supports--> v0f\n- v09 --refutes--> v08\n- v11 --refutes--> v02\n- v11 --supports--> v0e\n- v05 --refutes--> v0d\n- v00 --refutes--> v0a\n- v09 --supports--> v12\n- v03 --supports--> v0b\n- v04 --supports--> v05\n- v12 --supports--> v04\n- v09 --refutes--> v06\n- v0d --refutes--> v11\n- v10 --supports--> v0c\n- v07 --supports--> v01\n- v10 --supports--> v03\n- v0b --refutes--> v00\n- v07 --refutes--> v10\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"52c266ea0f184bfb8115209da9e0346b0da11a1531de87dd3c7105dff3a77aa8","schema":"livegraph.boolean-propagation.v1","spec_hash":"031b137caf336edcd6ca9083383fb81bed697ae6656372dbc53a398032730204","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-ad8ff6abad4c3ad818d4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v00\n- v12 --refutes--> v03\n- v08 --supports--> v0c\n- v02 --supports--> v04\n- v07 --refutes--> v0d\n- v0f --supports--> v07\n- v01 --supports--> v0b\n- v07 --supports--> v10\n- v0a --refutes--> v12\n- v0b --refutes--> v02\n- v0b --supports--> v09\n- v0a --supports--> v06\n- v03 --supports--> v0f\n- v0c --refutes--> v11\n- v00 --refutes--> v08\n- v0a --supports--> v05\n- v0c --refutes--> v0e\n- v11 --refutes--> v0a\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f249923d40b37fc8dec5a2ee46d2e05bd2370c6fd0216406998869cfe30c9e59","schema":"livegraph.boolean-propagation.v1","spec_hash":"ad8ff6abad4c3ad818d492ecb483df8b2bc063efeca68aa51fb13d47bf796871","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-7d99bb1eae92e2319c46","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v06\n- v0b --refutes--> v12\n- v06 --refutes--> v10\n- v06 --supports--> v00\n- v03 --refutes--> v11\n- v11 --refutes--> v04\n- v09 --supports--> v0f\n- v0a --supports--> v0c\n- v01 --supports--> v0b\n- v00 --supports--> v01\n- v06 --refutes--> v08\n- v05 --refutes--> v03\n- v07 --supports--> v02\n- v04 --supports--> v0e\n- v05 --refutes--> v0d\n- v0b --supports--> v05\n- v07 --supports--> v0a\n- v0f --supports--> v07\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"eef643eef8582092b4afbc22466c810038615eda6b74b7382aa03eaf389f035c","schema":"livegraph.boolean-propagation.v1","spec_hash":"7d99bb1eae92e2319c463d79cbb5ab90bb9a4cf465076f4e89c6f8ec22e909b1","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-4c6bfd49599ac03c2828","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v0e\n- v09 --refutes--> v01\n- v0e --refutes--> v0c\n- v03 --supports--> v00\n- v08 --refutes--> v10\n- v0a --supports--> v03\n- v04 --supports--> v09\n- v04 --refutes--> v07\n- v06 --refutes--> v0a\n- v00 --supports--> v0f\n- v08 --refutes--> v12\n- v10 --refutes--> v04\n- v0a --supports--> v05\n- v0b --refutes--> v0d\n- v00 --refutes--> v08\n- v0b --refutes--> v02\n- v0d --refutes--> v11\n- v0c --refutes--> v0b\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"80c47cb0539309f81f764721e7ce70d915c7185a7b1379a318086f2a6286cae7","schema":"livegraph.boolean-propagation.v1","spec_hash":"4c6bfd49599ac03c282820e95415593e3f1963948fecdad07ae7e2b6c35718f3","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-8dd3d31a6214157cf640","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v10\n- v12 --supports--> v0a\n- v01 --supports--> v0c\n- v01 --refutes--> v00\n- v02 --refutes--> v0f\n- v10 --supports--> v01\n- v08 --refutes--> v04\n- v08 --supports--> v12\n- v12 --refutes--> v06\n- v0e --refutes--> v09\n- v0b --refutes--> v0d\n- v11 --supports--> v0b\n- v0d --refutes--> v08\n- v00 --refutes--> v05\n- v0b --supports--> v03\n- v0f --supports--> v0e\n- v0e --supports--> v07\n- v00 --refutes--> v11\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"97ae29a35dd8d13b793aa662221545f08c7218bfd5248458d43b70429b260289","schema":"livegraph.boolean-propagation.v1","spec_hash":"8dd3d31a6214157cf640d3691638db8a3f0628b2a1741802df8d32aa3c02a4f4","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-8e2f12fab3dbbdc8b575","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --supports--> v01\n- v11 --supports--> v12\n- v0c --supports--> v04\n- v07 --refutes--> v09\n- v03 --supports--> v0f\n- v0a --supports--> v08\n- v07 --refutes--> v00\n- v00 --refutes--> v05\n- v10 --refutes--> v02\n- v08 --supports--> v06\n- v05 --supports--> v0b\n- v02 --refutes--> v03\n- v11 --supports--> v0d\n- v05 --supports--> v0c\n- v0c --refutes--> v0e\n- v0c --refutes--> v11\n- v0d --refutes--> v0a\n- v0f --refutes--> v07\n\n[VERIFIED ROOT]\nv10 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"29c2eeb7ec998179e72cd929dd25e13583ecd25f83792ffc5f04ab2167b3efca","schema":"livegraph.boolean-propagation.v1","spec_hash":"8e2f12fab3dbbdc8b575357ae00e627cd80309ad4c320e69dc8826190a855435","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-49654d4ec92995586848","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v09\n- v0b --refutes--> v06\n- v0d --refutes--> v01\n- v11 --refutes--> v04\n- v12 --refutes--> v0c\n- v01 --refutes--> v10\n- v07 --refutes--> v0b\n- v0b --refutes--> v0d\n- v09 --refutes--> v03\n- v0f --supports--> v08\n- v09 --refutes--> v02\n- v00 --refutes--> v0f\n- v08 --refutes--> v0a\n- v12 --supports--> v0e\n- v0a --refutes--> v11\n- v01 --refutes--> v00\n- v05 --supports--> v07\n- v12 --supports--> v05\n\n[VERIFIED ROOT]\nv12 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0acd8836df147b670f844255c45eb6557dbad11af0f461057dd2e8b155c02afd","schema":"livegraph.boolean-propagation.v1","spec_hash":"49654d4ec929955868481a2f3c5e6b30e5d02bdc18524d4b8078706c18b79183","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v12\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-a4089d904bac85e827f8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v08\n- v0c --supports--> v07\n- v0b --supports--> v10\n- v05 --refutes--> v11\n- v03 --refutes--> v12\n- v0b --refutes--> v03\n- v00 --supports--> v0c\n- v06 --supports--> v05\n- v05 --refutes--> v0e\n- v0b --refutes--> v04\n- v01 --supports--> v02\n- v08 --supports--> v00\n- v0c --refutes--> v0b\n- v12 --refutes--> v0d\n- v09 --refutes--> v01\n- v08 --refutes--> v0f\n- v02 --refutes--> v0a\n- v12 --refutes--> v06\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"046cae9768cffcccb3096f171066c93906aa982182d3b2aa5c222fb542c2a24e","schema":"livegraph.boolean-propagation.v1","spec_hash":"a4089d904bac85e827f8702a0fcbaf11538da17be87986decee6fb71a1bc3f7a","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-eca6b586e21784984cdc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v11\n- v08 --refutes--> v05\n- v04 --supports--> v10\n- v02 --supports--> v06\n- v11 --refutes--> v00\n- v11 --refutes--> v0f\n- v0c --refutes--> v0d\n- v0b --refutes--> v03\n- v03 --refutes--> v01\n- v09 --supports--> v08\n- v02 --refutes--> v07\n- v04 --refutes--> v12\n- v06 --refutes--> v0c\n- v11 --refutes--> v0e\n- v0a --supports--> v09\n- v0f --refutes--> v0a\n- v01 --refutes--> v02\n- v0c --supports--> v04\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"884e4026afa163bb3ebf0c5bda459e5554422d1358c9fc29d87b9e17c4ccc9b1","schema":"livegraph.boolean-propagation.v1","spec_hash":"eca6b586e21784984cdcbf28e30447cf7037657fb30bf1f136e2bac6b73a654b","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-ecebe60819f8eca52914","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v05\n- v0e --refutes--> v0b\n- v08 --supports--> v11\n- v0c --refutes--> v00\n- v00 --refutes--> v09\n- v09 --supports--> v01\n- v0e --supports--> v03\n- v11 --refutes--> v0d\n- v11 --refutes--> v02\n- v06 --refutes--> v07\n- v03 --refutes--> v08\n- v0a --supports--> v12\n- v04 --refutes--> v10\n- v0d --supports--> v0f\n- v09 --supports--> v06\n- v04 --supports--> v0e\n- v0a --supports--> v0c\n- v07 --supports--> v04\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"128d8d905140803a94ff994a097bdf820d044bc3588deeb02956b89b412b9fef","schema":"livegraph.boolean-propagation.v1","spec_hash":"ecebe60819f8eca52914661c30a51c3ef53271c53965c8e333d1b567ee7715db","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-3991592f7d91d03354b2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v04\n- v04 --refutes--> v06\n- v12 --refutes--> v0d\n- v10 --refutes--> v00\n- v08 --supports--> v10\n- v03 --refutes--> v0f\n- v05 --refutes--> v12\n- v0e --refutes--> v0a\n- v0a --refutes--> v02\n- v12 --supports--> v01\n- v0c --supports--> v09\n- v04 --supports--> v0b\n- v01 --supports--> v0c\n- v09 --refutes--> v08\n- v0a --supports--> v07\n- v04 --supports--> v03\n- v02 --refutes--> v05\n- v08 --refutes--> v11\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"de5ca79ff6676d48d4d99962d95a2ac6af8da916a1c50aad017d31bc3ce67259","schema":"livegraph.boolean-propagation.v1","spec_hash":"3991592f7d91d03354b2b7d4f22782b631d5feef9521281ea66179916e226d1a","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-402c2824b949aa8bd89a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v05\n- v10 --refutes--> v04\n- v07 --refutes--> v00\n- v0a --refutes--> v0d\n- v02 --supports--> v10\n- v04 --refutes--> v0c\n- v09 --refutes--> v07\n- v08 --refutes--> v0b\n- v06 --supports--> v09\n- v07 --refutes--> v13\n- v07 --supports--> v12\n- v08 --supports--> v03\n- v05 --supports--> v0a\n- v00 --refutes--> v02\n- v0a --supports--> v0e\n- v0a --refutes--> v0f\n- v05 --refutes--> v11\n- v03 --supports--> v06\n- v0d --refutes--> v01\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a962d275494c6b0c23dcafb7c50205b6cdbf5b942b292973f434479f595d1098","schema":"livegraph.boolean-propagation.v1","spec_hash":"402c2824b949aa8bd89a2f1f7154f73f9f373b42a300dfd4d4e40a209ac339fe","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":13,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-56541016a7bdacab4799","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v0b\n- v13 --supports--> v00\n- v07 --refutes--> v0a\n- v06 --supports--> v01\n- v08 --refutes--> v02\n- v0a --supports--> v12\n- v0e --refutes--> v06\n- v02 --supports--> v03\n- v09 --refutes--> v0c\n- v12 --refutes--> v09\n- v09 --refutes--> v0f\n- v06 --supports--> v07\n- v0a --refutes--> v10\n- v0f --supports--> v08\n- v11 --refutes--> v0d\n- v03 --supports--> v05\n- v0b --refutes--> v13\n- v13 --refutes--> v11\n- v12 --refutes--> v04\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"49154898ca5a073c71c395eb17ef504b1f1cae238c6784990dae2269c47accb2","schema":"livegraph.boolean-propagation.v1","spec_hash":"56541016a7bdacab4799aba489c6e4d2bf1a868784b0b299b18cd6b10b230056","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-656295cda46ee71858d6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --refutes--> v0c\n- v06 --refutes--> v10\n- v10 --refutes--> v07\n- v0f --supports--> v12\n- v00 --supports--> v0b\n- v0c --supports--> v08\n- v12 --supports--> v11\n- v12 --refutes--> v05\n- v03 --supports--> v13\n- v00 --supports--> v01\n- v10 --refutes--> v0d\n- v0e --refutes--> v00\n- v12 --refutes--> v04\n- v0f --refutes--> v02\n- v0d --supports--> v0e\n- v09 --refutes--> v03\n- v0b --supports--> v09\n- v03 --refutes--> v0a\n- v08 --supports--> v0f\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"36fee0be56ffb900e982a9869c02b4cef794a441df4872ce5e2d45bae4b434d0","schema":"livegraph.boolean-propagation.v1","spec_hash":"656295cda46ee71858d69fe2242eb9a3351889e5814a635fa6ad08b1879cc49e","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-a34947d4beba2ab3f6fa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v06\n- v0c --supports--> v07\n- v0d --supports--> v12\n- v04 --supports--> v00\n- v13 --supports--> v0f\n- v07 --refutes--> v05\n- v02 --supports--> v04\n- v0f --supports--> v02\n- v10 --refutes--> v0b\n- v05 --refutes--> v09\n- v12 --supports--> v13\n- v10 --refutes--> v03\n- v09 --supports--> v0d\n- v13 --supports--> v08\n- v0c --refutes--> v0a\n- v0d --supports--> v01\n- v0b --refutes--> v0c\n- v06 --refutes--> v0e\n- v09 --refutes--> v11\n\n[VERIFIED ROOT]\nv10 = FALSE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"be703f266a790a2d42a7a814e2a2c4b7e628850db8f4fa699c276772c53dcc2a","schema":"livegraph.boolean-propagation.v1","spec_hash":"a34947d4beba2ab3f6faa1a356bba1d4fa915a1dad5acddfcf2a030e7876c88a","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-ac244c767b1b43d4c0cb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v02\n- v12 --supports--> v05\n- v13 --supports--> v00\n- v0a --refutes--> v0f\n- v09 --refutes--> v13\n- v0b --refutes--> v06\n- v0e --supports--> v01\n- v0e --supports--> v0b\n- v09 --supports--> v11\n- v0e --refutes--> v08\n- v07 --supports--> v0d\n- v04 --supports--> v0e\n- v0d --supports--> v0c\n- v0c --refutes--> v04\n- v12 --supports--> v10\n- v06 --supports--> v03\n- v02 --supports--> v12\n- v0a --supports--> v09\n- v05 --refutes--> v0a\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"158e4a1a70dee2327c0066f38ae52c2af4e89ac7406c6fa1c1f9451b623c6041","schema":"livegraph.boolean-propagation.v1","spec_hash":"ac244c767b1b43d4c0cb4ba97bbd143fcfbe1b4f0c0bd458ff431dba44096b34","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-2a82b847bc29332c2934","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v13\n- v01 --supports--> v0e\n- v0f --refutes--> v02\n- v02 --supports--> v0b\n- v09 --refutes--> v03\n- v08 --supports--> v06\n- v04 --supports--> v07\n- v06 --refutes--> v12\n- v01 --supports--> v0a\n- v0c --supports--> v00\n- v02 --refutes--> v0d\n- v12 --refutes--> v11\n- v04 --refutes--> v0f\n- v10 --refutes--> v0c\n- v0b --supports--> v10\n- v0c --supports--> v01\n- v0f --supports--> v05\n- v0a --supports--> v09\n- v03 --refutes--> v08\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3d4415ee866f512dbd79a9ed879cc83f2a073d6deb3c8b29e0717176d53956ce","schema":"livegraph.boolean-propagation.v1","spec_hash":"2a82b847bc29332c29349f5dc92a049f961e268f782197eb1961e391c51b56c6","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-d796de0a35ee1387c9d9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v01\n- v04 --refutes--> v07\n- v13 --supports--> v10\n- v12 --refutes--> v05\n- v06 --supports--> v02\n- v13 --refutes--> v0b\n- v01 --refutes--> v08\n- v13 --refutes--> v09\n- v06 --refutes--> v0f\n- v00 --supports--> v12\n- v13 --refutes--> v0a\n- v10 --supports--> v03\n- v05 --refutes--> v0d\n- v03 --supports--> v11\n- v0d --supports--> v13\n- v13 --supports--> v0e\n- v01 --supports--> v04\n- v11 --supports--> v0c\n- v02 --refutes--> v00\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7bb04bf76d3406fc10a3bea53ba8122c73a1752e72b8e8eef2301498bc9a496f","schema":"livegraph.boolean-propagation.v1","spec_hash":"d796de0a35ee1387c9d9ac298745db4ba961a596225c99e70da1e86a7e65ae6e","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-a957450dbdd108f5efb7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v0b\n- v00 --refutes--> v0f\n- v02 --refutes--> v03\n- v02 --refutes--> v05\n- v09 --supports--> v11\n- v10 --supports--> v09\n- v03 --refutes--> v13\n- v00 --refutes--> v08\n- v12 --supports--> v04\n- v13 --refutes--> v01\n- v03 --refutes--> v06\n- v0c --supports--> v0e\n- v12 --supports--> v00\n- v01 --refutes--> v0d\n- v0e --refutes--> v02\n- v0b --refutes--> v10\n- v00 --supports--> v07\n- v08 --refutes--> v0c\n- v02 --supports--> v0a\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f8b9735ec1b29deb313c771d0517480c6f8daf9d47548daafdba130c94f71885","schema":"livegraph.boolean-propagation.v1","spec_hash":"a957450dbdd108f5efb7ee23d348c8836f6fa9bb0ba90443ca6e34c59d1bf6ac","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-45a6550be5d132e30547","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v11\n- v08 --supports--> v06\n- v03 --refutes--> v12\n- v02 --refutes--> v0a\n- v0c --refutes--> v0d\n- v07 --refutes--> v04\n- v13 --refutes--> v07\n- v03 --refutes--> v02\n- v13 --supports--> v0f\n- v08 --supports--> v00\n- v0b --supports--> v03\n- v0d --refutes--> v0e\n- v0b --refutes--> v09\n- v0c --supports--> v10\n- v04 --supports--> v01\n- v0e --refutes--> v08\n- v04 --supports--> v05\n- v00 --refutes--> v13\n- v05 --refutes--> v0b\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5a5f1194acd3b660621b7c376f566b74ea505a80b9b4a216acbd9dcc6d752bf5","schema":"livegraph.boolean-propagation.v1","spec_hash":"45a6550be5d132e30547af33f42304ffe2614a06cd6fc305c56a4fadf376b992","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-7e585036e4c527db6868","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v0e\n- v0f --supports--> v12\n- v11 --supports--> v08\n- v01 --supports--> v10\n- v0a --supports--> v06\n- v0e --supports--> v0c\n- v0d --supports--> v0f\n- v0e --refutes--> v09\n- v0b --refutes--> v0d\n- v00 --supports--> v05\n- v08 --supports--> v07\n- v0e --refutes--> v13\n- v05 --supports--> v04\n- v0e --supports--> v0b\n- v10 --supports--> v0a\n- v0f --supports--> v11\n- v07 --supports--> v02\n- v0b --refutes--> v03\n- v06 --refutes--> v00\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3f27de7a275273d21366344c9de3c50179531019ed6628d726384dbc81bc5f1f","schema":"livegraph.boolean-propagation.v1","spec_hash":"7e585036e4c527db6868fcd4a3d28f1801ee250a0ab48e9be1df1e728affe5e5","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":13,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-fdb1e6975c11cb6a3733","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v0b\n- v04 --refutes--> v0f\n- v0e --supports--> v0d\n- v0f --refutes--> v01\n- v0d --supports--> v10\n- v13 --supports--> v02\n- v05 --supports--> v06\n- v10 --supports--> v0a\n- v01 --refutes--> v12\n- v09 --refutes--> v13\n- v02 --supports--> v0e\n- v0d --supports--> v00\n- v08 --supports--> v0c\n- v01 --refutes--> v09\n- v11 --supports--> v03\n- v08 --supports--> v11\n- v05 --refutes--> v08\n- v03 --supports--> v04\n- v01 --refutes--> v07\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6cda64f787244f2bdba450104f7731b9e737d986887fc6b7905bb24dd9640f0c","schema":"livegraph.boolean-propagation.v1","spec_hash":"fdb1e6975c11cb6a37335da831cbb3c590002be187934641df71cc8a595c8b21","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-4cc515cc86ef68e80651","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v0b\n- v0e --supports--> v04\n- v11 --refutes--> v03\n- v0e --refutes--> v0f\n- v02 --refutes--> v13\n- v00 --supports--> v0c\n- v13 --supports--> v09\n- v0f --refutes--> v01\n- v09 --supports--> v00\n- v01 --supports--> v0a\n- v13 --refutes--> v07\n- v00 --refutes--> v0d\n- v10 --refutes--> v05\n- v03 --supports--> v10\n- v06 --refutes--> v11\n- v0a --supports--> v06\n- v11 --refutes--> v08\n- v03 --refutes--> v12\n- v05 --refutes--> v02\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1c963a14f1e95735baa060ee81dac99617ae79e275dd22f2c1e3e8afe391431c","schema":"livegraph.boolean-propagation.v1","spec_hash":"4cc515cc86ef68e80651c9ff569eac5958763ef37c2291cccb74abd01129e539","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-3dba8163fae38f53c94f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v0f\n- v0b --refutes--> v01\n- v07 --refutes--> v0a\n- v0f --refutes--> v07\n- v13 --supports--> v00\n- v0a --refutes--> v08\n- v02 --supports--> v05\n- v08 --supports--> v09\n- v0c --supports--> v03\n- v05 --refutes--> v06\n- v0f --supports--> v12\n- v08 --supports--> v02\n- v00 --refutes--> v11\n- v11 --refutes--> v0c\n- v03 --refutes--> v10\n- v06 --refutes--> v0b\n- v03 --supports--> v04\n- v03 --refutes--> v0d\n- v00 --supports--> v0e\n\n[VERIFIED ROOT]\nv13 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cf1e7c302ded6a616dc7f9c9689972705e731340f3be5f3064ed2515c9052428","schema":"livegraph.boolean-propagation.v1","spec_hash":"3dba8163fae38f53c94f98bd8349a2e22692ff4379d9be6a269421529095e0b3","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-b1cb78682f74299dfa00","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v12\n- v06 --supports--> v0d\n- v13 --supports--> v11\n- v0e --supports--> v09\n- v0f --refutes--> v04\n- v0d --supports--> v13\n- v02 --refutes--> v05\n- v0a --supports--> v0e\n- v01 --refutes--> v03\n- v0b --refutes--> v10\n- v13 --supports--> v0c\n- v13 --refutes--> v0f\n- v02 --refutes--> v07\n- v10 --supports--> v0a\n- v0f --supports--> v08\n- v03 --supports--> v02\n- v05 --refutes--> v06\n- v0e --supports--> v01\n- v00 --supports--> v0b\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"271f3ecf0d6227be10f401fcd723da3d795e2f3cdf70c164126988c89c5f9907","schema":"livegraph.boolean-propagation.v1","spec_hash":"b1cb78682f74299dfa00b008fc784d05ac1965a9a799f490b881f020952b6cac","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-33e5202ea306d5c4f2fb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v13\n- v11 --supports--> v12\n- v08 --supports--> v07\n- v0f --refutes--> v06\n- v03 --refutes--> v10\n- v0c --refutes--> v0f\n- v10 --refutes--> v0c\n- v02 --refutes--> v00\n- v01 --refutes--> v0e\n- v09 --refutes--> v01\n- v07 --supports--> v05\n- v13 --supports--> v0d\n- v07 --refutes--> v09\n- v13 --supports--> v02\n- v07 --supports--> v0b\n- v11 --refutes--> v04\n- v01 --refutes--> v0a\n- v02 --supports--> v03\n- v06 --refutes--> v08\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cc2c2ddc93cbfa2f212d6280771f27a974554d9857850175fa0bb38082d5b722","schema":"livegraph.boolean-propagation.v1","spec_hash":"33e5202ea306d5c4f2fbf36a8e96e53a41749b2ec86ee50f18b2365e558b4343","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-8a9458ab5f7926d8c0e3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v09\n- v07 --refutes--> v0d\n- v12 --supports--> v07\n- v00 --refutes--> v10\n- v0a --supports--> v12\n- v13 --refutes--> v0f\n- v02 --refutes--> v04\n- v0e --supports--> v06\n- v00 --supports--> v03\n- v06 --supports--> v08\n- v02 --refutes--> v13\n- v13 --supports--> v00\n- v11 --refutes--> v0c\n- v06 --supports--> v05\n- v11 --refutes--> v01\n- v09 --supports--> v0e\n- v05 --refutes--> v0a\n- v07 --supports--> v11\n- v01 --supports--> v0b\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1f5b8c7305b9c5c495dc0feff45e1d5da3d7e3cb1645895a8a86f816bb060c33","schema":"livegraph.boolean-propagation.v1","spec_hash":"8a9458ab5f7926d8c0e3003d8a832ba9d27624f64990ebbe7b088f22ea6b90c1","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-18fcb2a98816a6ee744d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v11\n- v13 --supports--> v08\n- v0c --supports--> v07\n- v0b --refutes--> v0f\n- v06 --refutes--> v09\n- v05 --supports--> v02\n- v0a --supports--> v0d\n- v0c --refutes--> v0a\n- v0b --refutes--> v06\n- v08 --supports--> v10\n- v01 --supports--> v0b\n- v11 --supports--> v13\n- v0b --refutes--> v12\n- v02 --supports--> v00\n- v06 --supports--> v03\n- v09 --refutes--> v04\n- v04 --refutes--> v0c\n- v08 --refutes--> v01\n- v09 --supports--> v0e\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4c85ac7869741b4e416cc0ce61d0c59f012e4da12d29dbca0d17752fbcf6d793","schema":"livegraph.boolean-propagation.v1","spec_hash":"18fcb2a98816a6ee744ddfeb7aa4b77ab19672abcae20eac0a5594dda73d307e","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-abb3b0555722c81f7a56","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v05\n- v01 --supports--> v13\n- v02 --supports--> v09\n- v09 --refutes--> v07\n- v13 --refutes--> v06\n- v13 --supports--> v03\n- v06 --supports--> v0d\n- v12 --refutes--> v10\n- v11 --supports--> v0b\n- v0c --refutes--> v04\n- v0e --supports--> v0f\n- v0f --supports--> v00\n- v06 --refutes--> v02\n- v0b --supports--> v0c\n- v07 --refutes--> v11\n- v0f --refutes--> v01\n- v02 --refutes--> v08\n- v11 --supports--> v0a\n- v04 --refutes--> v12\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b2d16f3690991b590fb9df78c8b211b25a0d97ae0f0438d8f14d5a92e0c0d5ac","schema":"livegraph.boolean-propagation.v1","spec_hash":"abb3b0555722c81f7a563b9285bb0041053d5177588653eb92d0d38bc712ebfd","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-a1c94609983e4cf23e3a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v07\n- v03 --refutes--> v13\n- v0f --supports--> v03\n- v13 --refutes--> v09\n- v00 --refutes--> v11\n- v10 --refutes--> v0a\n- v01 --supports--> v02\n- v00 --refutes--> v0b\n- v13 --supports--> v0d\n- v0a --refutes--> v0f\n- v0e --supports--> v10\n- v08 --refutes--> v04\n- v01 --refutes--> v12\n- v02 --supports--> v0e\n- v00 --refutes--> v0c\n- v07 --supports--> v00\n- v05 --refutes--> v08\n- v05 --refutes--> v06\n- v13 --supports--> v05\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3d7f3c201172a4f896d4aa2185fb5daee20a81bad758eb3b3b4f6adcc85700cc","schema":"livegraph.boolean-propagation.v1","spec_hash":"a1c94609983e4cf23e3a0ff3ea6ca589adb1c1e664d5539bd1bacaf6f20d18c4","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-b54b820be9677cadfc08","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v05\n- v08 --supports--> v06\n- v11 --supports--> v0a\n- v06 --refutes--> v0c\n- v05 --supports--> v0d\n- v04 --supports--> v11\n- v0d --supports--> v08\n- v02 --supports--> v07\n- v0f --supports--> v09\n- v0e --supports--> v0f\n- v0f --refutes--> v13\n- v02 --supports--> v0b\n- v04 --refutes--> v01\n- v0b --supports--> v03\n- v11 --supports--> v02\n- v0f --supports--> v04\n- v0f --supports--> v12\n- v10 --supports--> v00\n- v0c --refutes--> v10\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3cfab6f0ae352ffeecd836fd7e7522681cd9e8a4feb779602301204d363f1f48","schema":"livegraph.boolean-propagation.v1","spec_hash":"b54b820be9677cadfc089fcf9d610b64bd4684c54426ccf0ce5e24db340dfb1c","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":13,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-7595d85bf11623ea34eb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --refutes--> v03\n- v12 --supports--> v0e\n- v03 --supports--> v05\n- v0a --refutes--> v07\n- v04 --refutes--> v11\n- v03 --supports--> v02\n- v08 --refutes--> v0f\n- v02 --supports--> v0a\n- v08 --supports--> v04\n- v0b --refutes--> v0c\n- v02 --refutes--> v06\n- v08 --supports--> v01\n- v10 --refutes--> v13\n- v0e --supports--> v0d\n- v0c --refutes--> v10\n- v10 --refutes--> v00\n- v0d --supports--> v0b\n- v12 --supports--> v09\n- v07 --refutes--> v08\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6627845b7175236cbcd00640cec549c3abb61a542f532f46c1ce2f95a314a56d","schema":"livegraph.boolean-propagation.v1","spec_hash":"7595d85bf11623ea34ebf997e1b59cff137954bd26f763602dcf494e09cb602a","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-eb3280f60b27f90c4618","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v05\n- v0f --refutes--> v06\n- v0a --refutes--> v0e\n- v02 --supports--> v12\n- v0b --supports--> v02\n- v0d --supports--> v03\n- v00 --supports--> v07\n- v0d --supports--> v09\n- v0f --supports--> v0b\n- v03 --supports--> v0a\n- v02 --refutes--> v13\n- v00 --refutes--> v01\n- v0e --refutes--> v0c\n- v05 --refutes--> v00\n- v01 --supports--> v08\n- v13 --supports--> v11\n- v08 --refutes--> v0f\n- v08 --refutes--> v10\n- v0e --supports--> v04\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ddf18338feb04d9f816da88290dccff176f97f0c95042d7522c4168cf755d46c","schema":"livegraph.boolean-propagation.v1","spec_hash":"eb3280f60b27f90c461859044f58b892b6d310ea1cee65ef29526c8445a70255","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-777b139e36e824c6c018","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v0f\n- v0a --supports--> v0b\n- v10 --supports--> v02\n- v0c --refutes--> v07\n- v0d --refutes--> v08\n- v00 --supports--> v03\n- v0a --supports--> v00\n- v03 --supports--> v0e\n- v05 --supports--> v0c\n- v09 --refutes--> v12\n- v02 --refutes--> v01\n- v09 --refutes--> v04\n- v04 --supports--> v10\n- v09 --refutes--> v13\n- v03 --supports--> v0d\n- v0f --refutes--> v05\n- v09 --refutes--> v11\n- v07 --refutes--> v0a\n- v02 --supports--> v06\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"755af4a647f18080ac39d42c68978ddbfb59f37d092ee5c3f891cde90adad169","schema":"livegraph.boolean-propagation.v1","spec_hash":"777b139e36e824c6c01841458848e050fe24d963a91b2e357243202739e9b6bc","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":13,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-69f9a9b9e4dcdee6820a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v0b\n- v0b --refutes--> v09\n- v0d --supports--> v08\n- v02 --refutes--> v12\n- v10 --supports--> v0c\n- v13 --refutes--> v11\n- v03 --refutes--> v01\n- v05 --supports--> v10\n- v07 --refutes--> v0e\n- v11 --refutes--> v02\n- v02 --refutes--> v00\n- v0e --supports--> v05\n- v06 --supports--> v03\n- v0b --supports--> v0f\n- v05 --refutes--> v04\n- v0b --refutes--> v07\n- v01 --supports--> v13\n- v0c --refutes--> v0d\n- v02 --supports--> v0a\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"27642c72c2f88636e35ad629c6c06489252d8ea2ca48da3ee5ff1acd6fda6a22","schema":"livegraph.boolean-propagation.v1","spec_hash":"69f9a9b9e4dcdee6820acd8c84f4395df4f46c655bb526c77bdade871b487c5e","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-df8ac8748fd8ce70ae77","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v00\n- v0b --supports--> v0d\n- v0c --refutes--> v08\n- v04 --supports--> v09\n- v07 --refutes--> v0c\n- v07 --refutes--> v13\n- v0e --refutes--> v0b\n- v0c --refutes--> v01\n- v0f --refutes--> v04\n- v02 --refutes--> v0f\n- v0f --supports--> v10\n- v07 --refutes--> v12\n- v0a --refutes--> v02\n- v0d --supports--> v11\n- v05 --supports--> v03\n- v00 --refutes--> v07\n- v11 --refutes--> v05\n- v0d --refutes--> v06\n- v01 --supports--> v0a\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"09b6e9b4c9fee75c97d0dd02ff252eb7e35b2a29b35b40a7f97d4df90404383e","schema":"livegraph.boolean-propagation.v1","spec_hash":"df8ac8748fd8ce70ae779f0357e86663b4e0025397d983cc925a3d49e7803097","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":13,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-3f51cf2999d805068feb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v05\n- v07 --supports--> v08\n- v03 --supports--> v00\n- v04 --supports--> v0b\n- v04 --supports--> v02\n- v09 --supports--> v13\n- v06 --refutes--> v11\n- v01 --supports--> v07\n- v05 --supports--> v06\n- v04 --supports--> v0c\n- v11 --supports--> v01\n- v07 --supports--> v0e\n- v02 --refutes--> v03\n- v07 --refutes--> v12\n- v13 --refutes--> v10\n- v00 --refutes--> v0a\n- v09 --refutes--> v0f\n- v12 --supports--> v0d\n- v13 --supports--> v04\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"29d901e18581560b375076839cf964d33dd4f50803483e14792bb06a1fdad525","schema":"livegraph.boolean-propagation.v1","spec_hash":"3f51cf2999d805068feb40a2b7fa1cf57b5340591d7287dcb079743ce1a6d18c","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-5b429b70110a1b9f4674","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v04\n- v00 --supports--> v0c\n- v0c --supports--> v13\n- v09 --supports--> v0a\n- v0d --refutes--> v07\n- v10 --refutes--> v08\n- v08 --supports--> v02\n- v09 --refutes--> v0b\n- v0e --refutes--> v03\n- v13 --refutes--> v06\n- v02 --refutes--> v09\n- v07 --refutes--> v10\n- v01 --supports--> v00\n- v04 --supports--> v0d\n- v07 --supports--> v0f\n- v0b --refutes--> v01\n- v09 --refutes--> v11\n- v02 --refutes--> v12\n- v13 --refutes--> v05\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6012c0609bd58d5dbf8baf71db9b6ac2e5b18ec3275a7cf7175f4e4cce8fc7bd","schema":"livegraph.boolean-propagation.v1","spec_hash":"5b429b70110a1b9f467438af815e2d7a3aaa9c3bbe9d87580df60d4373186394","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-a31bb3c61421d6c2ce27","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v13\n- v0e --supports--> v0a\n- v04 --supports--> v09\n- v00 --refutes--> v0d\n- v09 --supports--> v02\n- v04 --supports--> v10\n- v0f --supports--> v0c\n- v0c --supports--> v00\n- v0b --supports--> v0f\n- v12 --supports--> v06\n- v05 --supports--> v08\n- v05 --refutes--> v07\n- v08 --refutes--> v0b\n- v0d --refutes--> v01\n- v08 --supports--> v03\n- v06 --refutes--> v04\n- v0a --supports--> v05\n- v0d --refutes--> v12\n- v0f --refutes--> v11\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dfcae22244567a62f04c1d5695cf412a146412138d8208ec9b79b0088480e87e","schema":"livegraph.boolean-propagation.v1","spec_hash":"a31bb3c61421d6c2ce2742a8a4b4f488d0b3a8591c861be4cdff5c038804bd74","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-77f53d6abdce0ebf2c71","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v13\n- v12 --refutes--> v0d\n- v0a --refutes--> v07\n- v11 --refutes--> v06\n- v0d --refutes--> v04\n- v02 --refutes--> v0c\n- v0f --refutes--> v02\n- v00 --supports--> v09\n- v0e --refutes--> v0a\n- v05 --refutes--> v10\n- v01 --supports--> v05\n- v0d --refutes--> v11\n- v01 --refutes--> v03\n- v07 --supports--> v0f\n- v02 --supports--> v00\n- v06 --refutes--> v0e\n- v09 --supports--> v08\n- v0f --refutes--> v0b\n- v05 --refutes--> v12\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a998e34197f099053fe576aa37faf1981b05c3ca5037a7efa4b2ffa7eed66f72","schema":"livegraph.boolean-propagation.v1","spec_hash":"77f53d6abdce0ebf2c712d71c2951df950be6cbcd72c6a4cd6bddde8de594aae","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-50bc68ea3344c2e44799","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v0e\n- v07 --refutes--> v08\n- v00 --supports--> v0f\n- v10 --supports--> v09\n- v0b --supports--> v10\n- v05 --supports--> v01\n- v08 --refutes--> v00\n- v11 --refutes--> v05\n- v0c --refutes--> v0d\n- v0c --refutes--> v0b\n- v01 --refutes--> v0a\n- v09 --supports--> v03\n- v07 --supports--> v12\n- v11 --refutes--> v02\n- v03 --supports--> v07\n- v11 --refutes--> v13\n- v06 --refutes--> v0c\n- v07 --supports--> v04\n- v0a --supports--> v06\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"412035cd49e0f67d2600a4647a7e3e30706381106516cc8ecaf6caa44b8204ac","schema":"livegraph.boolean-propagation.v1","spec_hash":"50bc68ea3344c2e44799199aca2cff1c38530a4bd46100e0cdb76177ed551519","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":13,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-2c4c58dd521b3548e635","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v07\n- v0f --supports--> v11\n- v00 --supports--> v01\n- v0f --refutes--> v03\n- v0c --supports--> v10\n- v00 --refutes--> v12\n- v13 --refutes--> v0a\n- v04 --supports--> v0d\n- v07 --supports--> v0f\n- v0a --supports--> v06\n- v0c --refutes--> v09\n- v0c --refutes--> v00\n- v0d --supports--> v05\n- v0b --supports--> v0e\n- v00 --refutes--> v04\n- v03 --supports--> v0b\n- v02 --supports--> v08\n- v05 --refutes--> v02\n- v06 --supports--> v0c\n\n[VERIFIED ROOT]\nv13 = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f2d98fa96a1a087c06b9797d5fc051c6aaae90e8ff075282ac467717ec160a79","schema":"livegraph.boolean-propagation.v1","spec_hash":"2c4c58dd521b3548e635cac680734ae8c7e2832721f1e7d536ab56cb6998819f","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-498bc75a028001b7c88e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v0f\n- v0e --refutes--> v06\n- v11 --refutes--> v0d\n- v07 --refutes--> v0b\n- v0d --supports--> v05\n- v00 --refutes--> v07\n- v04 --refutes--> v0c\n- v0a --refutes--> v13\n- v0c --supports--> v02\n- v0e --supports--> v0a\n- v12 --refutes--> v04\n- v0f --refutes--> v11\n- v12 --supports--> v10\n- v03 --supports--> v00\n- v0d --refutes--> v08\n- v0c --refutes--> v09\n- v0b --supports--> v0e\n- v0d --supports--> v12\n- v02 --supports--> v01\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"152626d10a759d00c9ff0b02cd8c660bb366281c4336c971dd7bdf0b73c2af9d","schema":"livegraph.boolean-propagation.v1","spec_hash":"498bc75a028001b7c88eb5dbba74a0e967073c83e870a3fba5e91ac501c167bc","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-97e7455870e48b4c5a67","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v07\n- v12 --refutes--> v0d\n- v00 --supports--> v13\n- v0c --refutes--> v0b\n- v07 --supports--> v08\n- v06 --refutes--> v0c\n- v01 --refutes--> v10\n- v05 --refutes--> v01\n- v04 --refutes--> v03\n- v0c --supports--> v0e\n- v0b --supports--> v04\n- v0c --supports--> v0f\n- v03 --supports--> v12\n- v13 --refutes--> v06\n- v03 --refutes--> v09\n- v08 --supports--> v00\n- v12 --refutes--> v05\n- v01 --refutes--> v11\n- v08 --supports--> v0a\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1d655b05760d7113b87111ee7fa99e43105cb38902b15f2da5915e65fe2de64a","schema":"livegraph.boolean-propagation.v1","spec_hash":"97e7455870e48b4c5a670b29a58d531fa589b13e8c701a6d92d579e0961707cf","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-c84488551dc28abe4be7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v0c\n- v10 --supports--> v11\n- v0f --supports--> v05\n- v0d --supports--> v06\n- v13 --refutes--> v09\n- v08 --supports--> v0d\n- v00 --supports--> v08\n- v13 --refutes--> v01\n- v05 --supports--> v02\n- v08 --supports--> v0b\n- v06 --supports--> v0e\n- v02 --refutes--> v04\n- v08 --refutes--> v0a\n- v01 --refutes--> v00\n- v00 --supports--> v03\n- v07 --refutes--> v0f\n- v04 --refutes--> v13\n- v05 --supports--> v12\n- v11 --supports--> v07\n\n[VERIFIED ROOT]\nv10 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"00ce75ec6a3f6e8a6b3e9c58650085e6d98895a198925ba5a0a691d7adef223b","schema":"livegraph.boolean-propagation.v1","spec_hash":"c84488551dc28abe4be7b3bc88d99ef8b4ee37e04b3076da7c8878ef9e932ca7","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-7d39e88b40f34fd6d453","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v0c\n- v09 --refutes--> v0e\n- v10 --refutes--> v04\n- v02 --supports--> v13\n- v0f --supports--> v06\n- v06 --refutes--> v0a\n- v00 --refutes--> v0d\n- v0f --supports--> v11\n- v09 --supports--> v03\n- v00 --supports--> v08\n- v09 --supports--> v12\n- v04 --supports--> v05\n- v0c --supports--> v02\n- v0c --refutes--> v01\n- v0d --supports--> v10\n- v13 --supports--> v07\n- v05 --supports--> v0f\n- v00 --supports--> v0b\n- v12 --refutes--> v00\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8352793e183f80e05af8a2b253fe2ddd117ea963c5a2d6fa5e085734e646889a","schema":"livegraph.boolean-propagation.v1","spec_hash":"7d39e88b40f34fd6d453f34b061902a24219004959c325e598badd82e41d8016","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-6dec6c728de2f765a47f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v03\n- v02 --refutes--> v0c\n- v00 --supports--> v0e\n- v11 --supports--> v10\n- v02 --supports--> v0b\n- v04 --refutes--> v08\n- v11 --refutes--> v12\n- v04 --refutes--> v0a\n- v09 --refutes--> v04\n- v0b --refutes--> v0d\n- v13 --supports--> v11\n- v00 --supports--> v01\n- v0f --supports--> v13\n- v0a --supports--> v00\n- v07 --supports--> v05\n- v06 --refutes--> v02\n- v0b --refutes--> v07\n- v10 --supports--> v09\n- v0e --supports--> v06\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a9eabe7f3af0f8d674fde18cc6877b3bdd916ffc9569a740a9962275c202a684","schema":"livegraph.boolean-propagation.v1","spec_hash":"6dec6c728de2f765a47f83bdaa4dd2408a22adb6951b6d026da4de929c4e39bf","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-137674b5e0a7b7d28c32","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --supports--> v02\n- v0a --supports--> v04\n- v07 --supports--> v0c\n- v00 --supports--> v06\n- v0b --supports--> v0f\n- v08 --refutes--> v00\n- v08 --supports--> v09\n- v06 --refutes--> v07\n- v02 --supports--> v0b\n- v04 --refutes--> v05\n- v08 --supports--> v11\n- v10 --refutes--> v01\n- v0e --refutes--> v12\n- v04 --supports--> v08\n- v06 --supports--> v03\n- v0c --supports--> v13\n- v12 --supports--> v0a\n- v0b --supports--> v0d\n- v0f --supports--> v0e\n\n[VERIFIED ROOT]\nv10 = TRUE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"91eaeee4eb5b20bdb6e5af35fe10e27a5748fbd599cb13436db324894bcfbd8f","schema":"livegraph.boolean-propagation.v1","spec_hash":"137674b5e0a7b7d28c3220d3f857aea3fe61510cc19a90766435b34bd82e0a5e","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-08feac6c437447360268","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v07\n- v11 --supports--> v03\n- v09 --supports--> v10\n- v02 --refutes--> v0c\n- v05 --supports--> v01\n- v0c --refutes--> v05\n- v12 --refutes--> v06\n- v09 --supports--> v04\n- v03 --supports--> v00\n- v08 --refutes--> v02\n- v12 --supports--> v0e\n- v01 --refutes--> v0f\n- v0f --supports--> v0d\n- v06 --refutes--> v0a\n- v0f --supports--> v12\n- v00 --refutes--> v09\n- v12 --supports--> v0b\n- v12 --refutes--> v13\n- v04 --supports--> v08\n\n[VERIFIED ROOT]\nv11 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"07fb6e79e9d763a6b861ce2bb0af9bd4a03e86e06ae007de81ae92aa39f45ba3","schema":"livegraph.boolean-propagation.v1","spec_hash":"08feac6c437447360268100560919c50508ab0fb694c35e407af5c33d0671c2f","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-ab0661226d3eeb1c0760","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v0c\n- v06 --supports--> v05\n- v07 --supports--> v00\n- v01 --supports--> v09\n- v10 --supports--> v0a\n- v04 --supports--> v06\n- v12 --refutes--> v08\n- v08 --supports--> v02\n- v13 --refutes--> v0e\n- v11 --refutes--> v0f\n- v03 --refutes--> v0d\n- v0f --refutes--> v10\n- v07 --supports--> v12\n- v08 --refutes--> v11\n- v0d --refutes--> v01\n- v13 --refutes--> v07\n- v01 --refutes--> v04\n- v10 --refutes--> v0b\n- v0b --refutes--> v03\n\n[VERIFIED ROOT]\nv13 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"49a3ca883763d4d84681e21bf518af05910be25418ce885c4947428c6d3fa9a9","schema":"livegraph.boolean-propagation.v1","spec_hash":"ab0661226d3eeb1c07608fa0cb6c6d7928e7d9de54db3cadf23027eb5d818c43","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v13\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-cdfe497bce4c69c17bed","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v02\n- v0f --refutes--> v00\n- v08 --refutes--> v04\n- v00 --supports--> v12\n- v0a --refutes--> v09\n- v0e --supports--> v08\n- v0d --refutes--> v07\n- v06 --supports--> v11\n- v13 --refutes--> v0b\n- v0b --supports--> v0a\n- v01 --supports--> v0e\n- v11 --supports--> v05\n- v0a --refutes--> v0f\n- v13 --supports--> v03\n- v12 --supports--> v01\n- v10 --supports--> v0c\n- v11 --refutes--> v13\n- v0d --supports--> v10\n- v10 --supports--> v06\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"46a9539ed5b025bc2454c27963e535a9d23981c97fe76e67fa6aa960b1d58842","schema":"livegraph.boolean-propagation.v1","spec_hash":"cdfe497bce4c69c17bedb9d5c8fc2a2c72ce5f337bc10f1ba82bd4f6f32203ea","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-b8d3b494e3149a344d95","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v07\n- v0a --supports--> v0d\n- v12 --supports--> v00\n- v01 --supports--> v0c\n- v13 --supports--> v0f\n- v0c --refutes--> v0b\n- v0c --supports--> v0e\n- v07 --supports--> v01\n- v08 --refutes--> v06\n- v0e --supports--> v13\n- v01 --supports--> v11\n- v12 --refutes--> v09\n- v05 --refutes--> v04\n- v07 --supports--> v03\n- v06 --supports--> v02\n- v0f --refutes--> v05\n- v04 --refutes--> v0a\n- v13 --supports--> v10\n- v09 --supports--> v08\n\n[VERIFIED ROOT]\nv12 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3a1456904f0d7a3e7c69239e4b781f4290284e0c84ec5d0865e1f2d283338c76","schema":"livegraph.boolean-propagation.v1","spec_hash":"b8d3b494e3149a344d95fc527dd04d8742f098c7132b80e6138fe3a5b3079059","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v12\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-40ffccc3081189bd45d1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v0f\n- v05 --refutes--> v0e\n- v12 --supports--> v02\n- v12 --supports--> v08\n- v03 --supports--> v0d\n- v11 --refutes--> v0c\n- v00 --supports--> v06\n- v0c --supports--> v04\n- v02 --refutes--> v09\n- v05 --supports--> v0b\n- v03 --supports--> v0a\n- v00 --supports--> v12\n- v0a --supports--> v10\n- v03 --refutes--> v07\n- v0f --refutes--> v03\n- v13 --supports--> v01\n- v0a --supports--> v05\n- v0e --refutes--> v11\n- v09 --refutes--> v13\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"de6047e2deadd874bc2e8fc2001f7265aa3444d9acf5102497f37115ceaec8e6","schema":"livegraph.boolean-propagation.v1","spec_hash":"40ffccc3081189bd45d1d9a42b14b031e97578accf6af8f1b5f04e81b201dc39","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-e1d0e116e8877599afe9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v03\n- v09 --refutes--> v02\n- v04 --supports--> v12\n- v10 --refutes--> v13\n- v0e --supports--> v06\n- v0b --supports--> v0f\n- v13 --supports--> v0b\n- v06 --refutes--> v10\n- v0e --refutes--> v0d\n- v05 --refutes--> v08\n- v02 --supports--> v05\n- v05 --supports--> v0e\n- v0f --supports--> v00\n- v01 --refutes--> v09\n- v12 --supports--> v01\n- v0f --supports--> v0a\n- v09 --supports--> v0c\n- v00 --refutes--> v11\n- v01 --supports--> v07\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9385155d3606140b80d44d92993b3b111d8c3d0a7f668055203bdf5550bbe52a","schema":"livegraph.boolean-propagation.v1","spec_hash":"e1d0e116e8877599afe9205645cbbf7cc7fc8662d776400bd2fbc3cb5cd122c0","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-e3cf933967b5a7b21c3e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v02\n- v01 --refutes--> v08\n- v0a --supports--> v01\n- v09 --supports--> v0d\n- v0a --supports--> v0c\n- v02 --refutes--> v0f\n- v11 --supports--> v12\n- v12 --supports--> v07\n- v0f --refutes--> v0e\n- v0d --supports--> v04\n- v07 --supports--> v10\n- v0f --supports--> v11\n- v0d --refutes--> v0b\n- v13 --refutes--> v09\n- v10 --refutes--> v06\n- v11 --refutes--> v05\n- v00 --supports--> v0a\n- v10 --refutes--> v13\n- v07 --refutes--> v03\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"be100b4df898962ccc1190369db8c01f8bb991d480235ba58831316b9d88037f","schema":"livegraph.boolean-propagation.v1","spec_hash":"e3cf933967b5a7b21c3ea45a4c996de59753263fe8207acb91916fb0e0c16561","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-fcf594a6fa19f5df6086","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v02\n- v10 --supports--> v06\n- v10 --supports--> v04\n- v05 --refutes--> v12\n- v0f --refutes--> v0c\n- v09 --refutes--> v0f\n- v11 --refutes--> v10\n- v0c --refutes--> v08\n- v03 --supports--> v0e\n- v12 --refutes--> v07\n- v12 --supports--> v0d\n- v0c --supports--> v0a\n- v0e --supports--> v05\n- v04 --supports--> v13\n- v09 --refutes--> v00\n- v02 --supports--> v09\n- v0b --refutes--> v01\n- v0b --supports--> v03\n- v08 --supports--> v11\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f333d70184f8dca29ea721f6c348c0286d1884f693d2a7377a906bcab0dc072b","schema":"livegraph.boolean-propagation.v1","spec_hash":"fcf594a6fa19f5df60869d908800877f3d1dc4d2c454483fea27c36a46da9ab0","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-8545895063e98d605feb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v0a\n- v02 --supports--> v01\n- v02 --supports--> v06\n- v03 --supports--> v02\n- v0f --refutes--> v05\n- v05 --refutes--> v0e\n- v0e --supports--> v0b\n- v09 --supports--> v0c\n- v01 --supports--> v11\n- v13 --refutes--> v03\n- v0b --supports--> v08\n- v01 --supports--> v00\n- v08 --refutes--> v07\n- v0e --supports--> v10\n- v08 --refutes--> v04\n- v12 --refutes--> v0f\n- v00 --supports--> v0d\n- v0a --refutes--> v12\n- v00 --supports--> v09\n\n[VERIFIED ROOT]\nv13 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6a98493e6a82b33b80c746d7915aee649c4596a2b6342b489e03ea41cbd60e06","schema":"livegraph.boolean-propagation.v1","spec_hash":"8545895063e98d605febc6514d9a6cd59994a482715f394ddf339eab8fe740f2","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v13\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-801d8c32927465cb474e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v08\n- v0f --supports--> v01\n- v0a --refutes--> v10\n- v0b --refutes--> v09\n- v09 --supports--> v02\n- v0d --supports--> v0c\n- v13 --refutes--> v0f\n- v13 --refutes--> v05\n- v0c --supports--> v03\n- v0f --supports--> v07\n- v10 --refutes--> v12\n- v03 --refutes--> v11\n- v11 --refutes--> v00\n- v0c --refutes--> v06\n- v04 --refutes--> v0a\n- v11 --refutes--> v0b\n- v09 --supports--> v0e\n- v0f --refutes--> v0d\n- v12 --supports--> v13\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"da4b672f063ccde514fa0409af877ca6b44331cfd5deec25e62ff50fac939981","schema":"livegraph.boolean-propagation.v1","spec_hash":"801d8c32927465cb474ee2fecb94549ac8a51cc19bd27d06f5f9b90250f11f0a","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-2fa1ccf0f141c0e7aa56","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v07\n- v04 --refutes--> v0b\n- v11 --refutes--> v0e\n- v0c --supports--> v10\n- v0f --supports--> v01\n- v0e --refutes--> v04\n- v01 --supports--> v0a\n- v12 --refutes--> v08\n- v0b --supports--> v09\n- v0a --supports--> v11\n- v05 --supports--> v0f\n- v13 --refutes--> v05\n- v0e --supports--> v06\n- v13 --supports--> v0d\n- v08 --supports--> v0c\n- v0b --refutes--> v12\n- v08 --refutes--> v00\n- v04 --refutes--> v03\n- v10 --supports--> v02\n\n[VERIFIED ROOT]\nv13 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7ae782157db7fc9622dd0e523165e132ecd4711ecf90253db21721835f9fd2f9","schema":"livegraph.boolean-propagation.v1","spec_hash":"2fa1ccf0f141c0e7aa56c8a96b4217a5ead7431ec0ab0dd1f8fddeffda1da1a3","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-e9a15963662e542ef892","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v0b\n- v05 --supports--> v09\n- v12 --refutes--> v13\n- v00 --refutes--> v12\n- v13 --refutes--> v0a\n- v10 --supports--> v03\n- v0b --refutes--> v0f\n- v07 --supports--> v06\n- v13 --supports--> v0e\n- v08 --supports--> v00\n- v02 --supports--> v11\n- v07 --supports--> v10\n- v05 --supports--> v02\n- v0f --refutes--> v07\n- v01 --supports--> v08\n- v0e --refutes--> v04\n- v0e --refutes--> v05\n- v12 --refutes--> v0c\n- v01 --supports--> v0d\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f1903bdc0568fce955fe8875cfb43f91dfb5a5338029243136602f4a0038dc29","schema":"livegraph.boolean-propagation.v1","spec_hash":"e9a15963662e542ef89287fee24022046d4549eaf29e2ef6ffbc0c6fe8e0d795","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-0bdc3f11b9a4f11b77b2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v06\n- v05 --supports--> v03\n- v00 --refutes--> v07\n- v10 --refutes--> v01\n- v0a --refutes--> v10\n- v11 --refutes--> v0c\n- v02 --supports--> v05\n- v0b --supports--> v09\n- v00 --refutes--> v0d\n- v09 --supports--> v11\n- v04 --supports--> v0b\n- v0e --refutes--> v0a\n- v07 --refutes--> v04\n- v0c --supports--> v08\n- v10 --refutes--> v0f\n- v10 --refutes--> v13\n- v13 --refutes--> v12\n- v05 --supports--> v00\n- v13 --supports--> v02\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f0f547a1196c39022efdf59b3b6538b402c0bd52a60da8520f10ec61aee9bef8","schema":"livegraph.boolean-propagation.v1","spec_hash":"0bdc3f11b9a4f11b77b2744c035779fd10446cb49ad8030792562023f4e6d94b","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-7675913621adebc3f504","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v0c\n- v05 --refutes--> v04\n- v0c --supports--> v0b\n- v0e --refutes--> v0f\n- v0c --supports--> v01\n- v09 --refutes--> v12\n- v13 --supports--> v00\n- v10 --refutes--> v0a\n- v00 --refutes--> v10\n- v10 --supports--> v03\n- v10 --supports--> v0d\n- v08 --refutes--> v09\n- v03 --supports--> v0e\n- v12 --supports--> v02\n- v06 --refutes--> v08\n- v03 --refutes--> v07\n- v0b --supports--> v06\n- v0f --refutes--> v05\n- v0f --refutes--> v11\n\n[VERIFIED ROOT]\nv13 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3c9b31af32874ae816bdbbd96031e71d178f8dcaf52adb7bb33ef3db26929231","schema":"livegraph.boolean-propagation.v1","spec_hash":"7675913621adebc3f5042c9476189a9fc0529b0ff0b30941673e87f20bc72183","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v13\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-6c5c86e6e89d5fa79d66","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --refutes--> v0e\n- v02 --supports--> v0d\n- v0b --refutes--> v03\n- v03 --refutes--> v12\n- v0c --supports--> v01\n- v0f --refutes--> v07\n- v08 --supports--> v06\n- v03 --refutes--> v0c\n- v07 --refutes--> v00\n- v07 --refutes--> v10\n- v01 --refutes--> v13\n- v0e --supports--> v04\n- v08 --supports--> v05\n- v0f --supports--> v09\n- v0b --refutes--> v11\n- v0d --supports--> v08\n- v06 --refutes--> v0f\n- v0c --refutes--> v0a\n- v10 --supports--> v0b\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c56e433a99a07a323b1f6b46324567b870b85db25f797d2e8eceda0324de40f0","schema":"livegraph.boolean-propagation.v1","spec_hash":"6c5c86e6e89d5fa79d668bb81313a0fdbfa8c1cdbccd08eed6de90b56cd9d7cf","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-7f39c78937aaac4ce72e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v0e\n- v04 --supports--> v03\n- v07 --supports--> v06\n- v0f --supports--> v09\n- v10 --refutes--> v01\n- v0c --refutes--> v07\n- v01 --supports--> v0f\n- v04 --refutes--> v08\n- v02 --refutes--> v05\n- v03 --supports--> v0d\n- v09 --refutes--> v0c\n- v0d --supports--> v10\n- v06 --supports--> v0b\n- v0e --supports--> v13\n- v0b --supports--> v11\n- v03 --refutes--> v0a\n- v01 --refutes--> v00\n- v0e --refutes--> v12\n- v0e --refutes--> v02\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a950c33ae2088ed9a53c8714145ce2e2e5457827da17238ed5729992de9f3d1d","schema":"livegraph.boolean-propagation.v1","spec_hash":"7f39c78937aaac4ce72e34d93fd2f379144c9e7c85bf543ae6b822b1558c94e8","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":13,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-1f7b4b6ac25d974d50d6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v00\n- v10 --refutes--> v0d\n- v03 --supports--> v12\n- v06 --refutes--> v0e\n- v0a --refutes--> v01\n- v00 --refutes--> v06\n- v06 --refutes--> v03\n- v12 --refutes--> v09\n- v03 --supports--> v0b\n- v09 --supports--> v13\n- v11 --refutes--> v10\n- v01 --supports--> v11\n- v10 --supports--> v07\n- v0d --refutes--> v08\n- v05 --refutes--> v02\n- v13 --supports--> v0a\n- v0d --supports--> v05\n- v04 --refutes--> v0c\n- v05 --supports--> v0f\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fd2272150bbdef8c5ab6fdf7fadc779cb55ab625a17e55e5b5f43e751cd6ee29","schema":"livegraph.boolean-propagation.v1","spec_hash":"1f7b4b6ac25d974d50d606cc04223ad19fb26d5bd8a0812ef2006698568c6f2f","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-76554ed80679896b4b1c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --refutes--> v0b\n- v07 --refutes--> v0f\n- v04 --supports--> v0e\n- v0a --supports--> v05\n- v12 --refutes--> v11\n- v04 --refutes--> v12\n- v05 --supports--> v07\n- v03 --supports--> v09\n- v07 --supports--> v02\n- v09 --supports--> v04\n- v05 --refutes--> v00\n- v0b --refutes--> v03\n- v0f --refutes--> v0d\n- v08 --supports--> v0a\n- v12 --refutes--> v10\n- v0d --supports--> v06\n- v06 --supports--> v13\n- v12 --refutes--> v0c\n- v07 --supports--> v01\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9888c6a441d1e736893cf7940697fcce860ba6cfea1936e968cad59f372fc1a5","schema":"livegraph.boolean-propagation.v1","spec_hash":"76554ed80679896b4b1c34b4a6423cc0c295be4bc314045dcbb0d50ff7576d70","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-49eebe2c814f8cbe42c8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v0a\n- v11 --supports--> v00\n- v11 --refutes--> v04\n- v0c --refutes--> v0d\n- v0d --refutes--> v12\n- v05 --refutes--> v08\n- v0d --supports--> v13\n- v10 --supports--> v0c\n- v07 --refutes--> v09\n- v0c --supports--> v06\n- v0a --refutes--> v0e\n- v13 --supports--> v0b\n- v03 --supports--> v01\n- v08 --supports--> v10\n- v03 --supports--> v07\n- v0b --supports--> v0f\n- v0e --refutes--> v03\n- v08 --refutes--> v02\n- v00 --refutes--> v05\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"92be534c616405b96c8b94e5e23b5fe1c921960f412a9bb04f7ce8f4dd92caf3","schema":"livegraph.boolean-propagation.v1","spec_hash":"49eebe2c814f8cbe42c8b26e9f4de4c8f4c0b6392e9de0907a9adde7c25caad3","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-04fe28f2eec9c3a07b8b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v02\n- v05 --refutes--> v10\n- v0a --supports--> v0c\n- v01 --supports--> v07\n- v00 --supports--> v0f\n- v09 --supports--> v03\n- v12 --refutes--> v00\n- v07 --supports--> v0b\n- v01 --refutes--> v04\n- v0c --refutes--> v08\n- v13 --supports--> v05\n- v08 --supports--> v06\n- v06 --refutes--> v13\n- v0f --refutes--> v0d\n- v03 --refutes--> v01\n- v12 --refutes--> v11\n- v05 --supports--> v12\n- v07 --refutes--> v0a\n- v0f --refutes--> v0e\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8769c05fba0d9b0af5b96eb70579a1e353bdd550e3b389627226eff0dcf62639","schema":"livegraph.boolean-propagation.v1","spec_hash":"04fe28f2eec9c3a07b8bdd724c9ee9db3bf007050d31fb1ada5ddf7b2cb70f71","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-d052f8c88060dbcd0b6d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v03\n- v01 --refutes--> v04\n- v10 --refutes--> v02\n- v05 --refutes--> v07\n- v02 --refutes--> v13\n- v04 --refutes--> v11\n- v07 --refutes--> v0c\n- v0f --refutes--> v01\n- v0b --supports--> v0a\n- v08 --supports--> v00\n- v0a --supports--> v12\n- v0c --supports--> v0f\n- v13 --supports--> v06\n- v04 --supports--> v0b\n- v08 --refutes--> v05\n- v0e --supports--> v10\n- v07 --supports--> v09\n- v0a --supports--> v0e\n- v05 --supports--> v0d\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b2f57ec4b5028490202cdd37f01443fa376560e0d87bffc12fbc27e6d6324c86","schema":"livegraph.boolean-propagation.v1","spec_hash":"d052f8c88060dbcd0b6d9316367e14db5dcbcd9dea0b40aa01233bfea9d7eece","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-c4cbd1b71895e521128c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v0e\n- v04 --supports--> v12\n- v09 --refutes--> v01\n- v03 --refutes--> v02\n- v02 --supports--> v07\n- v07 --supports--> v05\n- v00 --refutes--> v0b\n- v07 --supports--> v0a\n- v04 --refutes--> v13\n- v11 --refutes--> v03\n- v12 --supports--> v08\n- v12 --refutes--> v0d\n- v03 --refutes--> v06\n- v05 --supports--> v09\n- v0e --supports--> v04\n- v03 --supports--> v0c\n- v0d --refutes--> v10\n- v0b --supports--> v11\n- v05 --refutes--> v0f\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5ebc07925290b3d8b5d10e0357cfb579ea401bc743018dbf3317b6be2d8efb06","schema":"livegraph.boolean-propagation.v1","spec_hash":"c4cbd1b71895e521128c20f0d75aee9ebad291703af09cd59e099344d1753f47","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-d396122369cc02ac4fbe","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v02\n- v11 --supports--> v05\n- v0c --refutes--> v04\n- v09 --refutes--> v0b\n- v12 --supports--> v13\n- v08 --supports--> v0d\n- v10 --refutes--> v0e\n- v12 --supports--> v0c\n- v06 --supports--> v08\n- v10 --supports--> v0a\n- v10 --supports--> v00\n- v0b --refutes--> v11\n- v0c --supports--> v09\n- v07 --refutes--> v0f\n- v03 --supports--> v07\n- v0e --supports--> v12\n- v07 --supports--> v01\n- v0f --refutes--> v06\n- v05 --supports--> v03\n\n[VERIFIED ROOT]\nv10 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"293c143a983c8c2c80d7efcd150dc569aeb129c502d8a61ffb833e679b83ee56","schema":"livegraph.boolean-propagation.v1","spec_hash":"d396122369cc02ac4fbe0393b708917078942f26c2d0ec7172dc1b80e318a76a","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-eab42c51d90c5a972c9e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v04\n- v10 --refutes--> v0f\n- v04 --supports--> v02\n- v0d --refutes--> v13\n- v06 --supports--> v11\n- v05 --supports--> v00\n- v05 --supports--> v12\n- v09 --supports--> v0b\n- v03 --refutes--> v0d\n- v11 --supports--> v0c\n- v0d --refutes--> v07\n- v01 --supports--> v05\n- v08 --supports--> v06\n- v0d --refutes--> v0e\n- v0d --supports--> v0a\n- v0c --refutes--> v10\n- v04 --refutes--> v03\n- v0f --refutes--> v09\n- v00 --supports--> v08\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cf54f97db4989e5ca33ec7042d65c92a329ab15d79fb415b59e996f2d3b278ef","schema":"livegraph.boolean-propagation.v1","spec_hash":"eab42c51d90c5a972c9ec170e382d5fc48e17ce98fda95fb000cd0bd4f89f4cb","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":13,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-3c4a38dbf80d9cc55b17","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v01\n- v05 --supports--> v10\n- v00 --supports--> v0f\n- v05 --refutes--> v0d\n- v0a --refutes--> v00\n- v09 --supports--> v13\n- v0e --refutes--> v0a\n- v13 --supports--> v05\n- v13 --supports--> v11\n- v12 --supports--> v04\n- v06 --supports--> v0b\n- v05 --supports--> v0c\n- v0e --supports--> v02\n- v0b --supports--> v08\n- v07 --refutes--> v0e\n- v04 --supports--> v09\n- v08 --refutes--> v12\n- v06 --supports--> v03\n- v0f --supports--> v06\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"69d529cd7f6722bb06cd29a14e165d26f3bdc32a5ebf482c691346bfa52e147e","schema":"livegraph.boolean-propagation.v1","spec_hash":"3c4a38dbf80d9cc55b173b6fa8159bd6caab052b63c7ed5472bbf443104697c0","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-35bbaeb344b9c950040c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v11\n- v13 --supports--> v0a\n- v07 --supports--> v03\n- v0b --refutes--> v12\n- v11 --supports--> v0f\n- v02 --supports--> v0e\n- v01 --refutes--> v0c\n- v03 --refutes--> v00\n- v02 --refutes--> v13\n- v05 --supports--> v0d\n- v01 --supports--> v06\n- v0c --refutes--> v08\n- v0b --refutes--> v09\n- v0c --refutes--> v05\n- v0f --supports--> v01\n- v03 --supports--> v10\n- v0a --refutes--> v07\n- v0d --refutes--> v02\n- v12 --supports--> v04\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9cce38fe173f2123fff5c92beaa243c7e08d237ba239ce99f1f7b7fcad865ae1","schema":"livegraph.boolean-propagation.v1","spec_hash":"35bbaeb344b9c950040c50a4c26dc6b16966ab52761b50a4cc855b51607af01a","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-4ddb5256b99881171e88","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v01\n- v0c --supports--> v04\n- v0d --refutes--> v09\n- v02 --supports--> v07\n- v12 --supports--> v0d\n- v02 --supports--> v0b\n- v10 --supports--> v0a\n- v05 --refutes--> v0e\n- v11 --supports--> v06\n- v0a --supports--> v05\n- v08 --refutes--> v00\n- v0c --supports--> v03\n- v01 --supports--> v11\n- v11 --supports--> v02\n- v0b --supports--> v08\n- v0c --supports--> v0f\n- v03 --supports--> v10\n- v0d --supports--> v13\n- v08 --refutes--> v0c\n\n[VERIFIED ROOT]\nv12 = FALSE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"643b3b4ee80a157c7f8a2b90282973f79b2c2f5e4d10a819e6cda477ebff883e","schema":"livegraph.boolean-propagation.v1","spec_hash":"4ddb5256b99881171e88e878219cacc4e74f82e3cddaf29602bd2cef0ccf8de4","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v12\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-5303eb428af7425c3f60","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v0e\n- v0f --supports--> v13\n- v01 --supports--> v06\n- v13 --refutes--> v03\n- v08 --supports--> v10\n- v01 --supports--> v0b\n- v07 --supports--> v0c\n- v07 --supports--> v12\n- v0b --refutes--> v07\n- v11 --supports--> v01\n- v02 --refutes--> v0a\n- v0c --supports--> v0f\n- v0b --supports--> v0d\n- v08 --refutes--> v00\n- v03 --supports--> v05\n- v05 --supports--> v08\n- v0f --supports--> v04\n- v0a --refutes--> v11\n- v10 --supports--> v09\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"58dc42614a3125cec38af4b464554f064b16b46742b8ffc476a7155b0a9d4fb5","schema":"livegraph.boolean-propagation.v1","spec_hash":"5303eb428af7425c3f600ef089aa1138367b4cc6382c0c53069178a269e48e46","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-1155653ceeb2287c7d42","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v0e\n- v09 --supports--> v00\n- v12 --supports--> v07\n- v08 --supports--> v01\n- v0f --supports--> v02\n- v02 --supports--> v0b\n- v07 --refutes--> v08\n- v00 --supports--> v12\n- v08 --refutes--> v04\n- v02 --supports--> v05\n- v06 --refutes--> v10\n- v11 --refutes--> v0c\n- v03 --supports--> v11\n- v0e --refutes--> v0a\n- v0e --refutes--> v13\n- v0e --refutes--> v0f\n- v11 --supports--> v09\n- v00 --refutes--> v0d\n- v10 --refutes--> v03\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c2153773cab964dd5f02a931a64abf2261feb54ff1ca1333b0784da547369a0a","schema":"livegraph.boolean-propagation.v1","spec_hash":"1155653ceeb2287c7d42bca67c9e5986358bc1a55d12e20fc486216acf0b29eb","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-4cd61421133b81a5e522","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v05\n- v02 --supports--> v07\n- v00 --supports--> v0d\n- v05 --refutes--> v02\n- v03 --supports--> v00\n- v12 --supports--> v0e\n- v0a --supports--> v01\n- v01 --refutes--> v12\n- v0e --refutes--> v06\n- v0a --refutes--> v04\n- v07 --refutes--> v13\n- v0c --refutes--> v0a\n- v01 --refutes--> v0f\n- v06 --refutes--> v09\n- v13 --refutes--> v11\n- v0c --refutes--> v0b\n- v13 --refutes--> v08\n- v00 --supports--> v0c\n- v0e --supports--> v10\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f4e19e7792ff09bc7766333e6ed9df85654de6d002502e089d347e380e6bd791","schema":"livegraph.boolean-propagation.v1","spec_hash":"4cd61421133b81a5e522394f7b6070607f93e29c9a3223de50aa48ea62eb22e4","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-707036351a41d78b6b8e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v0a\n- v09 --supports--> v07\n- v05 --refutes--> v13\n- v12 --refutes--> v09\n- v03 --supports--> v01\n- v06 --supports--> v12\n- v03 --refutes--> v0d\n- v0c --refutes--> v0b\n- v04 --supports--> v06\n- v10 --supports--> v11\n- v10 --supports--> v0f\n- v0e --supports--> v04\n- v03 --refutes--> v10\n- v07 --supports--> v08\n- v00 --refutes--> v03\n- v11 --refutes--> v05\n- v06 --refutes--> v02\n- v0b --refutes--> v0e\n- v05 --refutes--> v0c\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"295cab4337dcf6138b4b9cdf16167d587a8223d04a583a98bacc55da046f1798","schema":"livegraph.boolean-propagation.v1","spec_hash":"707036351a41d78b6b8ecec356c269f2eb18d58c731eeb2b8886c4781674b206","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-533ec6f1c35766d05bec","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v04\n- v0a --supports--> v0f\n- v0e --supports--> v12\n- v09 --refutes--> v10\n- v0c --supports--> v0d\n- v04 --refutes--> v13\n- v02 --refutes--> v01\n- v10 --refutes--> v06\n- v11 --refutes--> v00\n- v0c --supports--> v09\n- v01 --refutes--> v0b\n- v13 --refutes--> v05\n- v00 --refutes--> v03\n- v03 --refutes--> v0e\n- v09 --refutes--> v07\n- v0a --refutes--> v11\n- v01 --supports--> v0a\n- v04 --supports--> v08\n- v05 --supports--> v0c\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"88b528ec6985ccc1ecf6cab715fd5370aaca2fd51ed0f7d52ebdfc35f9e5d91b","schema":"livegraph.boolean-propagation.v1","spec_hash":"533ec6f1c35766d05becfe34622e15d9cd5b3c9cc7e639114262e9556d71656e","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-aff975a5c44f32238d6f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --supports--> v08\n- v13 --refutes--> v00\n- v11 --supports--> v0d\n- v0e --refutes--> v01\n- v0c --refutes--> v0e\n- v04 --supports--> v13\n- v12 --supports--> v04\n- v06 --supports--> v11\n- v07 --refutes--> v0c\n- v03 --refutes--> v05\n- v0a --refutes--> v06\n- v08 --supports--> v09\n- v13 --supports--> v0a\n- v04 --supports--> v0b\n- v10 --supports--> v03\n- v03 --supports--> v02\n- v02 --supports--> v12\n- v03 --refutes--> v0f\n- v08 --refutes--> v07\n\n[VERIFIED ROOT]\nv10 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"18f586f7372ee7717d0308d2c22d98c178638fa439eb99ffa9a0997c523037bb","schema":"livegraph.boolean-propagation.v1","spec_hash":"aff975a5c44f32238d6ff030e8f8c7878ea94e9f3634cc946b53924481aa2bcc","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-ddc0cdde215a5e3c061f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v0f\n- v00 --supports--> v07\n- v0b --supports--> v11\n- v07 --supports--> v03\n- v13 --refutes--> v0c\n- v0a --refutes--> v02\n- v0f --refutes--> v12\n- v02 --supports--> v0e\n- v02 --refutes--> v10\n- v0f --supports--> v01\n- v08 --refutes--> v0d\n- v09 --refutes--> v04\n- v08 --supports--> v09\n- v06 --refutes--> v00\n- v01 --refutes--> v08\n- v0b --supports--> v05\n- v09 --supports--> v13\n- v05 --supports--> v06\n- v0e --supports--> v0b\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b546ca945d058e42898952ce91f12cac1b86ae907e885a4e701df7f2f3a24fbc","schema":"livegraph.boolean-propagation.v1","spec_hash":"ddc0cdde215a5e3c061f2494c120a1a9d65907b395975458a80c00ac2f599248","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-ff8932ae46474e437626","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --supports--> v09\n- v0e --supports--> v0c\n- v0d --supports--> v06\n- v11 --refutes--> v13\n- v12 --refutes--> v0a\n- v0b --supports--> v03\n- v0c --refutes--> v10\n- v02 --refutes--> v07\n- v03 --supports--> v12\n- v0a --supports--> v0d\n- v10 --refutes--> v02\n- v07 --refutes--> v0b\n- v12 --supports--> v05\n- v11 --supports--> v00\n- v03 --supports--> v04\n- v08 --supports--> v01\n- v08 --supports--> v0e\n- v0a --refutes--> v0f\n- v06 --refutes--> v11\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"89b63c077ab63810623d1d12ec8acc499502e3bbd3919b81eb2f0b1158b4ecf9","schema":"livegraph.boolean-propagation.v1","spec_hash":"ff8932ae46474e437626513fc8ab69fd14c050b106bea7bb21de7d4fc424d056","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-ab423925e9dffcf1cd76","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v0a\n- v0c --supports--> v05\n- v0b --supports--> v04\n- v08 --refutes--> v0d\n- v0c --refutes--> v08\n- v13 --refutes--> v0c\n- v12 --refutes--> v13\n- v0e --refutes--> v0b\n- v04 --supports--> v0f\n- v0c --refutes--> v06\n- v0f --supports--> v12\n- v10 --refutes--> v02\n- v13 --refutes--> v00\n- v0d --supports--> v01\n- v11 --supports--> v0e\n- v10 --refutes--> v07\n- v03 --supports--> v09\n- v01 --supports--> v03\n- v03 --supports--> v10\n\n[VERIFIED ROOT]\nv11 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"235285220396de6b0da5ab3c07edb4d6aed427a96d600f67ea4cf99729852de7","schema":"livegraph.boolean-propagation.v1","spec_hash":"ab423925e9dffcf1cd76aaf7e5ab8b33f714667605b299641c67f4c3f16ca2e7","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-9eee56aea88691e106d9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --supports--> v13\n- v05 --refutes--> v06\n- v0d --supports--> v02\n- v01 --supports--> v11\n- v06 --refutes--> v0b\n- v04 --refutes--> v12\n- v0d --refutes--> v0a\n- v02 --supports--> v0f\n- v04 --refutes--> v0d\n- v0c --supports--> v0e\n- v03 --refutes--> v09\n- v11 --refutes--> v03\n- v10 --supports--> v08\n- v0e --supports--> v01\n- v02 --supports--> v0c\n- v01 --refutes--> v00\n- v08 --supports--> v05\n- v0b --refutes--> v04\n- v08 --refutes--> v07\n\n[VERIFIED ROOT]\nv10 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5aa4e4d8a22f06805d2f8e19f09fe6570b11cb57d270fc5ec519bc11a78d4fee","schema":"livegraph.boolean-propagation.v1","spec_hash":"9eee56aea88691e106d9a0ab5fba0b4ff82e092406da740af1d2f86ce4b9fe6d","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":13,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-7d10ecb31dbce3500bbe","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v0a\n- v0b --supports--> v13\n- v11 --supports--> v0b\n- v0c --refutes--> v0d\n- v07 --supports--> v08\n- v05 --refutes--> v09\n- v00 --refutes--> v05\n- v10 --supports--> v07\n- v00 --supports--> v04\n- v0c --refutes--> v0f\n- v10 --refutes--> v02\n- v03 --supports--> v11\n- v0e --supports--> v12\n- v08 --refutes--> v00\n- v09 --refutes--> v03\n- v12 --refutes--> v10\n- v0e --supports--> v06\n- v10 --refutes--> v01\n- v0f --supports--> v0e\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ed6e9c51d5264d9a3413c66d168a93e94c1d17852c4caa2c9818f0aa0d728be9","schema":"livegraph.boolean-propagation.v1","spec_hash":"7d10ecb31dbce3500bbe77cacc4e1e1a3c2ced00961c658838e3f9c6fb20aacb","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-8ba400ff4b36a24a39aa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v10\n- v10 --refutes--> v11\n- v0c --supports--> v0b\n- v0e --supports--> v0a\n- v04 --refutes--> v03\n- v0c --supports--> v0d\n- v11 --supports--> v02\n- v0e --supports--> v00\n- v02 --refutes--> v05\n- v0e --refutes--> v01\n- v05 --supports--> v09\n- v0b --supports--> v07\n- v11 --supports--> v12\n- v0a --supports--> v13\n- v08 --refutes--> v0c\n- v13 --refutes--> v08\n- v05 --supports--> v0e\n- v03 --supports--> v06\n- v08 --supports--> v0f\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bfc0e28442bbfb54cfdc3018a1f5d8421db452c2e80de07e1c01905ddcd7343f","schema":"livegraph.boolean-propagation.v1","spec_hash":"8ba400ff4b36a24a39aac4f9f9431e29268ae03186edb46334a502632a76b095","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-df4ca654bd2e6ef387e7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v13\n- v10 --supports--> v05\n- v04 --refutes--> v0f\n- v0d --supports--> v08\n- v08 --refutes--> v03\n- v06 --refutes--> v04\n- v0e --supports--> v0a\n- v00 --supports--> v10\n- v09 --refutes--> v01\n- v0c --refutes--> v09\n- v08 --refutes--> v06\n- v0e --supports--> v12\n- v0f --refutes--> v0c\n- v01 --supports--> v0b\n- v0c --refutes--> v11\n- v05 --refutes--> v0d\n- v02 --supports--> v07\n- v0e --supports--> v02\n- v01 --refutes--> v0e\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"489ad610651943a4c84119d32f823da13dfaaeece08e353bebdab94c0cd26053","schema":"livegraph.boolean-propagation.v1","spec_hash":"df4ca654bd2e6ef387e7e1abb49898f9b12835e8422dd94121e2f0f51091eae1","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":13,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-f5a2b8356f6f87233715","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v00\n- v02 --supports--> v05\n- v0f --supports--> v0e\n- v05 --supports--> v0f\n- v03 --supports--> v11\n- v01 --supports--> v0b\n- v00 --refutes--> v08\n- v11 --supports--> v01\n- v00 --supports--> v03\n- v05 --supports--> v13\n- v0b --supports--> v06\n- v09 --supports--> v12\n- v12 --refutes--> v10\n- v12 --refutes--> v02\n- v06 --supports--> v0a\n- v12 --supports--> v0d\n- v0f --supports--> v07\n- v00 --supports--> v04\n- v0a --supports--> v09\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d944f8a44cda5df8e24961f2553f7e438461bb2c69db16df513b0861c002e745","schema":"livegraph.boolean-propagation.v1","spec_hash":"f5a2b8356f6f87233715a9da718d4d920450bcf1c61dbd9b8ea150942f762493","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-b8b2a0532b95bd7648c5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --supports--> v04\n- v07 --supports--> v13\n- v02 --refutes--> v0c\n- v0f --supports--> v09\n- v10 --refutes--> v0b\n- v08 --refutes--> v10\n- v01 --refutes--> v07\n- v00 --supports--> v0d\n- v12 --supports--> v02\n- v05 --refutes--> v12\n- v00 --supports--> v03\n- v0f --refutes--> v0a\n- v06 --supports--> v00\n- v0c --refutes--> v06\n- v13 --supports--> v05\n- v12 --supports--> v0e\n- v0f --supports--> v01\n- v04 --refutes--> v0f\n- v04 --supports--> v11\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d8ff3323bcd2bb17338bb47e1475ebfa3409af3b630bae4e97b8ec48f3b9bfa4","schema":"livegraph.boolean-propagation.v1","spec_hash":"b8b2a0532b95bd7648c5ef8fb9b44d700c3518a0e8e730dfd0bdbf1bc61b8454","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-b990a21f53567aaca8ef","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v11\n- v04 --supports--> v09\n- v00 --supports--> v04\n- v12 --refutes--> v0a\n- v0b --supports--> v0e\n- v08 --refutes--> v00\n- v0f --supports--> v03\n- v0b --refutes--> v08\n- v03 --refutes--> v0d\n- v0d --refutes--> v12\n- v06 --refutes--> v05\n- v12 --refutes--> v0c\n- v0d --supports--> v07\n- v02 --supports--> v10\n- v11 --supports--> v0f\n- v0a --refutes--> v01\n- v13 --refutes--> v0b\n- v02 --refutes--> v06\n- v06 --refutes--> v13\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7efc0bc8d715d058b6b68af3e5e7790c859e1b807d4e8f9fc4d758e602ca2d19","schema":"livegraph.boolean-propagation.v1","spec_hash":"b990a21f53567aaca8ef048e9e67d5bc50556d53d4f48f81b13f57a97149f2fd","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-33932e7d3b793fb81669","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --supports--> v0d\n- v03 --refutes--> v13\n- v02 --supports--> v0c\n- v0c --refutes--> v0b\n- v0f --supports--> v08\n- v04 --refutes--> v0a\n- v0b --supports--> v00\n- v08 --supports--> v05\n- v05 --refutes--> v09\n- v04 --refutes--> v11\n- v0d --supports--> v0e\n- v0b --supports--> v03\n- v0d --refutes--> v01\n- v08 --refutes--> v12\n- v0c --supports--> v06\n- v0a --supports--> v07\n- v09 --supports--> v10\n- v13 --refutes--> v04\n- v0a --supports--> v0f\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d932a3f0d7c8ca0caeeb19bc3848434333de3ffe6078d0b8f4c280a735e69353","schema":"livegraph.boolean-propagation.v1","spec_hash":"33932e7d3b793fb8166938d32b31332f16434d47263a1ded28467a6ee1b4a781","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-fe1d85235d178f7949ed","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v12\n- v03 --supports--> v0e\n- v09 --supports--> v00\n- v01 --supports--> v08\n- v0b --supports--> v0a\n- v12 --supports--> v07\n- v06 --supports--> v10\n- v09 --supports--> v13\n- v05 --refutes--> v0f\n- v00 --supports--> v11\n- v0a --refutes--> v05\n- v0e --supports--> v0b\n- v02 --refutes--> v03\n- v01 --supports--> v04\n- v00 --supports--> v06\n- v08 --supports--> v02\n- v10 --supports--> v0d\n- v0a --supports--> v0c\n- v10 --refutes--> v01\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"978c6fad9c72dcd8eb89f8db43aadadaec20d1fb6a8d60d4ecdff1734dac3ade","schema":"livegraph.boolean-propagation.v1","spec_hash":"fe1d85235d178f7949ed95b72132de3ccb048e866d476a2011962e7de266c9db","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-0f69786d3513baa65777","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v13\n- v0f --refutes--> v03\n- v13 --supports--> v12\n- v04 --refutes--> v10\n- v08 --supports--> v00\n- v02 --refutes--> v05\n- v0c --refutes--> v08\n- v06 --supports--> v02\n- v02 --supports--> v0e\n- v0f --refutes--> v04\n- v11 --refutes--> v07\n- v0c --refutes--> v09\n- v00 --refutes--> v0b\n- v13 --refutes--> v0c\n- v06 --refutes--> v0a\n- v10 --refutes--> v11\n- v0e --refutes--> v0f\n- v07 --refutes--> v0d\n- v0d --refutes--> v01\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"abb6e23f2d534eb6b5affd07d0729908027ee78119e1794ea87bf6bc2b984abb","schema":"livegraph.boolean-propagation.v1","spec_hash":"0f69786d3513baa657771db75186ab6c27850dc238273e0049e518736a9d508e","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-0d4ce292d62f1b0099fb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v04\n- v11 --refutes--> v0f\n- v06 --refutes--> v00\n- v03 --refutes--> v11\n- v04 --supports--> v13\n- v0a --supports--> v03\n- v13 --refutes--> v0b\n- v13 --refutes--> v01\n- v11 --refutes--> v0d\n- v11 --supports--> v05\n- v04 --supports--> v02\n- v07 --supports--> v06\n- v0c --supports--> v0a\n- v08 --refutes--> v0e\n- v01 --supports--> v07\n- v0d --supports--> v09\n- v01 --refutes--> v10\n- v06 --refutes--> v08\n- v0e --supports--> v0c\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"23b32f107883a4955fc3a16e1611706f296859ff5c56105828562fd285e0a59c","schema":"livegraph.boolean-propagation.v1","spec_hash":"0d4ce292d62f1b0099fb3bab253be61b87ddbd68e0c27531378af171f6c2391b","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-1f4b7bd41391a3fb8016","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v09\n- v0d --supports--> v01\n- v06 --refutes--> v0e\n- v03 --refutes--> v08\n- v0a --supports--> v0f\n- v06 --refutes--> v07\n- v0b --supports--> v10\n- v09 --supports--> v0c\n- v0a --refutes--> v04\n- v11 --supports--> v05\n- v12 --supports--> v13\n- v04 --refutes--> v03\n- v0d --refutes--> v00\n- v09 --refutes--> v02\n- v01 --refutes--> v11\n- v11 --supports--> v0a\n- v12 --refutes--> v06\n- v10 --refutes--> v0d\n- v0e --refutes--> v0b\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"daa1d937432590b9c4be452bce6b0abf29bd1add2aa7905e346d5c8bb7106372","schema":"livegraph.boolean-propagation.v1","spec_hash":"1f4b7bd41391a3fb80168675a96f82b0e37fa2e2ceaf4beffc6bdcf3733b719a","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-faa50a2c6804535540d6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v13\n- v0a --supports--> v02\n- v0c --supports--> v01\n- v0d --supports--> v06\n- v01 --refutes--> v00\n- v03 --supports--> v0c\n- v0d --supports--> v10\n- v02 --supports--> v11\n- v04 --refutes--> v0b\n- v01 --supports--> v09\n- v0e --refutes--> v0d\n- v02 --supports--> v04\n- v04 --refutes--> v0f\n- v0c --supports--> v12\n- v0f --refutes--> v07\n- v01 --supports--> v05\n- v13 --refutes--> v08\n- v08 --supports--> v0e\n- v10 --supports--> v03\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b29dc25d2accdb50bbe624e2078bf8c60845162a19c5281f231eaa66a554ca19","schema":"livegraph.boolean-propagation.v1","spec_hash":"faa50a2c6804535540d611049a17038b1ef7f37562727d3a74631d9293aa5d88","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-e65553888eeec0e4dac3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v01\n- v01 --supports--> v0f\n- v0d --refutes--> v05\n- v09 --supports--> v0b\n- v00 --supports--> v06\n- v0c --refutes--> v04\n- v13 --supports--> v0a\n- v02 --refutes--> v11\n- v13 --refutes--> v10\n- v08 --supports--> v00\n- v05 --supports--> v13\n- v0d --refutes--> v0e\n- v04 --refutes--> v02\n- v09 --supports--> v07\n- v12 --refutes--> v0d\n- v00 --supports--> v09\n- v13 --supports--> v08\n- v03 --supports--> v0c\n- v02 --refutes--> v12\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f41a415830b0d84d8ba993a5b3156dccdbdafb0f0f1b53e1c376833a6b31d0f3","schema":"livegraph.boolean-propagation.v1","spec_hash":"e65553888eeec0e4dac3c7d795a622d7c7b80d2174e47f3a313a71a0ad44cedf","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-07d9de95e4cbaafec588","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v10\n- v0c --refutes--> v0d\n- v0a --supports--> v0e\n- v0e --refutes--> v02\n- v0f --supports--> v0c\n- v09 --supports--> v12\n- v02 --refutes--> v0b\n- v0f --supports--> v11\n- v0a --refutes--> v06\n- v07 --supports--> v00\n- v12 --refutes--> v0a\n- v09 --refutes--> v01\n- v10 --supports--> v03\n- v03 --supports--> v13\n- v00 --refutes--> v09\n- v02 --refutes--> v04\n- v0b --supports--> v0f\n- v0d --refutes--> v05\n- v03 --supports--> v08\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ce1ce6b6ea323a513552f1bb0954502e020d437ead43be69aa99cb774bcda577","schema":"livegraph.boolean-propagation.v1","spec_hash":"07d9de95e4cbaafec5884a6890419678af9fb49ed92adc3c7f3f57eac1bb415b","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-ed98f852f0270b52d32d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v06\n- v00 --supports--> v04\n- v05 --refutes--> v09\n- v01 --refutes--> v0b\n- v11 --supports--> v02\n- v09 --refutes--> v03\n- v0c --supports--> v11\n- v08 --refutes--> v0a\n- v08 --refutes--> v0d\n- v03 --supports--> v0c\n- v11 --refutes--> v12\n- v01 --supports--> v07\n- v01 --refutes--> v0e\n- v07 --supports--> v08\n- v0f --supports--> v01\n- v0a --supports--> v13\n- v13 --supports--> v00\n- v12 --supports--> v0f\n- v05 --supports--> v10\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8bbb03d62038d8eefce73762ed397c1b59135123a74c235c250bafee674feda0","schema":"livegraph.boolean-propagation.v1","spec_hash":"ed98f852f0270b52d32d85c5bfdabd02943d8f0bf4eb80b6deef178c74a0147d","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-aff5be8649d5a1b26714","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v09\n- v00 --supports--> v03\n- v0a --refutes--> v0d\n- v08 --refutes--> v04\n- v0c --refutes--> v01\n- v12 --supports--> v06\n- v06 --refutes--> v0a\n- v0b --supports--> v11\n- v08 --supports--> v10\n- v00 --supports--> v05\n- v0a --supports--> v0c\n- v02 --supports--> v0e\n- v08 --supports--> v0b\n- v01 --refutes--> v02\n- v11 --refutes--> v00\n- v13 --refutes--> v12\n- v0b --refutes--> v07\n- v02 --refutes--> v0f\n- v09 --refutes--> v08\n\n[VERIFIED ROOT]\nv13 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5f28b3a55718fcc7e6a6865c6703ba79f44fccf0bbfea81ad5fafffbfca2ad88","schema":"livegraph.boolean-propagation.v1","spec_hash":"aff5be8649d5a1b26714d0dbe5ab19769cb5d0f2901428092442bc2aad112442","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-f61af56b89cd852e19f9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v0f\n- v07 --refutes--> v04\n- v11 --supports--> v02\n- v11 --supports--> v0d\n- v09 --supports--> v05\n- v08 --supports--> v10\n- v04 --supports--> v06\n- v06 --refutes--> v03\n- v01 --supports--> v0b\n- v09 --refutes--> v0a\n- v0f --supports--> v0e\n- v02 --supports--> v13\n- v08 --supports--> v0c\n- v10 --supports--> v12\n- v01 --refutes--> v11\n- v10 --supports--> v09\n- v00 --refutes--> v01\n- v03 --refutes--> v08\n- v0e --refutes--> v07\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"028d5a96cea37a2fcb9060a6fd5dca059962fc99e5f9b8504932b0e4ed20525b","schema":"livegraph.boolean-propagation.v1","spec_hash":"f61af56b89cd852e19f9107f10837c5cb4316d0afdbe2a84f30d18a547957c17","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":13,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-4aabd700c61f70d4a9b6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v07\n- v08 --supports--> v05\n- v09 --supports--> v0f\n- v0b --refutes--> v01\n- v10 --refutes--> v02\n- v13 --refutes--> v0c\n- v06 --supports--> v10\n- v13 --refutes--> v06\n- v0d --refutes--> v04\n- v11 --refutes--> v0d\n- v0a --supports--> v13\n- v05 --refutes--> v09\n- v0d --supports--> v0e\n- v04 --refutes--> v0b\n- v0f --supports--> v0a\n- v01 --supports--> v08\n- v06 --refutes--> v00\n- v10 --refutes--> v03\n- v0a --supports--> v12\n\n[VERIFIED ROOT]\nv11 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dfdf6b1ff3be999b4227c6c5304080279832675e4929040812f3e2070cd2d9a1","schema":"livegraph.boolean-propagation.v1","spec_hash":"4aabd700c61f70d4a9b6ae6acd35a0184857c50611b2a20d358716075ab45c35","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-a7c38df75ff542ea50e7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v0c\n- v02 --refutes--> v06\n- v0b --supports--> v05\n- v02 --refutes--> v04\n- v11 --supports--> v01\n- v06 --refutes--> v00\n- v12 --supports--> v09\n- v0c --supports--> v12\n- v01 --refutes--> v13\n- v10 --refutes--> v0a\n- v11 --refutes--> v0d\n- v05 --supports--> v03\n- v01 --refutes--> v0f\n- v03 --supports--> v08\n- v07 --supports--> v11\n- v09 --supports--> v02\n- v03 --refutes--> v0e\n- v00 --supports--> v10\n- v10 --supports--> v0b\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e23ddc704ee370534ef38d7d85bdccaaa2100287a00c7b8d28dfafb6673bec9d","schema":"livegraph.boolean-propagation.v1","spec_hash":"a7c38df75ff542ea50e711d102afa80d0bfceaa540a8cb014f5612e6b8f0da24","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-30d7eb3669a6e0c915f5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v07\n- v0f --refutes--> v11\n- v0e --refutes--> v03\n- v06 --refutes--> v02\n- v12 --refutes--> v10\n- v05 --refutes--> v0c\n- v00 --supports--> v0e\n- v09 --refutes--> v12\n- v12 --supports--> v05\n- v13 --supports--> v09\n- v05 --supports--> v0b\n- v0e --refutes--> v08\n- v01 --supports--> v0a\n- v0a --refutes--> v13\n- v0f --supports--> v00\n- v02 --supports--> v0f\n- v04 --supports--> v0d\n- v05 --supports--> v06\n- v03 --supports--> v04\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cdc11ffa0f267b48e64abe05ad0ee88ec5db6fe0bdc52577acc07bcf8fface56","schema":"livegraph.boolean-propagation.v1","spec_hash":"30d7eb3669a6e0c915f582b29f9d62852ce12aac07c6dcd518eb09e1e3692b96","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-853f9f7a6f297a3ce76d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v0f\n- v0b --refutes--> v0d\n- v0d --refutes--> v0e\n- v09 --refutes--> v05\n- v10 --refutes--> v0c\n- v0c --supports--> v04\n- v11 --supports--> v0b\n- v00 --refutes--> v0a\n- v04 --refutes--> v13\n- v05 --refutes--> v10\n- v08 --supports--> v01\n- v0d --refutes--> v02\n- v0a --refutes--> v09\n- v0d --refutes--> v07\n- v0b --refutes--> v03\n- v01 --refutes--> v00\n- v13 --supports--> v11\n- v0d --refutes--> v06\n- v04 --refutes--> v12\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4d24546ddce111ddaa7c1e471e8e29b34126348852d3df1cd863a801c6e72778","schema":"livegraph.boolean-propagation.v1","spec_hash":"853f9f7a6f297a3ce76d783eae1acbcb854868b2764bb334df782b0d2b6b40ea","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-a97cdd4a43d50eb4b083","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v12\n- v11 --refutes--> v07\n- v10 --supports--> v00\n- v0e --refutes--> v0c\n- v0b --refutes--> v02\n- v03 --supports--> v0a\n- v12 --refutes--> v0b\n- v06 --refutes--> v0d\n- v11 --supports--> v08\n- v05 --refutes--> v13\n- v0e --supports--> v0f\n- v0d --supports--> v0e\n- v0f --refutes--> v04\n- v0b --supports--> v11\n- v03 --refutes--> v09\n- v00 --supports--> v06\n- v07 --refutes--> v05\n- v05 --refutes--> v03\n- v12 --supports--> v01\n\n[VERIFIED ROOT]\nv10 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"95c398d6c7f2cf6bcc0835ebb50d4b568c41083951cdb61a043748b9a4c34797","schema":"livegraph.boolean-propagation.v1","spec_hash":"a97cdd4a43d50eb4b08360a5064982210118ee905d42606a87c02376ff7cd8d4","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-49927639269f597f1cd5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v0d\n- v06 --refutes--> v0f\n- v0a --supports--> v0b\n- v0a --supports--> v09\n- v07 --refutes--> v05\n- v0d --supports--> v0a\n- v0b --refutes--> v12\n- v0c --supports--> v10\n- v05 --refutes--> v06\n- v0b --refutes--> v03\n- v01 --refutes--> v0c\n- v0f --refutes--> v0e\n- v0f --supports--> v13\n- v0e --supports--> v08\n- v0c --supports--> v07\n- v00 --refutes--> v01\n- v00 --supports--> v02\n- v0a --supports--> v04\n- v12 --refutes--> v11\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7cfaabe35a2b3b98ec186e48aa5ad14e54f85ac7dde5acb5856d3c1e3a209595","schema":"livegraph.boolean-propagation.v1","spec_hash":"49927639269f597f1cd53866b30fa4bef1d4b41dc0d6a32459edd9fd68da63b1","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-b83337fa33873fce8bdb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v08\n- v12 --supports--> v00\n- v06 --supports--> v12\n- v0c --supports--> v07\n- v0c --refutes--> v0a\n- v05 --refutes--> v09\n- v11 --supports--> v0f\n- v02 --refutes--> v10\n- v13 --refutes--> v05\n- v00 --refutes--> v0d\n- v01 --refutes--> v0c\n- v10 --supports--> v13\n- v00 --refutes--> v03\n- v07 --refutes--> v02\n- v0d --refutes--> v04\n- v00 --supports--> v0e\n- v0f --supports--> v01\n- v0d --supports--> v11\n- v0c --refutes--> v0b\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"72060c6f22e9c860296bf6e1d4eaf4f78c80b69d74a437a1bdabcbac5bc342d7","schema":"livegraph.boolean-propagation.v1","spec_hash":"b83337fa33873fce8bdb92f8ff843b8bb769afd364f35c83289cf7e7232dd0f2","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-332fd4d1f094fe9a790a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v10\n- v11 --supports--> v01\n- v12 --supports--> v03\n- v02 --refutes--> v09\n- v0f --supports--> v05\n- v09 --refutes--> v07\n- v04 --refutes--> v08\n- v0d --supports--> v12\n- v0a --refutes--> v0d\n- v0e --supports--> v0a\n- v08 --supports--> v0f\n- v09 --refutes--> v06\n- v05 --refutes--> v0c\n- v0f --supports--> v0b\n- v01 --supports--> v04\n- v0a --supports--> v13\n- v12 --refutes--> v02\n- v05 --supports--> v0e\n- v00 --supports--> v11\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7f13f315567e8d2ea0d697a0ba2ad5a42b6725eaa5832dc0ef19a9b2d7100c1d","schema":"livegraph.boolean-propagation.v1","spec_hash":"332fd4d1f094fe9a790a187c345e6cd8e243808b19b2376da7b2ad41a9c8e8c3","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-6cee6f948dab56d06759","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v04\n- v07 --refutes--> v0c\n- v10 --supports--> v11\n- v0f --supports--> v0e\n- v0a --refutes--> v0d\n- v02 --supports--> v05\n- v0c --supports--> v09\n- v02 --supports--> v01\n- v0f --supports--> v10\n- v0c --supports--> v0b\n- v08 --supports--> v0f\n- v0d --supports--> v08\n- v05 --refutes--> v07\n- v00 --supports--> v12\n- v0c --refutes--> v00\n- v06 --refutes--> v02\n- v12 --supports--> v0a\n- v0d --refutes--> v03\n- v13 --refutes--> v06\n\n[VERIFIED ROOT]\nv13 = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8bf6c0f2b749fbb21124034c8bbfb54f1e60f1e4006495d80975d76e3e136c1a","schema":"livegraph.boolean-propagation.v1","spec_hash":"6cee6f948dab56d0675958287dd0878442ed2c566bc4da2213283f6b3a30320c","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-75299b23e35982c9ca62","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v00\n- v03 --refutes--> v06\n- v0a --refutes--> v04\n- v08 --supports--> v0b\n- v04 --supports--> v01\n- v05 --refutes--> v0e\n- v01 --refutes--> v05\n- v0e --refutes--> v0d\n- v08 --supports--> v0c\n- v12 --refutes--> v02\n- v02 --supports--> v0a\n- v08 --refutes--> v12\n- v0a --refutes--> v13\n- v02 --refutes--> v10\n- v07 --refutes--> v03\n- v05 --supports--> v0f\n- v09 --supports--> v11\n- v06 --refutes--> v08\n- v0d --supports--> v09\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b70bb2724f79a32479b1319e2b6aae8f2e56af9233bf8da145b572e7a9d1c6e5","schema":"livegraph.boolean-propagation.v1","spec_hash":"75299b23e35982c9ca62566fd45a26303ed059f76c1a61901709adc0f02799d5","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-f26b4a5580ca99cf00a8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v0f\n- v08 --refutes--> v0c\n- v03 --refutes--> v02\n- v00 --supports--> v06\n- v03 --refutes--> v0e\n- v06 --refutes--> v03\n- v04 --supports--> v07\n- v11 --supports--> v0d\n- v13 --supports--> v01\n- v10 --refutes--> v04\n- v0b --supports--> v13\n- v07 --supports--> v05\n- v08 --refutes--> v00\n- v0a --refutes--> v08\n- v06 --supports--> v09\n- v07 --refutes--> v11\n- v0d --refutes--> v0a\n- v08 --refutes--> v12\n- v13 --supports--> v10\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"39d5b24c2640c6a8c9941b36e1e413465d42d337681c7ea7fbb9f5712ff055d4","schema":"livegraph.boolean-propagation.v1","spec_hash":"f26b4a5580ca99cf00a83b546cfba0f5f45121b17b855fe64b2ddfeb712d39e5","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-cdcf7566eca1d14504e6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v0c\n- v05 --supports--> v10\n- v13 --supports--> v08\n- v0f --refutes--> v09\n- v03 --refutes--> v06\n- v0f --refutes--> v0b\n- v07 --supports--> v01\n- v00 --refutes--> v11\n- v10 --supports--> v03\n- v0d --supports--> v0f\n- v06 --supports--> v07\n- v00 --refutes--> v02\n- v0c --supports--> v13\n- v13 --refutes--> v00\n- v01 --refutes--> v0d\n- v01 --supports--> v0e\n- v03 --supports--> v0a\n- v06 --supports--> v12\n- v02 --supports--> v04\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5dffc7b903afb51e81a57c7b028ee41d6681cf4aaf306b37fe0cbf5f292331ff","schema":"livegraph.boolean-propagation.v1","spec_hash":"cdcf7566eca1d14504e6b98e2aec629e1afcc6427c6e56514201679f58425c2c","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-3bbfcd0f04af9ed4c760","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --supports--> v13\n- v00 --refutes--> v11\n- v0f --supports--> v07\n- v03 --supports--> v02\n- v09 --refutes--> v0d\n- v0a --supports--> v01\n- v0c --supports--> v09\n- v10 --supports--> v0e\n- v05 --supports--> v10\n- v07 --supports--> v06\n- v13 --supports--> v00\n- v00 --refutes--> v0a\n- v09 --refutes--> v05\n- v08 --refutes--> v0c\n- v05 --supports--> v04\n- v0b --refutes--> v12\n- v0a --supports--> v0f\n- v03 --supports--> v08\n- v12 --supports--> v03\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dbe5fb1fba0d9c679a9071ed31cf5413c0e8d1895c6a3f6d7484eb7c5bd2fb2d","schema":"livegraph.boolean-propagation.v1","spec_hash":"3bbfcd0f04af9ed4c760040e9720ce2b78cfcb4a80a11ddee8829d9e989b3f1c","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-4d7ba4fa1ca12aed28de","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v03\n- v10 --supports--> v05\n- v0c --refutes--> v04\n- v0b --supports--> v0e\n- v09 --supports--> v01\n- v0e --supports--> v10\n- v02 --refutes--> v08\n- v03 --supports--> v11\n- v12 --refutes--> v0d\n- v0d --supports--> v0f\n- v01 --refutes--> v13\n- v10 --supports--> v12\n- v08 --supports--> v00\n- v01 --supports--> v06\n- v0e --supports--> v0a\n- v06 --refutes--> v02\n- v11 --supports--> v0b\n- v02 --refutes--> v07\n- v0c --supports--> v09\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3ef8ce93e61aaa231111189d4862d7c7cef388742c09ebf6b1feea84d5f31ca0","schema":"livegraph.boolean-propagation.v1","spec_hash":"4d7ba4fa1ca12aed28de0fca019091063a2c17a9bdc0b4fda3858b3c863e3c7d","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-a33298d698d51c82ee3a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v0c\n- v05 --supports--> v07\n- v0d --supports--> v0e\n- v0a --refutes--> v0d\n- v10 --supports--> v08\n- v01 --supports--> v00\n- v11 --refutes--> v10\n- v05 --supports--> v01\n- v03 --supports--> v04\n- v0e --supports--> v03\n- v04 --refutes--> v05\n- v12 --refutes--> v02\n- v02 --refutes--> v11\n- v08 --refutes--> v09\n- v0b --supports--> v0a\n- v0d --supports--> v06\n- v01 --supports--> v12\n- v0d --refutes--> v13\n- v08 --refutes--> v0f\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bb80a1b661b3756820562eb44b17a278d3796193209e5edb2cb536d421822a7a","schema":"livegraph.boolean-propagation.v1","spec_hash":"a33298d698d51c82ee3ae22184c0c96d8c446825d49c8c1c717886285580c8cd","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-1fa55f78044b57f68bdc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v06\n- v09 --refutes--> v08\n- v07 --refutes--> v00\n- v04 --supports--> v0c\n- v09 --refutes--> v03\n- v05 --refutes--> v10\n- v13 --refutes--> v12\n- v10 --refutes--> v01\n- v06 --refutes--> v04\n- v03 --refutes--> v0d\n- v10 --refutes--> v0a\n- v00 --refutes--> v0e\n- v03 --supports--> v0b\n- v02 --refutes--> v13\n- v0d --supports--> v05\n- v0a --refutes--> v02\n- v0d --supports--> v0f\n- v02 --supports--> v11\n- v04 --refutes--> v07\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3425f7fcc0ac7d38d4b1c2c190df9c4eca91fc6743e235d3fcff6ac4df0cc7f9","schema":"livegraph.boolean-propagation.v1","spec_hash":"1fa55f78044b57f68bdc322cc21cc21fdd39ebce239d8d11f1440c0ca4a60065","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-1713c8262f3e98365be0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v10\n- v00 --supports--> v0a\n- v03 --supports--> v02\n- v0b --refutes--> v0e\n- v10 --refutes--> v03\n- v10 --refutes--> v06\n- v04 --refutes--> v0d\n- v03 --supports--> v07\n- v11 --supports--> v09\n- v04 --refutes--> v0c\n- v0e --supports--> v08\n- v01 --refutes--> v11\n- v0d --supports--> v13\n- v0f --refutes--> v0b\n- v0f --supports--> v05\n- v0a --refutes--> v01\n- v03 --refutes--> v00\n- v0d --refutes--> v12\n- v13 --refutes--> v0f\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3fbfc8791562aac08e8301e1c2d7939dd54ac45adab7691e97720c0302acb2af","schema":"livegraph.boolean-propagation.v1","spec_hash":"1713c8262f3e98365be014ad894516318077a62a01af78f79bb4d41e82632dd3","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-d883add09ea1819fa8cc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v0e\n- v11 --supports--> v05\n- v11 --supports--> v01\n- v12 --supports--> v00\n- v0e --supports--> v04\n- v08 --refutes--> v09\n- v0b --refutes--> v0f\n- v0a --supports--> v07\n- v08 --refutes--> v13\n- v10 --supports--> v0c\n- v00 --refutes--> v10\n- v0b --supports--> v08\n- v03 --refutes--> v0a\n- v04 --supports--> v12\n- v13 --refutes--> v0d\n- v02 --supports--> v11\n- v05 --supports--> v0b\n- v10 --refutes--> v06\n- v10 --refutes--> v02\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e9164494498568671d170f6658234e0192627b7e36dc89aa60a992f087aaf454","schema":"livegraph.boolean-propagation.v1","spec_hash":"d883add09ea1819fa8cc1ef07458d9f21f487fb01aed22b5839b03c981075083","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-3d2d461c503039efb1ce","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v08\n- v11 --refutes--> v05\n- v06 --supports--> v0a\n- v04 --refutes--> v10\n- v00 --refutes--> v01\n- v12 --refutes--> v00\n- v13 --refutes--> v03\n- v11 --supports--> v07\n- v08 --supports--> v11\n- v11 --refutes--> v0b\n- v0d --supports--> v04\n- v00 --refutes--> v0d\n- v10 --supports--> v0c\n- v12 --refutes--> v0f\n- v03 --supports--> v02\n- v0b --supports--> v09\n- v06 --refutes--> v13\n- v06 --refutes--> v0e\n- v09 --refutes--> v12\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3675c050113082ea723095112bcd91cb1e22b15d714f32b225d19b11ae62495b","schema":"livegraph.boolean-propagation.v1","spec_hash":"3d2d461c503039efb1cee0f84ba9a82fe310eee8c9c7cc84494f2992f0c3d520","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-018715d057ce5f7ba45d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v00\n- v08 --refutes--> v0f\n- v02 --supports--> v11\n- v02 --supports--> v12\n- v0d --refutes--> v13\n- v0a --refutes--> v0c\n- v10 --refutes--> v02\n- v0b --supports--> v0e\n- v0f --supports--> v04\n- v0e --supports--> v03\n- v0f --refutes--> v0d\n- v05 --refutes--> v08\n- v02 --supports--> v09\n- v11 --refutes--> v07\n- v03 --supports--> v06\n- v0f --refutes--> v01\n- v11 --supports--> v0a\n- v06 --refutes--> v05\n- v0c --supports--> v0b\n\n[VERIFIED ROOT]\nv10 = TRUE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"33941ed14ca6c5350b9698ef38e8b006d603d00d82f5d044bf90116ef516f3fd","schema":"livegraph.boolean-propagation.v1","spec_hash":"018715d057ce5f7ba45d72c447521cc28f37e8c22551325419fd4ce1a9d8537a","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":13,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-053aab3eda249aa3f8d2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --supports--> v03\n- v0f --refutes--> v10\n- v06 --supports--> v04\n- v06 --supports--> v08\n- v08 --refutes--> v07\n- v0d --refutes--> v12\n- v08 --supports--> v02\n- v10 --supports--> v09\n- v0f --refutes--> v00\n- v05 --supports--> v01\n- v10 --supports--> v13\n- v01 --supports--> v0a\n- v03 --supports--> v0e\n- v11 --supports--> v05\n- v0a --supports--> v0f\n- v02 --supports--> v0c\n- v02 --refutes--> v0d\n- v0d --refutes--> v0b\n- v12 --supports--> v11\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5fbf049b33845813793acb01d419cdfb65f941d2bdc6cf8fa40bf083021420b0","schema":"livegraph.boolean-propagation.v1","spec_hash":"053aab3eda249aa3f8d292a67bf1fb8114fe4ba5b73c949b84b8953f71e5c8bb","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-0c46826436917f966315","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v00\n- v0e --supports--> v11\n- v10 --supports--> v02\n- v0c --supports--> v0b\n- v02 --refutes--> v0d\n- v07 --supports--> v08\n- v02 --refutes--> v06\n- v0f --supports--> v0a\n- v08 --supports--> v13\n- v02 --supports--> v0c\n- v04 --supports--> v03\n- v0b --refutes--> v09\n- v01 --supports--> v10\n- v03 --supports--> v0f\n- v0b --supports--> v05\n- v11 --refutes--> v01\n- v08 --refutes--> v04\n- v09 --supports--> v07\n- v02 --supports--> v12\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d44671b00b8406e47e1e608b1769882b6d061f17ebf1ce8cc84c8fcf98a394c9","schema":"livegraph.boolean-propagation.v1","spec_hash":"0c46826436917f96631545ab78cf678914fcf9bd6b09656d0799440fb844c641","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":13,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-56bc9ce70184bfcbbcde","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v07\n- v08 --refutes--> v13\n- v02 --supports--> v04\n- v0c --refutes--> v0a\n- v0b --refutes--> v0d\n- v08 --supports--> v09\n- v03 --supports--> v0f\n- v00 --refutes--> v10\n- v0e --supports--> v01\n- v0a --supports--> v00\n- v12 --refutes--> v11\n- v0b --refutes--> v05\n- v05 --refutes--> v02\n- v00 --refutes--> v03\n- v03 --refutes--> v0b\n- v0e --supports--> v06\n- v13 --supports--> v12\n- v06 --supports--> v08\n- v11 --refutes--> v0c\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c0635ca14f9a6ce0c555dda9e43ca33e35853b130c98eb353b994e8713bb8ab4","schema":"livegraph.boolean-propagation.v1","spec_hash":"56bc9ce70184bfcbbcdef0d8d31094708e5d994d4f59b7af7a11cd5af814ad85","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-d318915e72a9068a444f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v08\n- v03 --refutes--> v05\n- v04 --refutes--> v03\n- v0b --refutes--> v11\n- v05 --supports--> v0d\n- v0e --refutes--> v04\n- v0f --supports--> v0e\n- v0a --refutes--> v06\n- v0b --refutes--> v01\n- v02 --refutes--> v13\n- v01 --refutes--> v12\n- v07 --supports--> v09\n- v08 --supports--> v0f\n- v0a --refutes--> v07\n- v0d --refutes--> v00\n- v01 --supports--> v0a\n- v07 --supports--> v10\n- v0d --refutes--> v02\n- v01 --refutes--> v0c\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0d7c0d9c2be9f8cb0b678c8bc14dbff4a2de70a5cc55fad98a7953321425d092","schema":"livegraph.boolean-propagation.v1","spec_hash":"d318915e72a9068a444f9db4c2477cdda7aa1eef84303afe55d00b4be7a2ceab","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":13,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-bdf2d3d45abf716cf99d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v0e\n- v10 --refutes--> v07\n- v02 --refutes--> v11\n- v04 --supports--> v01\n- v0f --supports--> v0a\n- v02 --supports--> v00\n- v03 --supports--> v0f\n- v02 --supports--> v08\n- v10 --refutes--> v12\n- v0a --supports--> v0b\n- v05 --refutes--> v13\n- v02 --supports--> v06\n- v00 --refutes--> v03\n- v13 --refutes--> v10\n- v0d --refutes--> v0c\n- v01 --refutes--> v0d\n- v09 --refutes--> v02\n- v0d --supports--> v05\n- v07 --refutes--> v09\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"887d650e6f4e4d68b59580b503587312cafd4b5d99c455e8c300be5c5ddec564","schema":"livegraph.boolean-propagation.v1","spec_hash":"bdf2d3d45abf716cf99dbaebe9d8fc5ba0cb4b8c634adceb2adbe7b5459d493c","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-ea4641206d98776d3f42","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v08\n- v12 --supports--> v13\n- v02 --supports--> v0d\n- v06 --refutes--> v0e\n- v13 --refutes--> v01\n- v0a --refutes--> v0c\n- v04 --supports--> v0b\n- v06 --refutes--> v05\n- v10 --supports--> v06\n- v07 --refutes--> v04\n- v05 --refutes--> v02\n- v06 --supports--> v03\n- v01 --refutes--> v09\n- v0b --refutes--> v00\n- v01 --supports--> v0f\n- v11 --supports--> v10\n- v0a --supports--> v12\n- v0b --refutes--> v11\n- v09 --supports--> v07\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"39d8336ed8b995e5c5486244ab83fd2ed75cd916860e4fc034ef3ba1b70450df","schema":"livegraph.boolean-propagation.v1","spec_hash":"ea4641206d98776d3f4278836b270c56e5953f56636d9d8778b1413f912e62d4","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-1f44a52e66cb47488b67","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v09\n- v11 --refutes--> v06\n- v08 --refutes--> v13\n- v02 --refutes--> v0a\n- v0e --refutes--> v03\n- v05 --supports--> v12\n- v0c --refutes--> v01\n- v00 --supports--> v11\n- v04 --supports--> v0c\n- v10 --refutes--> v04\n- v02 --supports--> v07\n- v02 --supports--> v0f\n- v0d --supports--> v02\n- v05 --refutes--> v00\n- v0c --refutes--> v0b\n- v09 --supports--> v05\n- v08 --supports--> v0e\n- v03 --refutes--> v10\n- v07 --refutes--> v08\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"53da2fa1feb259b69410aca835fdfd9a40e43fe68a7d0201fe750dce65f67be1","schema":"livegraph.boolean-propagation.v1","spec_hash":"1f44a52e66cb47488b6712b9beae12d7365897abc5d410c180afb586a5adcee8","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":13,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-aeec3d211bdd96f1bf6c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v0d\n- v0d --supports--> v00\n- v08 --refutes--> v10\n- v13 --supports--> v0b\n- v07 --supports--> v0c\n- v06 --refutes--> v09\n- v03 --supports--> v0e\n- v10 --supports--> v05\n- v10 --supports--> v06\n- v00 --refutes--> v02\n- v0e --refutes--> v01\n- v12 --refutes--> v03\n- v02 --refutes--> v0a\n- v02 --refutes--> v0f\n- v0a --supports--> v12\n- v03 --supports--> v04\n- v13 --supports--> v07\n- v0e --supports--> v08\n- v0c --refutes--> v11\n\n[VERIFIED ROOT]\nv13 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b1e64dbef82f0c0b101f13a7586a32bde909c9a381b2855996f56531f39431a9","schema":"livegraph.boolean-propagation.v1","spec_hash":"aeec3d211bdd96f1bf6c1d9c4f44fdf674f1e07848add9b86b827f99e40a66ad","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-f896e7757dfeb34be833","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v02\n- v0d --supports--> v04\n- v0e --refutes--> v12\n- v00 --supports--> v0b\n- v12 --refutes--> v10\n- v0b --supports--> v11\n- v02 --supports--> v07\n- v09 --refutes--> v05\n- v0a --supports--> v09\n- v0f --supports--> v0d\n- v0f --refutes--> v08\n- v05 --supports--> v0f\n- v11 --supports--> v03\n- v13 --supports--> v0c\n- v0a --supports--> v01\n- v0d --supports--> v06\n- v12 --supports--> v13\n- v07 --refutes--> v0e\n- v13 --refutes--> v00\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b08d1ea560e54b44cdf9053698157b66a06407284709cd8d0f6549557682a4e4","schema":"livegraph.boolean-propagation.v1","spec_hash":"f896e7757dfeb34be8333304b788b63a986f47048bef2d283b74f948afbd39ce","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-4b284c0ba91084b9b042","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v02\n- v00 --supports--> v0c\n- v0a --refutes--> v0b\n- v07 --refutes--> v03\n- v11 --supports--> v0a\n- v02 --supports--> v12\n- v0f --refutes--> v01\n- v0b --supports--> v05\n- v00 --refutes--> v06\n- v02 --refutes--> v08\n- v0c --supports--> v04\n- v0e --refutes--> v00\n- v0f --supports--> v13\n- v05 --refutes--> v0e\n- v0f --refutes--> v10\n- v07 --refutes--> v11\n- v0c --refutes--> v0d\n- v12 --refutes--> v09\n- v10 --supports--> v07\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b52996da18d63f7baa6826376f7be70d1dcb7daefdb0f417795d6f23cb0ccc2e","schema":"livegraph.boolean-propagation.v1","spec_hash":"4b284c0ba91084b9b0420b961054520bfe9b41890237b34bef0435c296b2ad75","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-c2ecd6d2847a17e3aa81","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v0f\n- v03 --supports--> v09\n- v0a --refutes--> v0b\n- v12 --refutes--> v04\n- v0c --supports--> v05\n- v05 --refutes--> v00\n- v13 --refutes--> v03\n- v01 --supports--> v0e\n- v00 --supports--> v10\n- v10 --supports--> v13\n- v0b --supports--> v08\n- v01 --supports--> v07\n- v0e --supports--> v0c\n- v13 --supports--> v06\n- v09 --refutes--> v0a\n- v01 --refutes--> v02\n- v04 --refutes--> v0d\n- v09 --supports--> v11\n- v0b --refutes--> v12\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ae1fd096688744d4049bd5466185578d69f4ef13a4dcca9df416b9cd9aa98287","schema":"livegraph.boolean-propagation.v1","spec_hash":"c2ecd6d2847a17e3aa81a9a4f708227478b555cbddd1137c03860d45d7bec440","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-5e6190f6b0a3e664d391","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v11\n- v09 --supports--> v0f\n- v01 --refutes--> v07\n- v13 --refutes--> v0c\n- v0a --refutes--> v0d\n- v03 --refutes--> v13\n- v0e --supports--> v09\n- v04 --refutes--> v03\n- v07 --supports--> v0e\n- v0d --supports--> v06\n- v07 --refutes--> v02\n- v0d --refutes--> v00\n- v13 --supports--> v08\n- v06 --refutes--> v04\n- v03 --supports--> v12\n- v0e --refutes--> v0b\n- v0f --supports--> v0a\n- v05 --supports--> v01\n- v0c --refutes--> v10\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4d5e37b180af00f1aaf4a01808f31279c703af22144f941a0aae8f88f70e83f4","schema":"livegraph.boolean-propagation.v1","spec_hash":"5e6190f6b0a3e664d391d8e7ff0aa500a71b1575fb81566f20848b5336437daf","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-11824d0b0bd6eb9bebbe","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v05\n- v0f --refutes--> v10\n- v0c --refutes--> v00\n- v12 --refutes--> v07\n- v00 --supports--> v02\n- v09 --refutes--> v13\n- v02 --supports--> v03\n- v05 --supports--> v06\n- v0d --supports--> v0c\n- v0a --supports--> v12\n- v09 --refutes--> v0f\n- v0f --refutes--> v01\n- v10 --supports--> v0a\n- v0d --supports--> v04\n- v08 --refutes--> v0e\n- v09 --refutes--> v11\n- v08 --refutes--> v0d\n- v06 --supports--> v09\n- v06 --supports--> v0b\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f468e4ce45941752b9f388ecb3b3ebb0565f6e63dccc3dc9d4c7b98bd847488d","schema":"livegraph.boolean-propagation.v1","spec_hash":"11824d0b0bd6eb9bebbed091549338570a8d5b2c730e54d8b9078a516fa2ab73","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-4dea8396ce7346a5548b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v10\n- v07 --supports--> v06\n- v02 --supports--> v11\n- v07 --refutes--> v09\n- v0d --refutes--> v07\n- v08 --refutes--> v0d\n- v09 --supports--> v0f\n- v09 --supports--> v0e\n- v12 --supports--> v04\n- v04 --supports--> v13\n- v03 --refutes--> v02\n- v00 --refutes--> v01\n- v01 --supports--> v12\n- v09 --supports--> v0c\n- v11 --supports--> v08\n- v00 --supports--> v0a\n- v09 --refutes--> v05\n- v05 --supports--> v0b\n- v13 --refutes--> v03\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"27ff975d5ae87316e943dbcc11b2bb7c514ee01d15e8f6bc031c94d962c423d8","schema":"livegraph.boolean-propagation.v1","spec_hash":"4dea8396ce7346a5548b4ef2c0bc58b2a8940904491ee46ebd5d6c74c07869bb","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-484810d6c8255ba9b81a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v03\n- v03 --supports--> v04\n- v01 --refutes--> v0a\n- v0a --refutes--> v00\n- v10 --supports--> v13\n- v05 --refutes--> v0e\n- v0d --refutes--> v09\n- v03 --supports--> v0b\n- v08 --supports--> v0c\n- v03 --supports--> v01\n- v02 --supports--> v11\n- v07 --supports--> v12\n- v0d --refutes--> v07\n- v0f --supports--> v0d\n- v11 --refutes--> v10\n- v05 --supports--> v02\n- v08 --refutes--> v0f\n- v10 --supports--> v08\n- v06 --refutes--> v05\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"934f5f8fd0ee7e2367b81fa689d71bbf1a3dafbbe49fc152a19c8a300b4cdb2d","schema":"livegraph.boolean-propagation.v1","spec_hash":"484810d6c8255ba9b81a0825b3a7fae3f0d85b3a9b61c277ab7ca4f88d85f67e","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-48e2cf365a57630fef4f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v0e\n- v0c --refutes--> v08\n- v09 --refutes--> v0d\n- v05 --supports--> v09\n- v03 --refutes--> v10\n- v00 --refutes--> v13\n- v03 --refutes--> v04\n- v09 --supports--> v0f\n- v0f --refutes--> v0b\n- v12 --supports--> v0c\n- v06 --supports--> v00\n- v05 --supports--> v02\n- v08 --supports--> v01\n- v04 --supports--> v12\n- v0b --supports--> v06\n- v01 --supports--> v05\n- v05 --supports--> v07\n- v00 --refutes--> v11\n- v11 --refutes--> v0a\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b4d6202e70ea5d1c0d35b8a92108297634bb80f4d04a7bc709ce59d5cc439669","schema":"livegraph.boolean-propagation.v1","spec_hash":"48e2cf365a57630fef4f4baf557615baa33abcbe73f0f325f1188ae1ca86b16c","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-f9573fcc8ddaffde9ae2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v04\n- v08 --supports--> v03\n- v09 --supports--> v13\n- v09 --refutes--> v11\n- v10 --supports--> v08\n- v02 --refutes--> v0f\n- v0d --supports--> v12\n- v08 --refutes--> v0b\n- v0c --refutes--> v00\n- v04 --refutes--> v10\n- v0b --refutes--> v02\n- v0a --refutes--> v0c\n- v09 --refutes--> v0e\n- v13 --supports--> v06\n- v0b --supports--> v07\n- v0f --refutes--> v0a\n- v00 --refutes--> v0d\n- v0d --refutes--> v09\n- v04 --refutes--> v05\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4ed665a1990dda0c22ffa66ebacf48fb84b3329b97771d1fbe4958bddbe4dc83","schema":"livegraph.boolean-propagation.v1","spec_hash":"f9573fcc8ddaffde9ae22cba0398e33e53d8a830638269b33c73b32e9767086a","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-25c2e3ab235383d8980d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v0d\n- v11 --supports--> v08\n- v0b --supports--> v11\n- v0d --supports--> v0c\n- v02 --supports--> v05\n- v0d --supports--> v03\n- v0d --refutes--> v12\n- v0e --supports--> v0f\n- v11 --supports--> v06\n- v07 --supports--> v10\n- v0a --supports--> v02\n- v0c --refutes--> v13\n- v0c --refutes--> v01\n- v04 --refutes--> v0e\n- v0b --supports--> v00\n- v09 --refutes--> v0b\n- v02 --refutes--> v04\n- v06 --supports--> v07\n- v0f --refutes--> v09\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4721948536a032abb187057aca238457906ada0395ee1f6a62ad9c737ceb507c","schema":"livegraph.boolean-propagation.v1","spec_hash":"25c2e3ab235383d8980d0b3096a188182d66861859ecbec28949df4a7f082daa","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":13,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-2d4d95f419b03a70942b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v13\n- v02 --refutes--> v00\n- v08 --supports--> v10\n- v01 --supports--> v0c\n- v0c --refutes--> v08\n- v01 --refutes--> v04\n- v0f --refutes--> v0b\n- v09 --refutes--> v0e\n- v10 --refutes--> v02\n- v01 --refutes--> v07\n- v02 --refutes--> v12\n- v0b --supports--> v06\n- v0e --refutes--> v01\n- v03 --refutes--> v0f\n- v0b --refutes--> v05\n- v05 --refutes--> v0a\n- v11 --supports--> v09\n- v09 --refutes--> v0d\n- v00 --refutes--> v03\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"042d021bcfd2e906e15180591c0a3465c935694774a53e006ab8b79d287a819c","schema":"livegraph.boolean-propagation.v1","spec_hash":"2d4d95f419b03a70942bc608f679902a6c8f6bb29d1b672faaa70760682cb801","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":13,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-0b8d3dbf8234aff75f04","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --supports--> v01\n- v0c --refutes--> v03\n- v0b --refutes--> v0c\n- v0c --refutes--> v12\n- v04 --refutes--> v0a\n- v03 --refutes--> v02\n- v04 --refutes--> v09\n- v0e --supports--> v0b\n- v08 --refutes--> v07\n- v06 --refutes--> v00\n- v08 --refutes--> v06\n- v09 --supports--> v05\n- v11 --supports--> v04\n- v13 --supports--> v11\n- v10 --refutes--> v0d\n- v03 --supports--> v13\n- v06 --refutes--> v0f\n- v05 --supports--> v10\n- v00 --supports--> v0e\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6fded137c9b81270c308d11b45b7eae2345fe0c0ac62a2c11f498edc4de5f1a1","schema":"livegraph.boolean-propagation.v1","spec_hash":"0b8d3dbf8234aff75f04d3d99f5aba36405ba48b05d88c64e28bdcc76c2c4b72","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-fb730158b3828c42a086","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v04\n- v06 --refutes--> v11\n- v09 --supports--> v12\n- v10 --refutes--> v09\n- v01 --refutes--> v0a\n- v10 --supports--> v05\n- v02 --supports--> v07\n- v04 --refutes--> v02\n- v09 --refutes--> v0e\n- v06 --refutes--> v01\n- v0d --refutes--> v13\n- v0d --supports--> v03\n- v00 --refutes--> v10\n- v0e --refutes--> v0d\n- v08 --refutes--> v0f\n- v02 --supports--> v00\n- v0a --refutes--> v0b\n- v0d --supports--> v06\n- v0f --refutes--> v0c\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"076e8f36af645eb41bbea0b9f2303e5139e60f8e139f9481257e48a383dff6a0","schema":"livegraph.boolean-propagation.v1","spec_hash":"fb730158b3828c42a08619d9c4959d6eadccac6fa32d21894d0e0d0bc3fb13fd","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":13,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-cedd47016f23a8e88a6e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v05\n- v11 --supports--> v07\n- v08 --refutes--> v00\n- v0e --supports--> v0c\n- v04 --supports--> v0d\n- v05 --refutes--> v0f\n- v0b --refutes--> v03\n- v0c --refutes--> v10\n- v04 --refutes--> v0a\n- v09 --refutes--> v06\n- v03 --supports--> v02\n- v03 --supports--> v01\n- v08 --supports--> v09\n- v06 --refutes--> v0b\n- v01 --refutes--> v12\n- v0f --supports--> v11\n- v05 --supports--> v13\n- v0a --refutes--> v0e\n- v01 --refutes--> v04\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"225ff3641dfbfaa219ecaef53271f63c54a7423c41050ec1803dea828258b152","schema":"livegraph.boolean-propagation.v1","spec_hash":"cedd47016f23a8e88a6ed0dcda4c63a89415621860a4da149a483a632a832eec","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":13,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-0f925674788204a3545d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v0a\n- v13 --supports--> v12\n- v0e --refutes--> v06\n- v09 --refutes--> v0f\n- v00 --refutes--> v10\n- v13 --refutes--> v07\n- v0b --supports--> v05\n- v11 --supports--> v08\n- v0e --supports--> v00\n- v02 --supports--> v0e\n- v0f --refutes--> v0b\n- v13 --supports--> v01\n- v05 --refutes--> v04\n- v0f --refutes--> v03\n- v04 --supports--> v13\n- v08 --refutes--> v02\n- v08 --supports--> v0c\n- v12 --refutes--> v11\n- v0d --supports--> v09\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"55ddb4221ce3301f0f0631978b516aa555c6e1afc0e8500a12b4b6dbf5d934a0","schema":"livegraph.boolean-propagation.v1","spec_hash":"0f925674788204a3545d09fcfdcbe7ebde72ee5c8f815e1a826fc2d3693cd0ea","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-4a2ddbf71312af23f38b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v01\n- v0b --supports--> v11\n- v13 --supports--> v04\n- v10 --refutes--> v08\n- v0d --supports--> v05\n- v0c --refutes--> v0f\n- v0c --refutes--> v07\n- v10 --refutes--> v03\n- v06 --supports--> v0d\n- v10 --supports--> v0e\n- v10 --refutes--> v09\n- v0f --supports--> v0b\n- v01 --refutes--> v12\n- v12 --supports--> v0c\n- v0b --supports--> v00\n- v00 --supports--> v0a\n- v03 --refutes--> v06\n- v00 --supports--> v13\n- v04 --supports--> v02\n\n[VERIFIED ROOT]\nv10 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e17d885ca3ec2097933d7c0f1bce439cc938bcbe3894659ea99cc09c8b4c77fd","schema":"livegraph.boolean-propagation.v1","spec_hash":"4a2ddbf71312af23f38b6a681351438beae21d6a016defe892542184102cda01","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-38fb517e42542ca341b5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v0d\n- v05 --supports--> v07\n- v0b --supports--> v09\n- v02 --supports--> v06\n- v10 --refutes--> v03\n- v0d --supports--> v0a\n- v00 --refutes--> v0f\n- v13 --supports--> v00\n- v11 --supports--> v0b\n- v0b --refutes--> v10\n- v00 --supports--> v0c\n- v02 --refutes--> v13\n- v07 --supports--> v11\n- v05 --supports--> v04\n- v02 --refutes--> v12\n- v01 --refutes--> v02\n- v08 --supports--> v05\n- v0c --refutes--> v08\n- v10 --refutes--> v0e\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c32ca53ff0d1198fc4c2376b611e816ae2cff0e101cdd541ace18be852087e69","schema":"livegraph.boolean-propagation.v1","spec_hash":"38fb517e42542ca341b5f18bf43201b8ca01e77794a3fe513cbdc284eb1196fc","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-73c476c705d06d61ff6e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v06\n- v06 --refutes--> v0b\n- v13 --supports--> v03\n- v02 --refutes--> v05\n- v02 --refutes--> v0e\n- v0b --supports--> v09\n- v11 --refutes--> v0f\n- v12 --refutes--> v13\n- v13 --refutes--> v10\n- v04 --supports--> v00\n- v02 --refutes--> v08\n- v09 --refutes--> v04\n- v02 --supports--> v01\n- v0c --refutes--> v0d\n- v05 --supports--> v11\n- v03 --supports--> v0c\n- v00 --supports--> v12\n- v0f --supports--> v07\n- v09 --refutes--> v0a\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9905c18f8b8dba0643830e58206a365c524de129f01a81d0d160f176a7f39536","schema":"livegraph.boolean-propagation.v1","spec_hash":"73c476c705d06d61ff6eaa5be76fc5c7c3ab9a5e7656b9bdb998ab274bb4b798","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-e669233418f5aadebaca","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v0e\n- v02 --refutes--> v0f\n- v03 --refutes--> v00\n- v00 --refutes--> v12\n- v07 --refutes--> v05\n- v01 --supports--> v07\n- v0a --supports--> v09\n- v08 --supports--> v0d\n- v05 --supports--> v0b\n- v0a --supports--> v02\n- v00 --refutes--> v06\n- v0f --refutes--> v0c\n- v0d --refutes--> v04\n- v0d --refutes--> v13\n- v04 --supports--> v03\n- v0f --refutes--> v11\n- v10 --refutes--> v01\n- v0c --supports--> v08\n- v05 --supports--> v0a\n\n[VERIFIED ROOT]\nv10 = FALSE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"321a596e9690962f38d831f9bb99f702628e86bd1163034cb82cf689e4708373","schema":"livegraph.boolean-propagation.v1","spec_hash":"e669233418f5aadebaca6e9808752bd915494c3f661132707ec7b49e5266d8bb","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":13,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-992c51ea44393d36b1b8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v04\n- v09 --refutes--> v0e\n- v07 --supports--> v0d\n- v10 --supports--> v07\n- v03 --supports--> v0f\n- v08 --refutes--> v12\n- v07 --supports--> v05\n- v02 --supports--> v0c\n- v03 --refutes--> v09\n- v04 --refutes--> v06\n- v02 --refutes--> v0a\n- v11 --refutes--> v00\n- v04 --supports--> v03\n- v13 --refutes--> v02\n- v09 --supports--> v01\n- v11 --refutes--> v10\n- v12 --refutes--> v11\n- v0b --supports--> v13\n- v0e --supports--> v0b\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2f27374a67400fa60fe83312dfcf60f1f9076f5f6c55f28f1efc4a26b84eff5b","schema":"livegraph.boolean-propagation.v1","spec_hash":"992c51ea44393d36b1b88b0897cedb956911ee2dcc7555d8cb04ab1a00f64105","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-4d9d17aed43879e1dcc6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v03\n- v04 --refutes--> v0d\n- v06 --refutes--> v0c\n- v00 --supports--> v08\n- v01 --supports--> v12\n- v0e --refutes--> v09\n- v11 --refutes--> v04\n- v07 --supports--> v0a\n- v0b --refutes--> v00\n- v0a --supports--> v11\n- v0e --refutes--> v0b\n- v12 --supports--> v05\n- v06 --supports--> v10\n- v0e --supports--> v0f\n- v0a --refutes--> v02\n- v00 --supports--> v01\n- v00 --refutes--> v13\n- v05 --refutes--> v06\n- v03 --refutes--> v07\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"55da8857e72f3003b05a7deaeefd1eccc275fb165422b9516fef715cea6572e2","schema":"livegraph.boolean-propagation.v1","spec_hash":"4d9d17aed43879e1dcc69cf51508c8b705a544f30b7e38e202974586d71f1d69","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-c54a09d4663373ec91b6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v11\n- v08 --refutes--> v03\n- v09 --supports--> v0a\n- v10 --supports--> v01\n- v0d --supports--> v00\n- v08 --refutes--> v02\n- v10 --refutes--> v0d\n- v06 --refutes--> v13\n- v0e --supports--> v08\n- v12 --supports--> v10\n- v05 --refutes--> v0c\n- v08 --supports--> v0b\n- v11 --supports--> v06\n- v08 --supports--> v04\n- v0b --refutes--> v09\n- v08 --refutes--> v0f\n- v13 --refutes--> v12\n- v0c --refutes--> v07\n- v09 --refutes--> v05\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c03654ec0915187457ae6c65988124eee7f7fe38865a489fd78fbf6edb74c0f4","schema":"livegraph.boolean-propagation.v1","spec_hash":"c54a09d4663373ec91b62e7a4c3b3f0b63f6b2d06dbe01960f1a154039696c2a","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-42e234bc63d1c5d38f9a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v11\n- v06 --refutes--> v0f\n- v0b --refutes--> v08\n- v04 --supports--> v05\n- v13 --refutes--> v0a\n- v09 --refutes--> v06\n- v0a --refutes--> v0d\n- v0f --refutes--> v02\n- v03 --refutes--> v10\n- v0a --refutes--> v00\n- v00 --supports--> v09\n- v05 --refutes--> v0e\n- v03 --supports--> v0b\n- v08 --refutes--> v04\n- v09 --supports--> v01\n- v0b --refutes--> v07\n- v11 --refutes--> v13\n- v08 --refutes--> v0c\n- v0f --supports--> v03\n\n[VERIFIED ROOT]\nv12 = FALSE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"245b7360cea7cd1b6a426054ae05aab8a97970f8f83eab859ce75761814dc9dd","schema":"livegraph.boolean-propagation.v1","spec_hash":"42e234bc63d1c5d38f9a66e4d5ba13c7724fc7563cb55b6d0fb039778982211c","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v12\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-dc157b0f6aeb17f9c569","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v00\n- v04 --refutes--> v0a\n- v07 --supports--> v10\n- v10 --supports--> v0d\n- v00 --supports--> v08\n- v04 --supports--> v09\n- v12 --refutes--> v06\n- v11 --supports--> v05\n- v0a --supports--> v0b\n- v11 --supports--> v01\n- v07 --supports--> v03\n- v0c --supports--> v04\n- v0d --supports--> v0e\n- v0b --supports--> v0f\n- v0b --refutes--> v12\n- v0a --supports--> v02\n- v0e --supports--> v13\n- v12 --supports--> v11\n- v08 --supports--> v07\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fcf7e8bb83bbc22306a5a4b314a7d395ab8d8f97b78cb163b81521890ae6f11c","schema":"livegraph.boolean-propagation.v1","spec_hash":"dc157b0f6aeb17f9c569ffa746cfb167e22cd261af6f66f746db6c2adee324d7","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-faab7b3dc1cf7daa880c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v01\n- v06 --refutes--> v05\n- v0b --supports--> v13\n- v09 --supports--> v0f\n- v01 --supports--> v11\n- v06 --supports--> v10\n- v11 --supports--> v12\n- v13 --supports--> v09\n- v09 --supports--> v0d\n- v09 --refutes--> v0c\n- v11 --refutes--> v0a\n- v10 --supports--> v0e\n- v0c --supports--> v03\n- v0a --refutes--> v08\n- v07 --refutes--> v04\n- v00 --refutes--> v0b\n- v07 --refutes--> v02\n- v08 --refutes--> v06\n- v10 --supports--> v07\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9779fc9e683f0e739e84e99c048028d7546824a934ce3005a19e30bba179782b","schema":"livegraph.boolean-propagation.v1","spec_hash":"faab7b3dc1cf7daa880cd53cb9a02b8b5f449cd8d60d6eb61e0e4b14bf1393ff","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-c0f394aee2fd210ca486","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --refutes--> v00\n- v0b --supports--> v03\n- v00 --supports--> v0f\n- v05 --refutes--> v13\n- v05 --supports--> v10\n- v05 --refutes--> v06\n- v0e --supports--> v05\n- v12 --refutes--> v0e\n- v00 --supports--> v08\n- v01 --refutes--> v0d\n- v0a --refutes--> v01\n- v07 --refutes--> v04\n- v0c --refutes--> v0a\n- v0d --refutes--> v12\n- v08 --supports--> v07\n- v0c --refutes--> v09\n- v0d --refutes--> v11\n- v04 --supports--> v0b\n- v0c --refutes--> v02\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"23a48fcf3d45d4e7ed1d1b4d4ac292c5ffca004843b9578cb4cc8a48081daf25","schema":"livegraph.boolean-propagation.v1","spec_hash":"c0f394aee2fd210ca4863f52c8f13c7bdc9bf690ae89f67d4006b77782f47bde","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-ff56a536a6c47f85d78a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v01\n- v06 --supports--> v0f\n- v01 --supports--> v10\n- v0d --refutes--> v06\n- v0e --supports--> v00\n- v0a --refutes--> v04\n- v09 --supports--> v11\n- v0f --refutes--> v02\n- v05 --refutes--> v0b\n- v0a --supports--> v03\n- v0b --supports--> v08\n- v01 --refutes--> v0c\n- v00 --supports--> v07\n- v0b --refutes--> v0d\n- v0f --supports--> v13\n- v10 --supports--> v12\n- v12 --refutes--> v0e\n- v11 --supports--> v05\n- v00 --supports--> v09\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"10fe12ec387e4281e5a1f1225153466ac8dbbda52da3e3630ae3b847a797e989","schema":"livegraph.boolean-propagation.v1","spec_hash":"ff56a536a6c47f85d78aa3748012b5a6e6d4ef803bf6862b47982daf8ba088c6","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-0479bf5bca7d6592a666","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v01\n- v12 --refutes--> v04\n- v01 --refutes--> v0c\n- v09 --supports--> v12\n- v03 --refutes--> v06\n- v02 --supports--> v07\n- v0c --refutes--> v0e\n- v11 --refutes--> v08\n- v12 --supports--> v00\n- v01 --supports--> v10\n- v0c --supports--> v13\n- v06 --refutes--> v0b\n- v07 --supports--> v03\n- v04 --refutes--> v11\n- v06 --supports--> v0d\n- v0f --supports--> v02\n- v0c --supports--> v0a\n- v0d --supports--> v05\n- v08 --supports--> v0f\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ddc6118b165c01a411691341dae9411988129403f1f7b618e86834a2b71da81d","schema":"livegraph.boolean-propagation.v1","spec_hash":"0479bf5bca7d6592a666de1a4ee2ed169739de4ded17e5cd69dcf3d5deab882f","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-164560dcecfa9d20f693","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v04\n- v04 --refutes--> v08\n- v08 --refutes--> v13\n- v13 --supports--> v0c\n- v01 --supports--> v03\n- v03 --supports--> v00\n- v0b --refutes--> v12\n- v03 --refutes--> v10\n- v13 --refutes--> v0a\n- v09 --refutes--> v0e\n- v09 --supports--> v11\n- v11 --supports--> v07\n- v07 --supports--> v0b\n- v12 --refutes--> v0d\n- v01 --refutes--> v05\n- v02 --supports--> v09\n- v11 --refutes--> v06\n- v0f --supports--> v02\n- v13 --refutes--> v01\n\n[VERIFIED ROOT]\nv0f = FALSE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bbe50c1a652d73974c8dd90f600ae02c9ae7a54d0fd537c6348158968cd47f4c","schema":"livegraph.boolean-propagation.v1","spec_hash":"164560dcecfa9d20f69394eadae42762e80022a21bc114ede3e418a16f8c515c","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-18f0937ac6e859c9b9b2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v0b\n- v05 --refutes--> v0f\n- v0c --refutes--> v09\n- v06 --refutes--> v08\n- v0e --supports--> v03\n- v12 --refutes--> v00\n- v08 --supports--> v0e\n- v0c --supports--> v02\n- v12 --supports--> v11\n- v0b --refutes--> v0c\n- v04 --supports--> v01\n- v04 --supports--> v07\n- v00 --refutes--> v13\n- v0f --refutes--> v12\n- v0f --refutes--> v0a\n- v00 --supports--> v0d\n- v00 --supports--> v04\n- v10 --refutes--> v06\n- v02 --supports--> v10\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"da6cbbe2e1d2d24235a7ecaa49f2bee182420df5f3998cd2316f825d2d7acb97","schema":"livegraph.boolean-propagation.v1","spec_hash":"18f0937ac6e859c9b9b2cdde1775d2847b38a6b5d549527dcbcf4c66237768a8","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-0c7d9c4be533193cb462","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v0d\n- v11 --supports--> v12\n- v03 --supports--> v0b\n- v0e --supports--> v10\n- v02 --supports--> v0c\n- v09 --refutes--> v05\n- v09 --refutes--> v0f\n- v03 --supports--> v09\n- v13 --supports--> v01\n- v11 --refutes--> v04\n- v07 --refutes--> v11\n- v11 --refutes--> v08\n- v02 --supports--> v0a\n- v08 --supports--> v02\n- v00 --refutes--> v13\n- v10 --supports--> v07\n- v0d --refutes--> v00\n- v0a --supports--> v03\n- v05 --supports--> v06\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bceba5e9cdb3f2bdb1881152f7abc17a47a509605a20282597d624627fba70b2","schema":"livegraph.boolean-propagation.v1","spec_hash":"0c7d9c4be533193cb462d2ee45348f56846099ee67b5e1c12f0d02582beea350","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-2dd83a618c2fdb0fe2fa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v05\n- v0c --refutes--> v0a\n- v08 --refutes--> v02\n- v0b --refutes--> v0e\n- v0d --supports--> v0c\n- v08 --refutes--> v00\n- v10 --refutes--> v01\n- v0a --supports--> v08\n- v05 --refutes--> v06\n- v0d --refutes--> v07\n- v12 --refutes--> v0d\n- v02 --supports--> v04\n- v11 --refutes--> v0b\n- v12 --supports--> v03\n- v01 --supports--> v09\n- v0a --supports--> v0f\n- v09 --supports--> v12\n- v06 --supports--> v11\n- v08 --refutes--> v13\n\n[VERIFIED ROOT]\nv10 = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7b4b5816b0f9e032937860991227c3de6f461e1170485837da5f9af60a24efeb","schema":"livegraph.boolean-propagation.v1","spec_hash":"2dd83a618c2fdb0fe2fa1c594c2d751d3cbb21af90d4cc72b3a2db787a02a491","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-0603adc9dab51e99d031","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v0c\n- v11 --supports--> v03\n- v10 --refutes--> v0e\n- v04 --supports--> v11\n- v0b --refutes--> v06\n- v07 --supports--> v0f\n- v0b --refutes--> v0d\n- v08 --refutes--> v00\n- v13 --refutes--> v05\n- v03 --refutes--> v09\n- v00 --refutes--> v12\n- v11 --supports--> v02\n- v05 --supports--> v08\n- v0f --refutes--> v0a\n- v0f --refutes--> v0b\n- v0f --refutes--> v01\n- v10 --supports--> v04\n- v12 --refutes--> v07\n- v06 --supports--> v10\n\n[VERIFIED ROOT]\nv13 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b7c2ae3ba7b4449f6f70ea7fb4af41f24b4e240b7b6d32c767253f0efdedb174","schema":"livegraph.boolean-propagation.v1","spec_hash":"0603adc9dab51e99d03179c1e4a5106a4af30a874464b695ea2071bcf905ef0d","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v13\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-c4c87a45d8f1d4ccd782","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v01\n- v0f --refutes--> v07\n- v0f --supports--> v12\n- v0a --refutes--> v05\n- v11 --refutes--> v13\n- v0a --refutes--> v0b\n- v02 --refutes--> v0e\n- v0f --refutes--> v0d\n- v10 --supports--> v08\n- v05 --supports--> v03\n- v09 --supports--> v0f\n- v01 --refutes--> v11\n- v12 --refutes--> v00\n- v13 --refutes--> v04\n- v0c --refutes--> v0a\n- v00 --supports--> v06\n- v0e --refutes--> v10\n- v10 --refutes--> v0c\n- v13 --refutes--> v02\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"10f33c6983d16ab9029c56205e5d28fb700159ec3d11b7c03490231e7c75fbeb","schema":"livegraph.boolean-propagation.v1","spec_hash":"c4c87a45d8f1d4ccd78213b9cec1597550a1394a38ca66f0fc2169147d27c803","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-dd42bd5e22c428dbe95f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v02\n- v08 --refutes--> v03\n- v11 --supports--> v00\n- v04 --supports--> v07\n- v0b --supports--> v05\n- v0c --supports--> v01\n- v03 --supports--> v09\n- v07 --refutes--> v11\n- v0b --supports--> v06\n- v0a --supports--> v0d\n- v05 --refutes--> v04\n- v03 --refutes--> v0a\n- v13 --refutes--> v08\n- v11 --refutes--> v0c\n- v0f --refutes--> v13\n- v0d --refutes--> v0e\n- v0d --supports--> v12\n- v0c --refutes--> v10\n- v12 --supports--> v0b\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"77c05d296ecb99d32d1494567adf042cc3de326ca06c1d4106f5bd16cb2dda38","schema":"livegraph.boolean-propagation.v1","spec_hash":"dd42bd5e22c428dbe95ff4b9770cc6ec0c6e02f09dac779290edc30d2f17b420","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-4cfada182c0673f939e5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v03\n- v12 --refutes--> v0a\n- v0b --refutes--> v11\n- v00 --supports--> v0d\n- v0a --supports--> v05\n- v0a --refutes--> v10\n- v10 --supports--> v02\n- v09 --refutes--> v13\n- v02 --supports--> v00\n- v11 --supports--> v0c\n- v0e --refutes--> v09\n- v06 --refutes--> v0f\n- v13 --refutes--> v06\n- v04 --refutes--> v0e\n- v01 --refutes--> v08\n- v06 --refutes--> v12\n- v00 --supports--> v01\n- v0e --supports--> v07\n- v11 --supports--> v04\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"56c5b85c0b3e863c87dbd23f951c1f79c2e1e78beddc625594ad81f35b1d0998","schema":"livegraph.boolean-propagation.v1","spec_hash":"4cfada182c0673f939e5e31b3e1f2e6fb45091ad11baa0ba3cbba327adaa85d1","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-dec4baccff5c5dff8371","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v13\n- v08 --supports--> v03\n- v0c --supports--> v00\n- v12 --supports--> v0b\n- v0c --refutes--> v09\n- v13 --refutes--> v07\n- v12 --refutes--> v05\n- v0a --supports--> v12\n- v11 --supports--> v0f\n- v0f --refutes--> v0c\n- v0e --supports--> v01\n- v03 --supports--> v02\n- v07 --supports--> v0d\n- v0d --supports--> v10\n- v0c --refutes--> v04\n- v12 --refutes--> v06\n- v03 --supports--> v0a\n- v0d --refutes--> v08\n- v09 --supports--> v0e\n\n[VERIFIED ROOT]\nv11 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7b197507ca4a69ad8bf0c7302752e01badda9896ac7bcff6f2c15fbd5809b8c7","schema":"livegraph.boolean-propagation.v1","spec_hash":"dec4baccff5c5dff8371b1e11b566a88dc2d23c434afd9456dc999431a36981e","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-4290f1eea3faa8230225","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v0a\n- v0f --supports--> v02\n- v07 --supports--> v0c\n- v0f --refutes--> v11\n- v01 --refutes--> v0d\n- v0d --supports--> v0f\n- v13 --refutes--> v09\n- v12 --supports--> v07\n- v08 --supports--> v00\n- v06 --refutes--> v12\n- v13 --supports--> v05\n- v03 --supports--> v01\n- v02 --supports--> v08\n- v0c --refutes--> v10\n- v0b --supports--> v03\n- v08 --refutes--> v13\n- v08 --refutes--> v04\n- v10 --supports--> v0b\n- v12 --refutes--> v0e\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"aee1b0aa1049ea8bf934bf4d5e173b92fd6bfead26cf6f58b05bbb27fd958db7","schema":"livegraph.boolean-propagation.v1","spec_hash":"4290f1eea3faa823022509bd5b023bc059bda4d581fe90197235fa09096d03a8","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-823d39336edc1e393907","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v10\n- v08 --supports--> v12\n- v01 --refutes--> v0e\n- v04 --refutes--> v05\n- v08 --refutes--> v07\n- v09 --supports--> v0a\n- v00 --refutes--> v01\n- v04 --supports--> v00\n- v08 --refutes--> v0f\n- v12 --refutes--> v09\n- v12 --supports--> v03\n- v0e --refutes--> v11\n- v0a --refutes--> v0b\n- v10 --refutes--> v06\n- v06 --refutes--> v02\n- v12 --refutes--> v13\n- v02 --supports--> v08\n- v04 --supports--> v0c\n- v0d --refutes--> v04\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"819efc899e6f1bd1e34db4836edbcda80fa1e9fa5a74c46fdc77bb0d850b1437","schema":"livegraph.boolean-propagation.v1","spec_hash":"823d39336edc1e3939071715abc281178fd41b7c0e40c24ec2774e870645372b","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-cbcfdbac1a2174ac6126","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v03\n- v06 --supports--> v0a\n- v01 --supports--> v05\n- v11 --supports--> v0b\n- v05 --supports--> v04\n- v12 --supports--> v0e\n- v11 --refutes--> v0f\n- v13 --refutes--> v10\n- v09 --refutes--> v0d\n- v06 --supports--> v0c\n- v09 --refutes--> v08\n- v02 --supports--> v07\n- v03 --refutes--> v00\n- v02 --refutes--> v01\n- v00 --supports--> v13\n- v08 --refutes--> v02\n- v12 --supports--> v06\n- v10 --refutes--> v11\n- v0b --supports--> v12\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"53945afccef2e991226c41af9f7e77b7f8a1caddf929426ce293030f9795e1b8","schema":"livegraph.boolean-propagation.v1","spec_hash":"cbcfdbac1a2174ac61262c6fd2d64ff3ecef4e0661536ca1190dce7883aca0bb","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-a90cbd93dcba35818fa9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v04\n- v04 --refutes--> v06\n- v0d --supports--> v00\n- v07 --supports--> v0b\n- v09 --supports--> v12\n- v0b --supports--> v11\n- v00 --refutes--> v05\n- v01 --refutes--> v10\n- v0b --refutes--> v02\n- v04 --refutes--> v0a\n- v06 --supports--> v0d\n- v12 --refutes--> v13\n- v0f --supports--> v0c\n- v01 --refutes--> v03\n- v12 --refutes--> v0f\n- v0c --supports--> v0e\n- v00 --supports--> v09\n- v0e --supports--> v07\n- v06 --supports--> v08\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c7aa512a057c3eadc74d42dd764c63038e78dcbd3b11751f5e63c48e5a8db86d","schema":"livegraph.boolean-propagation.v1","spec_hash":"a90cbd93dcba35818fa9d8c95bac04313cd94ec4e952fc77f363ac591f950b45","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-d8ca3477456fd9f66007","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v09\n- v03 --supports--> v12\n- v00 --supports--> v03\n- v0c --supports--> v0f\n- v0f --refutes--> v10\n- v0e --refutes--> v0a\n- v0b --refutes--> v04\n- v04 --refutes--> v08\n- v06 --supports--> v01\n- v04 --refutes--> v0c\n- v0e --refutes--> v06\n- v04 --supports--> v13\n- v12 --refutes--> v11\n- v01 --refutes--> v07\n- v0f --supports--> v0e\n- v0d --supports--> v02\n- v12 --refutes--> v05\n- v09 --refutes--> v0d\n- v05 --supports--> v0b\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9b9ac96a207538566d0895811bf7551bb02a25f70d3fd3d383b3a331c1b43ec3","schema":"livegraph.boolean-propagation.v1","spec_hash":"d8ca3477456fd9f660073a8d3388cf68eae5916420d1692ccccc865e8682f2d2","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-19782b73c1f761bd7f60","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v0f\n- v11 --supports--> v03\n- v06 --supports--> v01\n- v06 --refutes--> v09\n- v0c --refutes--> v06\n- v12 --supports--> v0e\n- v0b --supports--> v04\n- v0d --supports--> v05\n- v04 --supports--> v12\n- v13 --supports--> v0b\n- v0e --refutes--> v0c\n- v03 --supports--> v13\n- v11 --supports--> v02\n- v08 --supports--> v00\n- v00 --supports--> v10\n- v03 --refutes--> v0a\n- v11 --supports--> v07\n- v00 --refutes--> v11\n- v09 --supports--> v0d\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5a78da9be7660d6ce73e066247a7dfc9cdb90026bf94eac8fd47925c5776b1d2","schema":"livegraph.boolean-propagation.v1","spec_hash":"19782b73c1f761bd7f60a609ac68dc55e75381a2bc6b28c3d7adeeb6ee986065","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-b931d506b3eb0dfbd32f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v0c\n- v09 --supports--> v04\n- v0c --supports--> v11\n- v01 --refutes--> v0f\n- v09 --refutes--> v05\n- v13 --supports--> v0e\n- v00 --refutes--> v13\n- v12 --refutes--> v01\n- v11 --supports--> v0a\n- v0a --supports--> v0b\n- v0a --supports--> v09\n- v13 --refutes--> v12\n- v02 --refutes--> v0d\n- v02 --refutes--> v03\n- v06 --refutes--> v10\n- v09 --supports--> v07\n- v03 --refutes--> v06\n- v08 --refutes--> v02\n- v07 --supports--> v00\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1dac6583ef7c938548bffcbc392e9c627e0948e60d5983df8b83c79bf0073a71","schema":"livegraph.boolean-propagation.v1","spec_hash":"b931d506b3eb0dfbd32ffed0dd3a213f7a9018e19ff58f9787bf6bc0236f108b","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-345a93770e280b51ac25","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --refutes--> v01\n- v11 --supports--> v0d\n- v09 --supports--> v04\n- v04 --refutes--> v07\n- v0b --supports--> v0f\n- v0e --refutes--> v12\n- v0c --supports--> v06\n- v12 --refutes--> v13\n- v0c --refutes--> v09\n- v05 --supports--> v0b\n- v0d --supports--> v08\n- v11 --refutes--> v02\n- v11 --refutes--> v00\n- v0d --refutes--> v0e\n- v07 --supports--> v05\n- v07 --supports--> v0a\n- v09 --supports--> v10\n- v0f --supports--> v03\n- v03 --supports--> v11\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d5c0c182342526e13f3b012d5e23323f202ab06386e7258e81d526e9d0f56a56","schema":"livegraph.boolean-propagation.v1","spec_hash":"345a93770e280b51ac25a85a72385f56323054d53746d8d650d5b14fe150f4e7","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-d1878c9309a31a58c06f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --refutes--> v08\n- v0e --supports--> v0c\n- v07 --supports--> v0b\n- v13 --supports--> v02\n- v0b --refutes--> v09\n- v01 --refutes--> v0a\n- v0e --refutes--> v01\n- v10 --refutes--> v0e\n- v11 --refutes--> v06\n- v06 --supports--> v0d\n- v0f --supports--> v03\n- v0a --refutes--> v07\n- v03 --supports--> v04\n- v0a --supports--> v05\n- v0d --supports--> v10\n- v09 --supports--> v13\n- v03 --refutes--> v11\n- v0a --refutes--> v12\n- v11 --refutes--> v00\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"82cacc9e94e403b73712332abe3d0be79759164bf85cc4dc0ee4264074929efc","schema":"livegraph.boolean-propagation.v1","spec_hash":"d1878c9309a31a58c06f3d309192b64a04bc68b6faa49aaeed4e15bf552c47e6","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-51e0dcf71168251dab98","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v08\n- v00 --supports--> v13\n- v11 --supports--> v00\n- v01 --supports--> v06\n- v05 --refutes--> v03\n- v12 --refutes--> v10\n- v0e --refutes--> v0f\n- v09 --refutes--> v12\n- v04 --supports--> v01\n- v13 --refutes--> v09\n- v02 --refutes--> v0e\n- v10 --supports--> v0c\n- v01 --refutes--> v0d\n- v05 --refutes--> v0a\n- v10 --supports--> v05\n- v08 --supports--> v11\n- v0e --refutes--> v04\n- v13 --refutes--> v0b\n- v02 --refutes--> v07\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"64a52b82c49c3a63a0b4462e419a58c7feda3969d5ae3fdcb2d36d5f4d7bf985","schema":"livegraph.boolean-propagation.v1","spec_hash":"51e0dcf71168251dab9893f59c117fa45948c07bc25abcac1bc6cffc20a53e4d","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-9555542c441bbc77be17","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v0a\n- v11 --refutes--> v10\n- v10 --supports--> v09\n- v0c --refutes--> v0d\n- v0e --supports--> v11\n- v0b --supports--> v00\n- v0e --supports--> v03\n- v05 --supports--> v0c\n- v09 --refutes--> v13\n- v05 --supports--> v01\n- v04 --supports--> v0f\n- v09 --supports--> v07\n- v13 --refutes--> v0b\n- v0a --refutes--> v08\n- v0b --refutes--> v05\n- v08 --supports--> v04\n- v0e --refutes--> v06\n- v04 --refutes--> v12\n- v0f --refutes--> v02\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a273670cf88fcb7d27900553611fc33bad05355288d022518f2229959e269506","schema":"livegraph.boolean-propagation.v1","spec_hash":"9555542c441bbc77be1755cba92173422f4bf0469bf266e6b818ae79dc1f917c","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-91d27083b6f4f7948617","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v04\n- v04 --supports--> v12\n- v0f --supports--> v0b\n- v00 --supports--> v0f\n- v00 --supports--> v11\n- v08 --supports--> v01\n- v0f --refutes--> v0d\n- v09 --supports--> v05\n- v07 --refutes--> v09\n- v13 --refutes--> v08\n- v10 --supports--> v0e\n- v0f --refutes--> v03\n- v08 --refutes--> v02\n- v03 --supports--> v10\n- v10 --refutes--> v06\n- v05 --refutes--> v00\n- v08 --refutes--> v0a\n- v12 --refutes--> v13\n- v0c --refutes--> v07\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e6c31fc563078677e1afdd0d0a18b331267d63e3f7647f45f778e97e3ffa7c90","schema":"livegraph.boolean-propagation.v1","spec_hash":"91d27083b6f4f7948617acad8904d96ea43a059cd8bd810d653bdf684d9fd76a","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-d1066be2ca729e084a87","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v02\n- v02 --refutes--> v0c\n- v0a --refutes--> v07\n- v0f --refutes--> v0a\n- v0b --refutes--> v0e\n- v03 --refutes--> v01\n- v0c --supports--> v13\n- v13 --refutes--> v09\n- v09 --refutes--> v0d\n- v0b --supports--> v04\n- v03 --supports--> v06\n- v11 --refutes--> v0b\n- v06 --refutes--> v00\n- v0f --supports--> v05\n- v0b --supports--> v08\n- v11 --refutes--> v10\n- v08 --refutes--> v0f\n- v0d --refutes--> v11\n- v08 --refutes--> v12\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"481b34d7623a29fb7fb9a05b418f7a36a077c61e9e90da3823a9b3b4878901e1","schema":"livegraph.boolean-propagation.v1","spec_hash":"d1066be2ca729e084a872dfc4ca5e70f164ee66c66061bc1b1a6b282b832312a","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-93794014eda0310b436e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v01\n- v0c --refutes--> v00\n- v05 --refutes--> v0b\n- v01 --refutes--> v11\n- v10 --supports--> v0f\n- v0b --refutes--> v0d\n- v0e --supports--> v07\n- v02 --refutes--> v13\n- v0e --supports--> v02\n- v0c --supports--> v0a\n- v11 --refutes--> v06\n- v13 --refutes--> v09\n- v06 --refutes--> v10\n- v06 --supports--> v08\n- v0a --refutes--> v0e\n- v13 --refutes--> v12\n- v0b --refutes--> v03\n- v0b --supports--> v04\n- v03 --supports--> v0c\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"40953a3449b5209a77f5125aeb3ef82f343c56cfca610c1989d38bee6a198b4b","schema":"livegraph.boolean-propagation.v1","spec_hash":"93794014eda0310b436ee742ba9ec8dd23fcfde975e5b73c42be860b7b323d73","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-6a61fda7e8fab4ff54a4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v07\n- v09 --supports--> v0d\n- v0d --supports--> v0e\n- v0e --refutes--> v00\n- v0e --refutes--> v0c\n- v13 --refutes--> v09\n- v0e --supports--> v02\n- v06 --supports--> v10\n- v0b --refutes--> v03\n- v12 --supports--> v06\n- v08 --refutes--> v0b\n- v04 --supports--> v05\n- v02 --supports--> v08\n- v03 --supports--> v01\n- v11 --supports--> v13\n- v04 --supports--> v11\n- v01 --refutes--> v12\n- v09 --refutes--> v0a\n- v04 --refutes--> v0f\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c28cb20b9418f463f437aeec9a476b4ae915b1ed50c4765272b4f667c0fa3660","schema":"livegraph.boolean-propagation.v1","spec_hash":"6a61fda7e8fab4ff54a41119e72d5eead1e57c1d85055f9080f3aa1324635520","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-363490397cc59a616cee","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v07\n- v0f --supports--> v13\n- v03 --supports--> v02\n- v0f --refutes--> v0c\n- v13 --refutes--> v06\n- v03 --refutes--> v05\n- v04 --supports--> v03\n- v08 --supports--> v04\n- v10 --supports--> v0e\n- v06 --supports--> v10\n- v11 --refutes--> v08\n- v09 --refutes--> v11\n- v11 --supports--> v0a\n- v05 --supports--> v0b\n- v08 --refutes--> v0d\n- v00 --refutes--> v12\n- v0b --refutes--> v01\n- v0e --supports--> v09\n- v00 --refutes--> v0f\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e0935f94fef7a20f8d9cca9332088b551183b07b4d4ef5dc8499097581941dab","schema":"livegraph.boolean-propagation.v1","spec_hash":"363490397cc59a616cee6f043e95378c004b451fe36e6768ed8bbecdc508cdb0","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-ebbc0ad89b7e40d50fdb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v12\n- v00 --refutes--> v05\n- v08 --supports--> v11\n- v0b --supports--> v00\n- v02 --supports--> v0f\n- v00 --supports--> v0d\n- v0b --refutes--> v0a\n- v03 --refutes--> v0c\n- v06 --supports--> v03\n- v08 --supports--> v01\n- v0e --refutes--> v07\n- v03 --supports--> v08\n- v05 --supports--> v06\n- v04 --refutes--> v0e\n- v07 --supports--> v02\n- v02 --supports--> v09\n- v13 --supports--> v0b\n- v01 --supports--> v10\n- v0f --refutes--> v13\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"607bbc9cfabf7aadd91a0b92dc93517acf1b8bc8bad218d49e44d2bd1409119d","schema":"livegraph.boolean-propagation.v1","spec_hash":"ebbc0ad89b7e40d50fdb7723ef2cd5dfdd5a9d7a54052cba348f3ac922a30b61","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-1d184d46a83577bd207d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v0f\n- v05 --refutes--> v12\n- v11 --supports--> v0c\n- v03 --supports--> v01\n- v12 --supports--> v0a\n- v10 --supports--> v05\n- v0e --refutes--> v0d\n- v0c --supports--> v04\n- v10 --supports--> v06\n- v07 --refutes--> v11\n- v08 --supports--> v0e\n- v05 --supports--> v13\n- v10 --refutes--> v09\n- v08 --supports--> v02\n- v0d --refutes--> v03\n- v0a --refutes--> v07\n- v00 --refutes--> v10\n- v00 --refutes--> v0b\n- v01 --refutes--> v00\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3ca4f3e5f987688623e42dd68486abd95ec95ef5e22092bee7a2320fc73e38ea","schema":"livegraph.boolean-propagation.v1","spec_hash":"1d184d46a83577bd207d0e88da46278c6e5f39b5bf4fca09e18e4664cb948d5e","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-7212b1975275232e480e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v0b\n- v05 --supports--> v00\n- v10 --supports--> v03\n- v05 --refutes--> v07\n- v01 --refutes--> v12\n- v0c --supports--> v10\n- v08 --refutes--> v0f\n- v12 --supports--> v0d\n- v10 --supports--> v11\n- v00 --supports--> v0a\n- v0d --supports--> v05\n- v06 --refutes--> v0c\n- v13 --refutes--> v08\n- v0e --supports--> v02\n- v08 --supports--> v0e\n- v0b --supports--> v04\n- v10 --supports--> v01\n- v0d --supports--> v09\n- v00 --refutes--> v13\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7785da299e958c67e9f55648eb31b6058a22fa901adf6258819414288eaf4251","schema":"livegraph.boolean-propagation.v1","spec_hash":"7212b1975275232e480e754a3a1f2d8c0ce455700aeac4047dda816f0f6bb431","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-24348d4fb5acdfe89da0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v07\n- v07 --refutes--> v05\n- v00 --refutes--> v0e\n- v12 --supports--> v13\n- v11 --refutes--> v03\n- v05 --refutes--> v04\n- v01 --refutes--> v12\n- v04 --refutes--> v0d\n- v13 --refutes--> v09\n- v05 --refutes--> v02\n- v05 --supports--> v0c\n- v00 --refutes--> v0b\n- v09 --supports--> v08\n- v00 --supports--> v01\n- v0d --supports--> v11\n- v0a --refutes--> v06\n- v07 --refutes--> v0f\n- v10 --supports--> v00\n- v0a --supports--> v10\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"62fae4b7ed9e5045a59dfb140b237fa06d7636a64287cd1661b33421e1ee1580","schema":"livegraph.boolean-propagation.v1","spec_hash":"24348d4fb5acdfe89da0dc544ec3e6df4352d4c24af3b1fd5f07ab6696ff303a","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-faec81af54a871bdc642","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v0f\n- v02 --refutes--> v11\n- v13 --refutes--> v03\n- v04 --refutes--> v07\n- v0d --supports--> v0a\n- v0d --supports--> v08\n- v0b --refutes--> v0c\n- v07 --refutes--> v09\n- v09 --supports--> v13\n- v02 --refutes--> v12\n- v0c --refutes--> v10\n- v01 --refutes--> v02\n- v0a --refutes--> v05\n- v13 --supports--> v0b\n- v04 --refutes--> v06\n- v0f --refutes--> v0e\n- v0e --refutes--> v04\n- v0a --refutes--> v00\n- v10 --supports--> v01\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e2c272afd989395bf62bf10994e13e55157afc9eeafbdf5a022a562df2ec4324","schema":"livegraph.boolean-propagation.v1","spec_hash":"faec81af54a871bdc64296bdcf500026b9fdb3425b9c324139df5f477cb93e6a","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-b3bd0a39a3ba4e95cfa0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v0b\n- v0e --refutes--> v0a\n- v05 --supports--> v13\n- v04 --supports--> v11\n- v13 --supports--> v12\n- v02 --supports--> v07\n- v13 --refutes--> v10\n- v09 --supports--> v03\n- v03 --refutes--> v02\n- v0d --refutes--> v05\n- v06 --supports--> v08\n- v01 --supports--> v00\n- v09 --refutes--> v0f\n- v04 --refutes--> v01\n- v0c --refutes--> v09\n- v06 --refutes--> v0d\n- v01 --supports--> v0e\n- v0a --refutes--> v0c\n- v07 --refutes--> v06\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bf998086f285dd0df36cc4d50a1427ef2e82e4dff0af3d6ace68bf4543a10a24","schema":"livegraph.boolean-propagation.v1","spec_hash":"b3bd0a39a3ba4e95cfa06486f6b83c20c2466f7847fcde285d6a9aba077f56b6","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-8d0b9d96d1966518b2c5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v01\n- v08 --supports--> v0f\n- v06 --supports--> v00\n- v08 --refutes--> v04\n- v01 --refutes--> v0c\n- v0a --refutes--> v03\n- v07 --supports--> v0d\n- v0b --refutes--> v11\n- v11 --refutes--> v0a\n- v0f --supports--> v09\n- v03 --supports--> v10\n- v00 --refutes--> v07\n- v0d --refutes--> v05\n- v03 --supports--> v12\n- v0f --supports--> v02\n- v0f --supports--> v06\n- v05 --refutes--> v0e\n- v01 --supports--> v0b\n- v00 --supports--> v13\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f53a06551524d015af781a66fc1983352985f31d3fa15a7cc0b380c081e6b944","schema":"livegraph.boolean-propagation.v1","spec_hash":"8d0b9d96d1966518b2c5eded34f4233dea70f52767f7c3a702ed439a18058c33","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":13,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-32f61536eabd4a80db8b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v04\n- v11 --refutes--> v0b\n- v0a --refutes--> v11\n- v0d --supports--> v0f\n- v0e --supports--> v07\n- v08 --refutes--> v05\n- v06 --supports--> v0a\n- v01 --refutes--> v03\n- v09 --refutes--> v01\n- v12 --refutes--> v0d\n- v0d --refutes--> v08\n- v06 --supports--> v02\n- v0c --refutes--> v06\n- v04 --refutes--> v0c\n- v01 --refutes--> v10\n- v0e --supports--> v09\n- v11 --supports--> v12\n- v00 --refutes--> v13\n- v05 --supports--> v0e\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5d6bbf2a72ff0bcf88f4f0276742a2cc738dbde44818a9460468768995c8571f","schema":"livegraph.boolean-propagation.v1","spec_hash":"32f61536eabd4a80db8b3cd64672267ac545fda915384caf58c0d91c4a6c7c86","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-73f0d4a02f6cb62ceec1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v0c\n- v05 --supports--> v10\n- v03 --refutes--> v0b\n- v04 --refutes--> v11\n- v10 --supports--> v03\n- v12 --refutes--> v04\n- v0d --supports--> v02\n- v10 --refutes--> v0a\n- v11 --supports--> v08\n- v00 --supports--> v09\n- v10 --refutes--> v07\n- v11 --refutes--> v0f\n- v13 --refutes--> v12\n- v0b --refutes--> v0e\n- v08 --refutes--> v05\n- v08 --refutes--> v01\n- v0e --supports--> v00\n- v13 --refutes--> v06\n- v02 --supports--> v13\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"67440fbc16ced104bb3c7546eb97612000c368bbb7db58fb57fedb60521eb35d","schema":"livegraph.boolean-propagation.v1","spec_hash":"73f0d4a02f6cb62ceec1aa1638dbccd44e7f34fa69cb4850ca15a4f9e4e853df","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-d4579a5abe4f6f4e227e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --supports--> v08\n- v07 --supports--> v0c\n- v10 --supports--> v11\n- v10 --refutes--> v12\n- v0f --refutes--> v00\n- v0f --supports--> v01\n- v0b --supports--> v0f\n- v13 --supports--> v09\n- v0f --supports--> v05\n- v02 --supports--> v10\n- v0d --supports--> v03\n- v03 --refutes--> v02\n- v04 --supports--> v0b\n- v07 --supports--> v0e\n- v06 --supports--> v0d\n- v09 --refutes--> v0a\n- v0a --supports--> v07\n- v0f --refutes--> v06\n- v07 --refutes--> v04\n\n[VERIFIED ROOT]\nv13 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a9817faec81ca1dcd65ff74f4555dc1203432174b816b6de17bf724d27579cd2","schema":"livegraph.boolean-propagation.v1","spec_hash":"d4579a5abe4f6f4e227e403be7aaadbefd7e8d237c586363b1123b7a6d990aa4","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v13\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-a4cb94bb4b10299bdd96","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v13\n- v12 --supports--> v09\n- v04 --refutes--> v07\n- v11 --supports--> v0c\n- v04 --refutes--> v10\n- v00 --refutes--> v0a\n- v0d --supports--> v12\n- v13 --supports--> v0e\n- v01 --supports--> v04\n- v10 --supports--> v05\n- v0c --refutes--> v0d\n- v12 --refutes--> v08\n- v09 --supports--> v06\n- v10 --refutes--> v00\n- v03 --supports--> v11\n- v0a --refutes--> v0b\n- v0e --supports--> v01\n- v04 --refutes--> v02\n- v0e --supports--> v0f\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e06809bc2cb50f3929cc13be43ef1bf05a70b4e1503fcb0d34ca84de18e6c305","schema":"livegraph.boolean-propagation.v1","spec_hash":"a4cb94bb4b10299bdd96ef960e5aaef61514c2a2d4a490c809f5e36de5a4d72c","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-f4300c95d960c837b04a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v03\n- v0f --refutes--> v11\n- v09 --refutes--> v10\n- v00 --supports--> v01\n- v07 --refutes--> v12\n- v0f --supports--> v0d\n- v06 --supports--> v09\n- v05 --refutes--> v0a\n- v0e --supports--> v08\n- v04 --refutes--> v0e\n- v00 --refutes--> v06\n- v0f --supports--> v00\n- v0b --supports--> v0c\n- v0b --supports--> v02\n- v0a --refutes--> v04\n- v12 --refutes--> v13\n- v10 --refutes--> v0b\n- v08 --refutes--> v07\n- v12 --supports--> v0f\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3d4f8e4c1de028c546b833d9c19a45d4bce8aa5e432c705b82dffec212fc790d","schema":"livegraph.boolean-propagation.v1","spec_hash":"f4300c95d960c837b04afefef73c23f43e5bd19b7c480749126e68d2ab4b9b4c","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-ccf0eb070d09e3ebc757","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v08\n- v12 --supports--> v0c\n- v07 --supports--> v01\n- v0f --refutes--> v0a\n- v12 --supports--> v0e\n- v0a --supports--> v12\n- v03 --refutes--> v09\n- v07 --refutes--> v0b\n- v05 --refutes--> v00\n- v13 --supports--> v04\n- v07 --supports--> v11\n- v0d --supports--> v13\n- v04 --supports--> v0f\n- v13 --refutes--> v06\n- v01 --refutes--> v0d\n- v10 --refutes--> v05\n- v00 --supports--> v07\n- v0c --refutes--> v03\n- v00 --supports--> v02\n\n[VERIFIED ROOT]\nv10 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9bce7c86bc1110c47706023e743e648423d011f6d900a4417cbe65eca0f2eed8","schema":"livegraph.boolean-propagation.v1","spec_hash":"ccf0eb070d09e3ebc7576bd44686a351cef6b25ad3d0989c2ea1e3960b9d73fc","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-e67acdc48881e050e3c4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v13\n- v0f --supports--> v10\n- v01 --refutes--> v0f\n- v0e --refutes--> v05\n- v09 --supports--> v07\n- v06 --refutes--> v04\n- v00 --refutes--> v11\n- v03 --refutes--> v09\n- v00 --supports--> v02\n- v04 --refutes--> v00\n- v11 --supports--> v0a\n- v0d --supports--> v01\n- v04 --supports--> v0c\n- v07 --supports--> v0b\n- v10 --supports--> v12\n- v10 --supports--> v06\n- v11 --supports--> v03\n- v05 --supports--> v08\n- v07 --refutes--> v0e\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3671a29a2e36ef4fc444f470ca730daa6b28f45fa3b2460bc0e3d3eaaa524c19","schema":"livegraph.boolean-propagation.v1","spec_hash":"e67acdc48881e050e3c47860843a8e047bd73ccee444ceff319675f74a784151","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":13,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-7faf362c0b8dc8f00406","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --supports--> v0b\n- v07 --supports--> v08\n- v0f --refutes--> v06\n- v0e --refutes--> v01\n- v01 --supports--> v04\n- v01 --supports--> v0c\n- v05 --refutes--> v0e\n- v11 --refutes--> v0f\n- v0b --supports--> v09\n- v0a --supports--> v05\n- v0e --supports--> v03\n- v09 --refutes--> v11\n- v0d --refutes--> v12\n- v08 --refutes--> v00\n- v05 --refutes--> v10\n- v06 --refutes--> v02\n- v01 --refutes--> v0d\n- v06 --refutes--> v07\n- v08 --refutes--> v13\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f6036e14035abe517aa13c7b9f2c1f3101c13369fa4ce35346c517b3d636ff38","schema":"livegraph.boolean-propagation.v1","spec_hash":"7faf362c0b8dc8f004065db5deb7485524feb281396ced74589f54ecb4dcffef","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-a24d477ea3dd1bf952d2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v0b\n- v0b --refutes--> v10\n- v12 --supports--> v07\n- v0e --supports--> v06\n- v02 --supports--> v0a\n- v0a --supports--> v03\n- v09 --supports--> v08\n- v0e --refutes--> v05\n- v0b --supports--> v01\n- v07 --supports--> v0e\n- v07 --supports--> v0f\n- v12 --supports--> v0d\n- v08 --supports--> v02\n- v0c --supports--> v13\n- v08 --supports--> v04\n- v05 --supports--> v00\n- v13 --supports--> v09\n- v05 --refutes--> v11\n- v10 --supports--> v12\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"714571899b167ec907beb0ea423878cebf025d818786545f90482658eb962a79","schema":"livegraph.boolean-propagation.v1","spec_hash":"a24d477ea3dd1bf952d23f2c2d9b6160acdd8df8e06dbace0d0b1f410df5f3b0","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-2906f2659fb523e9a573","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v06\n- v07 --refutes--> v12\n- v0a --refutes--> v07\n- v09 --refutes--> v11\n- v08 --refutes--> v0c\n- v11 --supports--> v0d\n- v06 --refutes--> v0f\n- v00 --supports--> v01\n- v10 --refutes--> v09\n- v13 --refutes--> v04\n- v03 --supports--> v05\n- v04 --refutes--> v02\n- v00 --refutes--> v0a\n- v09 --supports--> v0b\n- v0d --supports--> v0e\n- v03 --supports--> v00\n- v04 --supports--> v08\n- v07 --refutes--> v10\n- v08 --refutes--> v03\n\n[VERIFIED ROOT]\nv13 = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1164db213c24dacaad88b150bb2b8a380a86a4361297b7f1b335f482de7eecbe","schema":"livegraph.boolean-propagation.v1","spec_hash":"2906f2659fb523e9a57318094079ba1520c21f775a87f237ecb29999c1dc6f17","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-2d8ca19e101c4ad03b49","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v0b\n- v05 --refutes--> v0a\n- v06 --supports--> v04\n- v0e --supports--> v0d\n- v0c --supports--> v00\n- v05 --refutes--> v09\n- v06 --supports--> v07\n- v0f --supports--> v08\n- v05 --supports--> v10\n- v00 --refutes--> v11\n- v07 --supports--> v0f\n- v0d --refutes--> v0c\n- v08 --supports--> v02\n- v02 --supports--> v13\n- v0b --supports--> v03\n- v00 --refutes--> v12\n- v09 --refutes--> v0e\n- v13 --supports--> v05\n- v07 --refutes--> v01\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bb78e5c2a095e29111695e768a5693af4713f86f6802137b0cef5a0312566e17","schema":"livegraph.boolean-propagation.v1","spec_hash":"2d8ca19e101c4ad03b497b3a01a885c5c61de4b786826fe3fa263135c573862e","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-100a444e848821f9923f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --supports--> v00\n- v03 --supports--> v02\n- v00 --refutes--> v0f\n- v0f --refutes--> v0e\n- v00 --supports--> v13\n- v03 --supports--> v0d\n- v01 --supports--> v0b\n- v04 --supports--> v09\n- v05 --supports--> v0c\n- v02 --refutes--> v06\n- v0c --supports--> v04\n- v09 --supports--> v01\n- v00 --supports--> v12\n- v06 --refutes--> v08\n- v0b --refutes--> v11\n- v03 --refutes--> v10\n- v0e --refutes--> v03\n- v11 --refutes--> v07\n- v0b --refutes--> v0a\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a3707b56a6b4eb024d4e34b2ead1b663ca0899bb2c8df38e0cbd0455cfed6ae3","schema":"livegraph.boolean-propagation.v1","spec_hash":"100a444e848821f9923f0e67b5b411ef65787f36b02e1c201da38349a708ab08","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":13,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-ee40de69ca47d661c69b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v12\n- v11 --supports--> v0e\n- v07 --refutes--> v03\n- v04 --refutes--> v06\n- v0c --supports--> v0b\n- v08 --supports--> v0c\n- v00 --supports--> v04\n- v01 --refutes--> v07\n- v13 --refutes--> v0a\n- v09 --refutes--> v13\n- v06 --supports--> v05\n- v0c --supports--> v10\n- v11 --supports--> v00\n- v06 --refutes--> v02\n- v0a --supports--> v08\n- v09 --refutes--> v0f\n- v05 --supports--> v09\n- v01 --refutes--> v0d\n- v03 --refutes--> v11\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"09e141b824b9d0105258218ffb6d8a70740314c9cd343f7469eee50f68cf91b4","schema":"livegraph.boolean-propagation.v1","spec_hash":"ee40de69ca47d661c69b56b811a92151c65cf4a31ec81050087dfb20fc899a4d","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-e8992206fff2a9101092","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v09\n- v0b --refutes--> v01\n- v12 --refutes--> v00\n- v06 --supports--> v05\n- v02 --refutes--> v0d\n- v0c --supports--> v03\n- v0f --supports--> v12\n- v09 --refutes--> v0e\n- v04 --supports--> v13\n- v02 --refutes--> v0c\n- v0e --refutes--> v07\n- v0b --supports--> v0f\n- v02 --supports--> v0a\n- v07 --supports--> v02\n- v0c --supports--> v04\n- v11 --supports--> v06\n- v13 --refutes--> v0b\n- v13 --supports--> v08\n- v00 --refutes--> v10\n\n[VERIFIED ROOT]\nv11 = FALSE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0a73b5a0860b27b14352b77f1e77c504f9757e8b9bbcf04132d3985cff7a1d84","schema":"livegraph.boolean-propagation.v1","spec_hash":"e8992206fff2a9101092d513b2f4294368f71f28ae1f997eb8ab84741a1933ce","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-6dbb4e0cd60d86b4a79f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v06\n- v12 --supports--> v0b\n- v01 --refutes--> v00\n- v09 --supports--> v01\n- v0b --supports--> v03\n- v0c --supports--> v12\n- v08 --refutes--> v10\n- v10 --refutes--> v02\n- v07 --refutes--> v05\n- v11 --supports--> v0f\n- v11 --supports--> v0c\n- v05 --supports--> v11\n- v06 --supports--> v09\n- v10 --supports--> v0a\n- v10 --supports--> v04\n- v00 --supports--> v08\n- v0b --supports--> v13\n- v10 --refutes--> v07\n- v10 --supports--> v0e\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"10984b05aa03a0a8d78afe90715c9b9a9af91d7afe3eafdafe98e056d2a69dc9","schema":"livegraph.boolean-propagation.v1","spec_hash":"6dbb4e0cd60d86b4a79f8055d81f95da960ddc5aced3b034e3625833005d22da","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-847ead4296c1bcb0a646","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --supports--> v06\n- v04 --refutes--> v07\n- v0d --supports--> v03\n- v0f --supports--> v11\n- v0c --refutes--> v0e\n- v0a --refutes--> v00\n- v09 --supports--> v01\n- v0e --supports--> v10\n- v11 --supports--> v02\n- v0d --refutes--> v0b\n- v04 --refutes--> v05\n- v0b --refutes--> v09\n- v11 --supports--> v08\n- v05 --refutes--> v0f\n- v02 --supports--> v12\n- v0e --supports--> v0a\n- v12 --supports--> v13\n- v01 --refutes--> v0c\n- v0a --refutes--> v04\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"33ead6d743f3fd7882f8d2212979de7511e5b7708171b3d9b02ce811f0f26726","schema":"livegraph.boolean-propagation.v1","spec_hash":"847ead4296c1bcb0a6463fd0d39dfe095db48f9495ca2be7e085452a14a9deca","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-31a7fcc726a8505f144e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v01\n- v03 --refutes--> v06\n- v08 --supports--> v0d\n- v12 --refutes--> v11\n- v01 --supports--> v13\n- v00 --refutes--> v0c\n- v04 --refutes--> v0f\n- v02 --supports--> v0a\n- v0b --refutes--> v02\n- v00 --supports--> v0e\n- v11 --refutes--> v07\n- v04 --refutes--> v09\n- v0d --supports--> v12\n- v11 --supports--> v00\n- v06 --refutes--> v10\n- v06 --supports--> v08\n- v02 --supports--> v04\n- v13 --refutes--> v03\n- v03 --supports--> v05\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"51c24f5f7aab4459844a82860271515db4ca07d63952918d9669ef985d38979d","schema":"livegraph.boolean-propagation.v1","spec_hash":"31a7fcc726a8505f144e83f21f5733aef29b8ccc905df8be502586cf096efd7b","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-e798be9f6605f0597df2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v0b\n- v05 --supports--> v08\n- v03 --refutes--> v06\n- v0d --supports--> v12\n- v11 --supports--> v05\n- v06 --supports--> v0d\n- v04 --supports--> v00\n- v05 --supports--> v0e\n- v07 --refutes--> v02\n- v05 --refutes--> v13\n- v09 --supports--> v01\n- v08 --supports--> v03\n- v0f --refutes--> v04\n- v07 --refutes--> v0c\n- v0d --refutes--> v10\n- v12 --supports--> v07\n- v08 --refutes--> v0a\n- v00 --supports--> v11\n- v01 --supports--> v0f\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2a455271535727a20e2e848c3d6930b84f9f150ec54b65caf427e240adf8073a","schema":"livegraph.boolean-propagation.v1","spec_hash":"e798be9f6605f0597df2bb64dc54aaa690f71a2a2a55469bec78f858785c6795","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-873c2f6290dbdf7baa88","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v01\n- v0a --supports--> v09\n- v05 --supports--> v11\n- v0e --supports--> v06\n- v04 --refutes--> v03\n- v02 --refutes--> v10\n- v11 --refutes--> v0e\n- v10 --supports--> v0b\n- v05 --refutes--> v0c\n- v0b --supports--> v08\n- v01 --refutes--> v05\n- v03 --supports--> v12\n- v03 --refutes--> v00\n- v08 --supports--> v0f\n- v03 --refutes--> v13\n- v08 --supports--> v0d\n- v12 --supports--> v0a\n- v06 --supports--> v02\n- v0b --supports--> v07\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ee04469f8f125c95206b6ab147b2587f25b8d7d97dcb7ff9124bc1903b9a0cb2","schema":"livegraph.boolean-propagation.v1","spec_hash":"873c2f6290dbdf7baa887ef88e270ec2535ad54720ee30c69f14446a9b748b6f","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-b04e945cc6bae0fa7d62","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v0a\n- v0a --supports--> v09\n- v13 --supports--> v12\n- v12 --refutes--> v08\n- v07 --supports--> v11\n- v11 --refutes--> v0e\n- v13 --refutes--> v0b\n- v0d --refutes--> v06\n- v01 --supports--> v00\n- v12 --refutes--> v10\n- v07 --refutes--> v02\n- v00 --supports--> v13\n- v08 --supports--> v04\n- v08 --refutes--> v05\n- v05 --supports--> v0c\n- v0e --refutes--> v01\n- v09 --refutes--> v07\n- v00 --refutes--> v0f\n- v06 --supports--> v03\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5eefa0a231bc09609958da6f14d6de47a9e6a013cab59e0d422d4ab8e602834e","schema":"livegraph.boolean-propagation.v1","spec_hash":"b04e945cc6bae0fa7d62206cbe42d6386249a67ca5963a8bbe830bbbaa03d465","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-ae9104f943d6a14b3854","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v0a\n- v07 --refutes--> v04\n- v01 --refutes--> v0f\n- v03 --supports--> v0b\n- v05 --refutes--> v10\n- v0d --refutes--> v00\n- v0c --refutes--> v0d\n- v03 --refutes--> v09\n- v0b --supports--> v12\n- v0a --supports--> v13\n- v07 --supports--> v0c\n- v08 --supports--> v02\n- v13 --supports--> v07\n- v07 --refutes--> v0e\n- v02 --refutes--> v06\n- v02 --refutes--> v11\n- v12 --supports--> v08\n- v0d --supports--> v01\n- v0f --refutes--> v03\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e18e8c0076b14d0bd747f290ba54652c65d718d233431a11b8078a976f250a8d","schema":"livegraph.boolean-propagation.v1","spec_hash":"ae9104f943d6a14b3854cb36dbfdcebec3afedf82233e0de3ff280cb988eb2b2","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-e31840627f8c40ac7c00","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v0e\n- v08 --refutes--> v11\n- v10 --supports--> v0c\n- v11 --supports--> v13\n- v12 --refutes--> v10\n- v0d --supports--> v12\n- v0b --refutes--> v00\n- v04 --refutes--> v02\n- v05 --refutes--> v06\n- v13 --refutes--> v01\n- v03 --supports--> v05\n- v00 --supports--> v03\n- v09 --supports--> v0f\n- v08 --refutes--> v07\n- v04 --refutes--> v0a\n- v05 --refutes--> v0d\n- v13 --supports--> v04\n- v02 --supports--> v0b\n- v09 --refutes--> v08\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7c627cf3a09e4ec77326f1a5115050f2283bfbe81662c9b845490bea0ed7c0d7","schema":"livegraph.boolean-propagation.v1","spec_hash":"e31840627f8c40ac7c000a50faf93864ac155d84602e24f8d366dbf89c3fc475","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-bdfe9f0b1450a6517d7b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v00\n- v05 --supports--> v07\n- v13 --supports--> v0a\n- v0f --supports--> v13\n- v13 --supports--> v08\n- v07 --supports--> v02\n- v0f --refutes--> v0d\n- v0b --supports--> v04\n- v12 --refutes--> v10\n- v03 --supports--> v0e\n- v08 --supports--> v0c\n- v01 --supports--> v06\n- v10 --refutes--> v0f\n- v03 --supports--> v12\n- v06 --supports--> v03\n- v02 --supports--> v0b\n- v0b --supports--> v01\n- v09 --supports--> v05\n- v03 --supports--> v11\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e445ea20acea1dfd6a297dab0e3c0862fd352e0ca5570abe033be1b81607d368","schema":"livegraph.boolean-propagation.v1","spec_hash":"bdfe9f0b1450a6517d7bab7e467446bb6a7e67bab171f5a2d28dd32b276dbb33","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-3cbbba3627f2f4ad0172","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v01\n- v0d --supports--> v0b\n- v02 --refutes--> v00\n- v07 --supports--> v0c\n- v0a --supports--> v11\n- v0b --refutes--> v0e\n- v0b --supports--> v06\n- v06 --supports--> v03\n- v00 --refutes--> v0f\n- v06 --refutes--> v0a\n- v0a --refutes--> v08\n- v04 --refutes--> v05\n- v10 --supports--> v04\n- v08 --refutes--> v09\n- v00 --supports--> v0d\n- v0a --refutes--> v12\n- v13 --supports--> v07\n- v09 --supports--> v13\n- v05 --refutes--> v02\n\n[VERIFIED ROOT]\nv10 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3e5dcdbd804af8ddad063bd8729342928ddf4a2faf829e188ab0523a132b66e9","schema":"livegraph.boolean-propagation.v1","spec_hash":"3cbbba3627f2f4ad0172acc87c27ce3d70d822ba8cae40f154b2b744c73b2942","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-b2857c0058ad61e3ab3c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v0b\n- v0d --supports--> v08\n- v03 --supports--> v01\n- v06 --refutes--> v09\n- v08 --supports--> v11\n- v07 --supports--> v00\n- v07 --refutes--> v13\n- v0f --supports--> v05\n- v13 --supports--> v10\n- v08 --supports--> v06\n- v0c --refutes--> v07\n- v12 --refutes--> v04\n- v04 --supports--> v0a\n- v0e --supports--> v12\n- v05 --supports--> v0c\n- v03 --supports--> v02\n- v04 --refutes--> v0d\n- v10 --supports--> v03\n- v06 --refutes--> v0f\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"db33ef4e025b5cd8b0251710af2344b28ba99ff838edcbc897135cf8fd3bb650","schema":"livegraph.boolean-propagation.v1","spec_hash":"b2857c0058ad61e3ab3cebcfe95c0c6aa973a7625c17af8ef65aaaa64657f24f","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-364f3f5a9eb54f4f6b5b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v04\n- v02 --refutes--> v0d\n- v04 --supports--> v02\n- v03 --refutes--> v0b\n- v06 --refutes--> v0a\n- v13 --refutes--> v07\n- v0c --refutes--> v0e\n- v0f --supports--> v10\n- v0a --supports--> v03\n- v12 --supports--> v0c\n- v02 --refutes--> v06\n- v12 --supports--> v08\n- v05 --supports--> v0f\n- v0e --refutes--> v00\n- v03 --refutes--> v11\n- v13 --refutes--> v12\n- v00 --refutes--> v09\n- v09 --supports--> v05\n- v0c --refutes--> v01\n\n[VERIFIED ROOT]\nv13 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f0cf40275f7b67af7ddfb20349b91e7aec7e2e921726060f4d921642fcff588a","schema":"livegraph.boolean-propagation.v1","spec_hash":"364f3f5a9eb54f4f6b5b2cfea87f90128e5e1c91c8b0da361b5cc6aa353364c9","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v13\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-786023ae5e1d8f19b05b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v0a\n- v00 --refutes--> v08\n- v08 --supports--> v0d\n- v12 --supports--> v06\n- v08 --supports--> v0f\n- v07 --supports--> v04\n- v0d --supports--> v03\n- v02 --refutes--> v07\n- v0c --supports--> v0e\n- v13 --refutes--> v00\n- v09 --supports--> v10\n- v02 --supports--> v0b\n- v04 --supports--> v12\n- v05 --refutes--> v02\n- v03 --refutes--> v11\n- v06 --supports--> v13\n- v06 --supports--> v01\n- v09 --refutes--> v05\n- v0c --supports--> v09\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"065b804f60878a7396e78e23df031aa1b99ea1f022e62eca31a06a4c4e36d92e","schema":"livegraph.boolean-propagation.v1","spec_hash":"786023ae5e1d8f19b05b3a35415f326e1080536b949bddc1a104e7bf037155bd","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-4ede5486a6258ed4953c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v05\n- v0d --supports--> v10\n- v09 --refutes--> v0a\n- v0d --refutes--> v13\n- v05 --supports--> v06\n- v00 --supports--> v08\n- v0a --supports--> v0c\n- v03 --supports--> v11\n- v11 --refutes--> v01\n- v06 --supports--> v07\n- v10 --supports--> v03\n- v10 --refutes--> v12\n- v0e --refutes--> v0b\n- v0a --supports--> v0e\n- v08 --supports--> v09\n- v06 --supports--> v0d\n- v0b --refutes--> v04\n- v05 --supports--> v0f\n- v05 --supports--> v02\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"519ec0fa7ac5fb9ac6370441ee288b47130d882cfe01a4b5b79901997bf93923","schema":"livegraph.boolean-propagation.v1","spec_hash":"4ede5486a6258ed4953c348e5bcfd8cb205efd751231deef22f1202cbf95be0e","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":13,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-c8d60a48beb245858ef1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v0c\n- v00 --supports--> v06\n- v0a --supports--> v0f\n- v08 --supports--> v11\n- v07 --refutes--> v10\n- v03 --supports--> v13\n- v04 --refutes--> v00\n- v06 --supports--> v08\n- v0d --supports--> v12\n- v03 --refutes--> v04\n- v10 --supports--> v09\n- v01 --refutes--> v03\n- v11 --refutes--> v0a\n- v08 --refutes--> v0b\n- v0e --supports--> v0d\n- v0d --supports--> v02\n- v10 --supports--> v0e\n- v12 --refutes--> v01\n- v06 --supports--> v05\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2202f09e7929e0ed4af961ccbf532817dc0ad79b1dc914a0a59fdc8935759afe","schema":"livegraph.boolean-propagation.v1","spec_hash":"c8d60a48beb245858ef122fdd94d4a882f30632e00825d1771d116602692bf2f","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-0d3a5947d005708a3c7d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v05\n- v11 --refutes--> v13\n- v0c --supports--> v06\n- v01 --refutes--> v0d\n- v08 --refutes--> v0c\n- v08 --supports--> v03\n- v02 --refutes--> v08\n- v0a --supports--> v01\n- v09 --refutes--> v11\n- v04 --refutes--> v09\n- v12 --supports--> v00\n- v0c --refutes--> v07\n- v0e --refutes--> v02\n- v01 --supports--> v04\n- v06 --supports--> v12\n- v08 --supports--> v0b\n- v10 --supports--> v0a\n- v00 --supports--> v10\n- v12 --supports--> v0f\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7e5297b92b69bf17cd318fb281f3752a1d1f268f6f7a5c496240fbe3a71662f0","schema":"livegraph.boolean-propagation.v1","spec_hash":"0d3a5947d005708a3c7dfd324138deb6d37ab154021e3d10400b1636e247a192","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-4634fd51f931d20779c7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v0b\n- v01 --refutes--> v0a\n- v13 --refutes--> v12\n- v0f --refutes--> v0c\n- v07 --supports--> v0f\n- v12 --supports--> v01\n- v11 --refutes--> v08\n- v01 --refutes--> v06\n- v09 --supports--> v07\n- v03 --refutes--> v0e\n- v02 --refutes--> v03\n- v0f --refutes--> v13\n- v06 --supports--> v0d\n- v05 --refutes--> v02\n- v0d --refutes--> v04\n- v0d --supports--> v00\n- v02 --supports--> v10\n- v11 --refutes--> v05\n- v00 --refutes--> v11\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5b3b86161510c5260f61012dc2910d5bd6be888ebd453667651afd71db943819","schema":"livegraph.boolean-propagation.v1","spec_hash":"4634fd51f931d20779c7a17387951d8a3473b119034297f8a2afc84f4e5e5bed","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-12699b20400e732b67bf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v01\n- v09 --supports--> v0f\n- v0e --refutes--> v09\n- v09 --supports--> v03\n- v06 --refutes--> v0d\n- v0b --supports--> v0c\n- v00 --supports--> v0b\n- v11 --refutes--> v04\n- v0f --supports--> v06\n- v08 --refutes--> v07\n- v04 --refutes--> v08\n- v12 --refutes--> v0a\n- v00 --supports--> v10\n- v12 --supports--> v05\n- v0c --supports--> v0e\n- v13 --refutes--> v11\n- v06 --refutes--> v13\n- v05 --refutes--> v00\n- v0e --supports--> v02\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a4fa45c306b8d46500af92040dcc8f335f6c08eb5995f94ee0e590855b5632c3","schema":"livegraph.boolean-propagation.v1","spec_hash":"12699b20400e732b67bf63694caa2cbbb195ccd6a36c74d825dff3aee549b6eb","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-da07079313548e004b05","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --supports--> v0f\n- v0f --refutes--> v08\n- v08 --supports--> v0c\n- v12 --supports--> v07\n- v04 --supports--> v10\n- v00 --supports--> v04\n- v06 --refutes--> v0b\n- v06 --supports--> v09\n- v07 --supports--> v0d\n- v0e --refutes--> v12\n- v0d --supports--> v06\n- v10 --refutes--> v13\n- v0b --supports--> v01\n- v08 --supports--> v02\n- v12 --supports--> v0a\n- v13 --refutes--> v03\n- v0b --supports--> v11\n- v0c --supports--> v0e\n- v0f --refutes--> v05\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f6b8f177344a03c5a9b1401a8666ad8a4dd438d6dfefe482c01efd7293508bad","schema":"livegraph.boolean-propagation.v1","spec_hash":"da07079313548e004b055493dcd5a5d837384a2f3551d979f4fd3a4874454909","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-af0c4353a6397736785d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v11\n- v04 --supports--> v0c\n- v05 --supports--> v01\n- v0b --supports--> v0f\n- v0b --supports--> v03\n- v12 --refutes--> v0a\n- v0d --supports--> v07\n- v0d --supports--> v09\n- v01 --supports--> v10\n- v02 --supports--> v08\n- v0c --supports--> v02\n- v11 --supports--> v04\n- v10 --refutes--> v06\n- v02 --refutes--> v0b\n- v05 --refutes--> v00\n- v0a --supports--> v0e\n- v0e --supports--> v05\n- v07 --refutes--> v13\n- v07 --refutes--> v12\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f3762da5d71b73fc3ba1f6438b6067cf93299d115202b428acd90bd3df09bb37","schema":"livegraph.boolean-propagation.v1","spec_hash":"af0c4353a6397736785d77423d212f2fd1a6bf9165539896534715e3a4fdf18c","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-ee68a2f0fa675e2fc725","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v05\n- v08 --supports--> v13\n- v0a --refutes--> v04\n- v01 --refutes--> v09\n- v02 --supports--> v08\n- v08 --supports--> v0c\n- v01 --refutes--> v12\n- v07 --supports--> v06\n- v00 --refutes--> v0f\n- v0c --supports--> v0e\n- v0b --supports--> v0d\n- v10 --supports--> v0b\n- v01 --supports--> v10\n- v0f --supports--> v01\n- v06 --refutes--> v00\n- v05 --supports--> v02\n- v0e --supports--> v07\n- v0e --refutes--> v03\n- v0a --refutes--> v11\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6e094d15fdf9026b4a1cd0c90d93f8a8d687d930863842a09c5206f6a0831197","schema":"livegraph.boolean-propagation.v1","spec_hash":"ee68a2f0fa675e2fc725556d2a37a66d6cba039f745db78affea7ec8d9b01b12","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":13,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-f1ce968e17225411cf18","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v03\n- v10 --supports--> v05\n- v03 --supports--> v06\n- v11 --supports--> v0c\n- v08 --refutes--> v0b\n- v0f --supports--> v10\n- v07 --supports--> v01\n- v0d --supports--> v09\n- v09 --refutes--> v02\n- v0a --supports--> v07\n- v0d --supports--> v12\n- v0c --refutes--> v0f\n- v07 --refutes--> v13\n- v03 --refutes--> v11\n- v02 --refutes--> v0e\n- v08 --refutes--> v0d\n- v05 --refutes--> v00\n- v0d --refutes--> v04\n- v00 --refutes--> v0a\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"46156f809e804b01a4c2b7ad6990bd3ec67abad22a48f47da25025226bc7104f","schema":"livegraph.boolean-propagation.v1","spec_hash":"f1ce968e17225411cf187e3e3db62b38f969f3e22df50247c085cdcb266a02d9","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":13,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-a9666aaac53de84b79e6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v04\n- v0e --refutes--> v06\n- v0c --supports--> v02\n- v0e --refutes--> v11\n- v05 --supports--> v0a\n- v02 --refutes--> v08\n- v0e --supports--> v10\n- v0c --refutes--> v07\n- v08 --refutes--> v09\n- v13 --refutes--> v0d\n- v04 --refutes--> v0c\n- v09 --refutes--> v00\n- v09 --supports--> v0e\n- v03 --supports--> v01\n- v0f --supports--> v03\n- v05 --refutes--> v12\n- v11 --supports--> v05\n- v01 --supports--> v13\n- v05 --supports--> v0b\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5981b2ee925c54918baaaae597954cfd4a0e9c6332e9b8c71938332b7326f4a7","schema":"livegraph.boolean-propagation.v1","spec_hash":"a9666aaac53de84b79e6f8f9005863e8e83c7582e66b1e13efc51fb61bf4004f","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-a630222be593ec7968e9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --supports--> v06\n- v0b --refutes--> v02\n- v07 --refutes--> v0c\n- v05 --refutes--> v10\n- v08 --supports--> v07\n- v03 --refutes--> v0e\n- v0e --supports--> v09\n- v0f --supports--> v04\n- v0f --supports--> v08\n- v05 --supports--> v03\n- v0b --supports--> v0d\n- v06 --refutes--> v01\n- v06 --supports--> v0f\n- v11 --supports--> v0b\n- v00 --supports--> v11\n- v0c --refutes--> v0a\n- v00 --supports--> v13\n- v0c --supports--> v05\n- v0d --refutes--> v12\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9fc409f7107b1b202767557367a8194d4321e96ffac71ebddd15f6d466ee0e53","schema":"livegraph.boolean-propagation.v1","spec_hash":"a630222be593ec7968e9ccea385e5667861c32bb43820c8e2f6e28dd715f6f71","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-6c4f341e545a783a60c8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --supports--> v0a\n- v00 --supports--> v0b\n- v11 --refutes--> v03\n- v13 --supports--> v02\n- v06 --refutes--> v0c\n- v06 --supports--> v0d\n- v12 --refutes--> v0e\n- v05 --supports--> v0f\n- v13 --supports--> v01\n- v03 --supports--> v06\n- v06 --refutes--> v08\n- v01 --supports--> v04\n- v0d --supports--> v09\n- v0b --supports--> v07\n- v0a --refutes--> v11\n- v0d --refutes--> v13\n- v0b --supports--> v05\n- v10 --refutes--> v12\n- v0f --supports--> v10\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b553ce676e98d64f6f72855f5ebc575656545cac8127f14e65626f1acba030af","schema":"livegraph.boolean-propagation.v1","spec_hash":"6c4f341e545a783a60c8c731f17cb7d9051be88996f81daaa8e7780396e72792","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-614fcb3b7b9e2aa04b83","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --supports--> v03\n- v07 --refutes--> v0d\n- v0d --refutes--> v0e\n- v0b --supports--> v06\n- v13 --refutes--> v10\n- v09 --refutes--> v01\n- v08 --supports--> v0f\n- v0f --refutes--> v12\n- v04 --refutes--> v05\n- v0d --refutes--> v04\n- v11 --refutes--> v0b\n- v06 --refutes--> v07\n- v0d --refutes--> v02\n- v04 --refutes--> v0c\n- v09 --refutes--> v0a\n- v04 --refutes--> v13\n- v01 --supports--> v00\n- v12 --refutes--> v09\n- v10 --supports--> v08\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e8d403f92da73407a529641f1f70b033b012c0a9c1922eb5ad543e8ede1b374f","schema":"livegraph.boolean-propagation.v1","spec_hash":"614fcb3b7b9e2aa04b83d85e6b090eae0289e5d5a85a2306a5dc34b721fdfc25","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-769c51700ca723b60108","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v13\n- v02 --refutes--> v01\n- v11 --refutes--> v0f\n- v12 --supports--> v07\n- v10 --supports--> v04\n- v04 --refutes--> v0b\n- v0c --refutes--> v0d\n- v0a --refutes--> v03\n- v02 --supports--> v0a\n- v11 --refutes--> v0e\n- v0b --supports--> v06\n- v0d --supports--> v11\n- v0e --supports--> v02\n- v09 --supports--> v12\n- v07 --supports--> v10\n- v0a --supports--> v09\n- v06 --supports--> v08\n- v06 --supports--> v00\n- v0d --supports--> v05\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0177026aebed98041f38b3788d1308e5ae7cb612397d0fb573c1028dd83ee4ae","schema":"livegraph.boolean-propagation.v1","spec_hash":"769c51700ca723b6010830bed634cc4fdb0383d3a6d3d32db95c800a2afeddb1","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-41f013a9ab1c68cb095a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --supports--> v06\n- v13 --supports--> v02\n- v01 --refutes--> v05\n- v0c --refutes--> v01\n- v09 --refutes--> v0d\n- v0b --refutes--> v08\n- v12 --refutes--> v13\n- v0b --refutes--> v04\n- v13 --refutes--> v03\n- v03 --supports--> v0c\n- v08 --refutes--> v07\n- v0a --refutes--> v0e\n- v0f --supports--> v09\n- v0f --refutes--> v00\n- v11 --refutes--> v10\n- v09 --refutes--> v0a\n- v07 --refutes--> v11\n- v0a --refutes--> v12\n- v10 --supports--> v0f\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cb981dfc7740cbaef90a31d94ecdb6cf39cc65b36f49ba1a6ca08f7d14f78fa1","schema":"livegraph.boolean-propagation.v1","spec_hash":"41f013a9ab1c68cb095a199ef074cfacd31325657b8800c0f392dc744392c91c","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-5e54db66dccce627e33f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v12\n- v13 --refutes--> v01\n- v03 --refutes--> v08\n- v13 --refutes--> v09\n- v08 --refutes--> v13\n- v01 --refutes--> v10\n- v0b --supports--> v02\n- v10 --refutes--> v00\n- v11 --supports--> v04\n- v06 --supports--> v0b\n- v0d --refutes--> v0a\n- v0a --refutes--> v0e\n- v12 --refutes--> v0c\n- v0b --supports--> v0d\n- v10 --supports--> v05\n- v12 --refutes--> v06\n- v0e --supports--> v03\n- v12 --supports--> v07\n- v0b --refutes--> v0f\n\n[VERIFIED ROOT]\nv11 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"461949810e3956fe2679b991e2cfb44cc788d714277b6b87a033dd06a4e8c65a","schema":"livegraph.boolean-propagation.v1","spec_hash":"5e54db66dccce627e33f36471d851c889e48008d07cb3bda25dc1317475109b4","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-46ea58a1769c97a4a027","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v0b\n- v01 --refutes--> v04\n- v06 --supports--> v01\n- v0f --refutes--> v05\n- v13 --supports--> v09\n- v0a --supports--> v0d\n- v08 --refutes--> v0e\n- v02 --refutes--> v03\n- v00 --supports--> v08\n- v08 --supports--> v13\n- v04 --refutes--> v0a\n- v13 --supports--> v06\n- v05 --refutes--> v0c\n- v0c --refutes--> v07\n- v0d --refutes--> v12\n- v0d --refutes--> v02\n- v0c --supports--> v10\n- v06 --supports--> v11\n- v03 --refutes--> v0f\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d3c3b94ec6f5cf0fea0026d1407fb19fb3ffe90d9441a8487de6b7a9418bbac9","schema":"livegraph.boolean-propagation.v1","spec_hash":"46ea58a1769c97a4a027a58741a2665b60fe1157171a9327812216c2af7f4694","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-a18389366a459ec45771","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v0b\n- v08 --refutes--> v0f\n- v0d --refutes--> v00\n- v0d --supports--> v12\n- v0f --refutes--> v10\n- v07 --refutes--> v0e\n- v0a --supports--> v01\n- v06 --supports--> v0c\n- v06 --refutes--> v04\n- v0d --supports--> v0a\n- v10 --refutes--> v09\n- v09 --supports--> v07\n- v09 --refutes--> v03\n- v05 --refutes--> v11\n- v0a --supports--> v05\n- v04 --supports--> v08\n- v0b --refutes--> v0d\n- v02 --refutes--> v06\n- v0b --supports--> v13\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5d3d39e84cf74cb4d6a59437129f1cb9a9527d7a957d3b4db05cda7b3f049648","schema":"livegraph.boolean-propagation.v1","spec_hash":"a18389366a459ec45771fc28ab87853d69446b8571b6af25fe29e6b302572931","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":13,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-c8cbc053dce726fc6d73","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v00\n- v05 --refutes--> v02\n- v0f --supports--> v0e\n- v13 --refutes--> v06\n- v08 --refutes--> v05\n- v06 --refutes--> v07\n- v13 --supports--> v10\n- v0f --supports--> v0a\n- v0e --supports--> v0d\n- v03 --supports--> v08\n- v0d --supports--> v11\n- v01 --supports--> v09\n- v00 --refutes--> v0b\n- v01 --supports--> v03\n- v04 --supports--> v01\n- v00 --supports--> v0f\n- v05 --supports--> v0c\n- v13 --refutes--> v12\n- v0c --supports--> v13\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5dfa625fe1710c3c99a47bd160b2f7aab1e1609c4082808b442e01a411e78024","schema":"livegraph.boolean-propagation.v1","spec_hash":"c8cbc053dce726fc6d733268fd2cceb5bbd0550cdd29f70616f4d1cbb8a42df7","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-d1f5d6f2054158f6f290","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v04\n- v07 --supports--> v0c\n- v0d --supports--> v11\n- v03 --supports--> v0a\n- v0e --refutes--> v09\n- v11 --supports--> v13\n- v07 --supports--> v0f\n- v09 --refutes--> v08\n- v02 --supports--> v05\n- v12 --refutes--> v02\n- v01 --supports--> v10\n- v10 --supports--> v03\n- v0b --supports--> v00\n- v0a --supports--> v0e\n- v0b --refutes--> v12\n- v11 --refutes--> v06\n- v11 --supports--> v01\n- v07 --refutes--> v0d\n- v05 --supports--> v07\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"65ff086d23456ae4ffa2b2f95470c83455e2f83b997272efedc2ef5d3bc229dd","schema":"livegraph.boolean-propagation.v1","spec_hash":"d1f5d6f2054158f6f2901247a2cbcdaf2e33ac8430b3355b84887f6e99d368c1","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-ca94b16e7db50cd24efd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v07\n- v09 --supports--> v08\n- v13 --supports--> v04\n- v04 --supports--> v12\n- v13 --supports--> v00\n- v0b --supports--> v11\n- v02 --refutes--> v13\n- v05 --refutes--> v0d\n- v06 --refutes--> v01\n- v12 --supports--> v10\n- v06 --refutes--> v0c\n- v07 --supports--> v0b\n- v0d --refutes--> v0f\n- v0d --supports--> v0a\n- v0f --supports--> v09\n- v12 --supports--> v05\n- v0b --refutes--> v03\n- v0c --supports--> v02\n- v09 --supports--> v0e\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2bee4debf5d2f9bd19be4f20262e0169a3ed048cc556d95a7c39524df0b99a6f","schema":"livegraph.boolean-propagation.v1","spec_hash":"ca94b16e7db50cd24efdb51e35d7b55c42f8398f28ff04d03b04fe1f22e6eca0","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-52970d19f82434ecb013","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v08\n- v10 --refutes--> v0a\n- v03 --refutes--> v13\n- v0a --refutes--> v09\n- v00 --supports--> v02\n- v01 --refutes--> v07\n- v0c --supports--> v06\n- v07 --supports--> v0e\n- v05 --supports--> v01\n- v00 --supports--> v0b\n- v0c --supports--> v05\n- v08 --supports--> v04\n- v08 --supports--> v0c\n- v10 --refutes--> v11\n- v08 --refutes--> v0f\n- v12 --refutes--> v0d\n- v02 --supports--> v10\n- v0e --refutes--> v00\n- v09 --refutes--> v12\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4914d04c2b79083f35823cf1ca1ce74d82440653d33f1ac3075ef878c8027373","schema":"livegraph.boolean-propagation.v1","spec_hash":"52970d19f82434ecb01343f6f8043d48b8b28cc70ebac8bda30f928b8e74a106","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-ee92a977992574cfff3d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --refutes--> v07\n- v0a --supports--> v06\n- v01 --supports--> v03\n- v0c --supports--> v0e\n- v05 --refutes--> v04\n- v05 --supports--> v08\n- v01 --supports--> v12\n- v0f --refutes--> v02\n- v07 --supports--> v0d\n- v0e --supports--> v10\n- v09 --refutes--> v0c\n- v08 --supports--> v0f\n- v07 --refutes--> v09\n- v0a --supports--> v05\n- v02 --refutes--> v00\n- v00 --supports--> v01\n- v10 --refutes--> v0a\n- v01 --refutes--> v11\n- v07 --supports--> v0b\n\n[VERIFIED ROOT]\nv13 = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0291cd5f4d40e0f9eb7443a6be3bc522f950ecd6bcbb9657644b35cf8fd18a6f","schema":"livegraph.boolean-propagation.v1","spec_hash":"ee92a977992574cfff3de5dd0bf07b789f0229fd439f354f8aa96361c10b85ea","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-cb1f97fb1844a3c07488","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v10\n- v09 --supports--> v11\n- v10 --refutes--> v00\n- v0c --supports--> v12\n- v10 --supports--> v06\n- v0a --refutes--> v03\n- v08 --supports--> v04\n- v01 --supports--> v0b\n- v11 --supports--> v0e\n- v06 --supports--> v02\n- v0c --refutes--> v08\n- v11 --supports--> v01\n- v07 --supports--> v0f\n- v04 --supports--> v07\n- v06 --supports--> v13\n- v05 --supports--> v0a\n- v0d --refutes--> v0c\n- v0f --refutes--> v05\n- v01 --supports--> v0d\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5f125b02be4234cac3d7e4c070c22d3c14d33ace9e0e4d4edb0dba4370e3c9aa","schema":"livegraph.boolean-propagation.v1","spec_hash":"cb1f97fb1844a3c07488b0d6f822f5acbab8256208099eb5bfb422239d89e41c","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":13,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-37570ab08c61fb422eca","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v02\n- v06 --refutes--> v11\n- v0f --refutes--> v03\n- v00 --refutes--> v07\n- v05 --supports--> v12\n- v09 --refutes--> v08\n- v06 --refutes--> v0d\n- v04 --refutes--> v01\n- v01 --supports--> v00\n- v0a --supports--> v04\n- v07 --refutes--> v06\n- v09 --supports--> v05\n- v0c --refutes--> v0f\n- v11 --supports--> v09\n- v00 --refutes--> v13\n- v10 --supports--> v0a\n- v05 --supports--> v0b\n- v0a --supports--> v0e\n- v0b --refutes--> v0c\n\n[VERIFIED ROOT]\nv10 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3b4bdf62e1166fde4b37909b69cb5500f6d78f052a285d179501ba2ff1f40335","schema":"livegraph.boolean-propagation.v1","spec_hash":"37570ab08c61fb422eca06de9cc743f60ee3a36142ca09baa177fabbd52562a1","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":13,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-6c979dd681815137155e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v07\n- v05 --refutes--> v13\n- v13 --refutes--> v00\n- v0f --supports--> v05\n- v08 --supports--> v06\n- v08 --supports--> v0a\n- v0c --supports--> v09\n- v04 --refutes--> v0e\n- v00 --supports--> v08\n- v12 --supports--> v10\n- v02 --refutes--> v12\n- v0e --refutes--> v0f\n- v12 --supports--> v0c\n- v0c --supports--> v03\n- v12 --supports--> v0d\n- v0a --refutes--> v02\n- v0b --supports--> v01\n- v01 --refutes--> v04\n- v0a --refutes--> v11\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"037827046bacaf997e7b071036869cc757f6b5c544cbd869cbf96c76659ae81a","schema":"livegraph.boolean-propagation.v1","spec_hash":"6c979dd681815137155e0384e05ad591739f028f53d011a016f062f9250db11f","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-882c063ee0772d70a316","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v06\n- v02 --refutes--> v09\n- v01 --supports--> v0c\n- v06 --supports--> v0f\n- v0e --supports--> v01\n- v11 --supports--> v0d\n- v04 --supports--> v11\n- v02 --refutes--> v04\n- v03 --refutes--> v02\n- v01 --refutes--> v13\n- v12 --refutes--> v00\n- v05 --supports--> v12\n- v12 --refutes--> v0e\n- v07 --refutes--> v10\n- v07 --supports--> v03\n- v13 --refutes--> v0b\n- v11 --supports--> v0a\n- v05 --supports--> v08\n- v0a --refutes--> v05\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"543154ebece188534d2c433e5ce7406eda159fe72213869b423fdb39a0e03e05","schema":"livegraph.boolean-propagation.v1","spec_hash":"882c063ee0772d70a316da03255a89cb4e364a473d2cd4697f9f152da5fa335f","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-3f669b214f80c7c4a405","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v0a\n- v00 --supports--> v0f\n- v0b --supports--> v00\n- v08 --supports--> v05\n- v03 --refutes--> v12\n- v0a --refutes--> v01\n- v01 --supports--> v10\n- v06 --supports--> v04\n- v00 --refutes--> v0e\n- v11 --refutes--> v08\n- v07 --refutes--> v02\n- v08 --refutes--> v0d\n- v01 --supports--> v03\n- v03 --refutes--> v0b\n- v04 --supports--> v07\n- v00 --refutes--> v09\n- v0f --refutes--> v13\n- v13 --refutes--> v11\n- v08 --supports--> v0c\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b42515dd564a7702e2e3991141cdc29b000a9ea5143b604a00b75f2d05b817ce","schema":"livegraph.boolean-propagation.v1","spec_hash":"3f669b214f80c7c4a405f1e6cb05b67dfc88a14856e184a70aac98b70a1148eb","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-b3191f9412d6c1555c1f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v08\n- v0f --refutes--> v12\n- v13 --refutes--> v0e\n- v0e --refutes--> v03\n- v04 --refutes--> v06\n- v00 --supports--> v09\n- v0f --refutes--> v10\n- v04 --refutes--> v0c\n- v11 --refutes--> v05\n- v0d --refutes--> v0a\n- v12 --supports--> v13\n- v0a --supports--> v0f\n- v0c --refutes--> v00\n- v00 --refutes--> v0b\n- v05 --refutes--> v04\n- v0b --refutes--> v07\n- v07 --supports--> v0d\n- v0b --refutes--> v01\n- v0c --supports--> v02\n\n[VERIFIED ROOT]\nv11 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cb86f6af9099f70835c6f9d9ae2296460722b8121d9f0ba4ad0e2cc2d70cf155","schema":"livegraph.boolean-propagation.v1","spec_hash":"b3191f9412d6c1555c1f11cfb0d49d5acc37b9fbf484299123cc612576f21f1c","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-5910897581c726b30844","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v0a\n- v05 --refutes--> v00\n- v12 --refutes--> v08\n- v02 --supports--> v03\n- v07 --supports--> v0d\n- v0f --supports--> v10\n- v0f --supports--> v06\n- v04 --refutes--> v05\n- v09 --supports--> v0f\n- v05 --refutes--> v0c\n- v07 --supports--> v0e\n- v00 --supports--> v0b\n- v0d --refutes--> v12\n- v0f --refutes--> v01\n- v12 --supports--> v11\n- v08 --supports--> v04\n- v06 --refutes--> v13\n- v13 --supports--> v02\n- v02 --refutes--> v07\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"597d7603e2cb31cf4dfbdaa37010bc182bc7805ad2f29df1fa87a9adff1bc059","schema":"livegraph.boolean-propagation.v1","spec_hash":"5910897581c726b30844b87e7e3ffbcd7ae5f732f9e7596f2eccc74324594243","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-f255d790c2d79c6542e7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v0a\n- v0a --supports--> v06\n- v03 --supports--> v00\n- v06 --supports--> v08\n- v0f --supports--> v01\n- v06 --supports--> v11\n- v10 --refutes--> v13\n- v06 --refutes--> v0e\n- v0b --supports--> v02\n- v03 --supports--> v0c\n- v0d --supports--> v09\n- v01 --supports--> v0d\n- v12 --refutes--> v04\n- v01 --refutes--> v05\n- v04 --supports--> v0b\n- v00 --refutes--> v12\n- v0d --refutes--> v03\n- v13 --supports--> v0f\n- v10 --supports--> v07\n\n[VERIFIED ROOT]\nv10 = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c184767aa1d09f60aeabed343fea5a221d38b286cf02311506b87676139cf00c","schema":"livegraph.boolean-propagation.v1","spec_hash":"f255d790c2d79c6542e7d88e1bef7bf650b56f7a309a27ade4fdd6ad1c42a3b6","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-2f1b36b927cf1af747cc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v06\n- v09 --supports--> v05\n- v04 --supports--> v01\n- v01 --supports--> v11\n- v09 --refutes--> v0c\n- v0e --refutes--> v09\n- v12 --supports--> v07\n- v0d --supports--> v12\n- v0a --supports--> v0e\n- v07 --refutes--> v04\n- v10 --refutes--> v00\n- v0f --refutes--> v10\n- v12 --supports--> v0b\n- v00 --supports--> v03\n- v06 --supports--> v13\n- v03 --refutes--> v0d\n- v07 --refutes--> v02\n- v06 --refutes--> v0f\n- v0f --refutes--> v08\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"44162311fd1ded0da557160a2815e7f6ae7c147f1e15190438d71cb1a222464d","schema":"livegraph.boolean-propagation.v1","spec_hash":"2f1b36b927cf1af747ccf90c6b1819607753c240d074d7ef80f798666cfd1af5","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-44ecc82a4bd62fd64b6e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v01\n- v02 --refutes--> v00\n- v0b --refutes--> v04\n- v13 --supports--> v0e\n- v0c --supports--> v08\n- v00 --refutes--> v0b\n- v04 --refutes--> v09\n- v04 --supports--> v12\n- v01 --refutes--> v07\n- v11 --refutes--> v10\n- v01 --refutes--> v0d\n- v09 --refutes--> v0f\n- v08 --refutes--> v0a\n- v00 --supports--> v06\n- v10 --supports--> v02\n- v07 --supports--> v13\n- v13 --refutes--> v0c\n- v0f --refutes--> v05\n- v0b --supports--> v03\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e9ac649657abbf219ed8524dec9113187c143eeeb90033c802a7bdd8e747bfe7","schema":"livegraph.boolean-propagation.v1","spec_hash":"44ecc82a4bd62fd64b6eb5261f7a1c35520d8bf6b7e1decb97dca7625474a4e0","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-435bb892be4d7d272d45","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v0c\n- v0a --supports--> v01\n- v11 --supports--> v07\n- v0c --refutes--> v13\n- v03 --refutes--> v0a\n- v01 --supports--> v05\n- v00 --supports--> v08\n- v05 --refutes--> v06\n- v09 --refutes--> v03\n- v0c --supports--> v0e\n- v0f --refutes--> v04\n- v13 --supports--> v0f\n- v13 --supports--> v10\n- v12 --refutes--> v0b\n- v04 --refutes--> v12\n- v00 --refutes--> v11\n- v11 --supports--> v0d\n- v03 --supports--> v02\n- v05 --supports--> v00\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0e29d935c91fd19d9c3ee8652225c5e811cd2b1edfb0dbb2818ddb5e69725467","schema":"livegraph.boolean-propagation.v1","spec_hash":"435bb892be4d7d272d458690b800d8410a0515accd4dc61d90b3c2498ae2d4de","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-ca2c7a738365c7af4a00","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v11\n- v0d --supports--> v12\n- v0f --refutes--> v06\n- v0d --refutes--> v02\n- v0c --supports--> v0e\n- v06 --supports--> v00\n- v03 --supports--> v05\n- v0a --refutes--> v13\n- v00 --refutes--> v04\n- v0f --supports--> v09\n- v0f --refutes--> v0b\n- v10 --refutes--> v0a\n- v0e --refutes--> v03\n- v10 --supports--> v07\n- v13 --supports--> v0f\n- v02 --supports--> v01\n- v03 --refutes--> v08\n- v04 --refutes--> v0c\n- v02 --refutes--> v10\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"731a525c066d28df117895e1cd7514cbaf91f46d601c4f9643bc34448ce23a0b","schema":"livegraph.boolean-propagation.v1","spec_hash":"ca2c7a738365c7af4a0003bf696d20c223621b19f2c21ce4b53f68765eeca5ef","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-c8a62236c7a3610f25b5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v11\n- v07 --refutes--> v03\n- v0c --refutes--> v0d\n- v01 --supports--> v0c\n- v0e --refutes--> v01\n- v00 --refutes--> v06\n- v0b --refutes--> v13\n- v04 --refutes--> v0e\n- v13 --refutes--> v12\n- v06 --supports--> v0a\n- v12 --supports--> v04\n- v0c --refutes--> v05\n- v06 --refutes--> v08\n- v07 --supports--> v02\n- v05 --supports--> v0f\n- v05 --refutes--> v10\n- v08 --refutes--> v07\n- v10 --refutes--> v00\n- v04 --supports--> v09\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ccabc648dceeea70304df962abc816f79d3cc539ea40c8701c0394f2f7218bc1","schema":"livegraph.boolean-propagation.v1","spec_hash":"c8a62236c7a3610f25b59686959b07ba76656e561c6e95d8bbd989cc8b852ec6","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-9ea39987feec5e623a5e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v04\n- v13 --refutes--> v03\n- v12 --supports--> v13\n- v08 --supports--> v0e\n- v0e --refutes--> v0a\n- v03 --supports--> v0d\n- v0d --supports--> v09\n- v0c --refutes--> v02\n- v01 --supports--> v00\n- v08 --refutes--> v0b\n- v07 --supports--> v01\n- v0e --supports--> v05\n- v09 --supports--> v08\n- v0c --refutes--> v07\n- v0c --supports--> v10\n- v01 --supports--> v06\n- v06 --supports--> v11\n- v04 --refutes--> v0f\n- v04 --refutes--> v0c\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c0f1734efc86d5538b0c863942318bc7a8e52e58bb9b06134ab5f1c7938f406c","schema":"livegraph.boolean-propagation.v1","spec_hash":"9ea39987feec5e623a5ecc31f9c6e368ba7317c56720c823056159422a91cdc3","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-e558e40e789e315f28e1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --supports--> v07\n- v0d --supports--> v04\n- v0c --refutes--> v12\n- v06 --supports--> v0b\n- v12 --refutes--> v02\n- v13 --supports--> v08\n- v00 --supports--> v0d\n- v0f --supports--> v05\n- v13 --supports--> v11\n- v09 --refutes--> v06\n- v02 --supports--> v03\n- v0d --supports--> v13\n- v0a --supports--> v00\n- v0d --supports--> v01\n- v0f --refutes--> v10\n- v0b --supports--> v0f\n- v12 --refutes--> v0e\n- v05 --refutes--> v0a\n- v08 --supports--> v0c\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"76b7ff14b9dbc44b936a6886193fb22742881efe66c6890bbf3f91e244e30ac2","schema":"livegraph.boolean-propagation.v1","spec_hash":"e558e40e789e315f28e12f39eb0cd28088272a02c46dd66e55ab9ba1f388c305","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-700e37735134776a1995","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v09\n- v07 --supports--> v01\n- v08 --supports--> v03\n- v0f --refutes--> v13\n- v0e --refutes--> v0b\n- v0a --supports--> v00\n- v0c --refutes--> v0f\n- v10 --supports--> v08\n- v10 --refutes--> v02\n- v0d --refutes--> v11\n- v03 --supports--> v0d\n- v04 --refutes--> v0e\n- v09 --supports--> v07\n- v0a --supports--> v04\n- v10 --refutes--> v06\n- v10 --supports--> v12\n- v0b --supports--> v0c\n- v11 --refutes--> v05\n- v0f --supports--> v10\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"401c750474143d86f1969d9cb580b256b3211374651508de1be7b2629109817d","schema":"livegraph.boolean-propagation.v1","spec_hash":"700e37735134776a19958c813c8e585837f794dd4f95091fb63e77827d50648a","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":13,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-aa164a97ce22a1cb0aec","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v12\n- v0e --refutes--> v0f\n- v0d --supports--> v06\n- v11 --refutes--> v0c\n- v0b --supports--> v00\n- v06 --supports--> v0e\n- v08 --supports--> v07\n- v10 --supports--> v03\n- v13 --supports--> v11\n- v09 --supports--> v0b\n- v06 --refutes--> v02\n- v0a --supports--> v10\n- v0d --refutes--> v01\n- v04 --refutes--> v0a\n- v0f --refutes--> v08\n- v03 --supports--> v0d\n- v08 --refutes--> v09\n- v0b --refutes--> v13\n- v09 --refutes--> v05\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c11023b1d528623924be913a78af8dd7ad80204c83676a85c5f565fa5bf40568","schema":"livegraph.boolean-propagation.v1","spec_hash":"aa164a97ce22a1cb0aec82ff5a018d74d70a2dca7d1df1e776bbf5a51fbc844a","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-cef876fa4401d8027c20","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v03\n- v13 --refutes--> v12\n- v09 --supports--> v0c\n- v0d --supports--> v11\n- v0c --supports--> v0a\n- v07 --refutes--> v10\n- v06 --supports--> v02\n- v0c --refutes--> v07\n- v0f --supports--> v13\n- v04 --refutes--> v01\n- v0d --supports--> v09\n- v07 --supports--> v08\n- v12 --supports--> v0d\n- v01 --refutes--> v0e\n- v03 --supports--> v00\n- v04 --supports--> v05\n- v00 --supports--> v0b\n- v0b --supports--> v0f\n- v06 --supports--> v04\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dc6c741953ae4fd68aa822a53388bcc02a83df019e60d8b51ac265592b9e4093","schema":"livegraph.boolean-propagation.v1","spec_hash":"cef876fa4401d8027c201c4973966395afccb1180e7d7cd06a7d7b578234d551","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-d1a6ef142b58241fc7b5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v0f\n- v00 --refutes--> v0d\n- v0c --refutes--> v07\n- v07 --refutes--> v0e\n- v05 --supports--> v08\n- v07 --supports--> v13\n- v08 --refutes--> v11\n- v13 --supports--> v05\n- v13 --refutes--> v06\n- v09 --refutes--> v0c\n- v10 --refutes--> v01\n- v02 --refutes--> v09\n- v10 --supports--> v02\n- v0d --refutes--> v0a\n- v0d --supports--> v0b\n- v0f --refutes--> v12\n- v12 --refutes--> v00\n- v02 --refutes--> v04\n- v03 --supports--> v10\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1feace06c557cd8f7ceb33b7fb6d56990c7534997cbb0ae3c7f501315b9f6b5f","schema":"livegraph.boolean-propagation.v1","spec_hash":"d1a6ef142b58241fc7b54a2fcc032792fd1be8c0e9247034acf358f4002fd409","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-b029860d1909552961eb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v07\n- v02 --supports--> v0c\n- v07 --refutes--> v13\n- v11 --supports--> v0d\n- v04 --supports--> v06\n- v0e --supports--> v03\n- v09 --refutes--> v0f\n- v0c --refutes--> v05\n- v0c --supports--> v00\n- v0c --supports--> v01\n- v0c --refutes--> v0e\n- v03 --supports--> v08\n- v03 --supports--> v12\n- v0d --refutes--> v09\n- v03 --refutes--> v10\n- v10 --refutes--> v0b\n- v0b --supports--> v11\n- v10 --refutes--> v0a\n- v06 --supports--> v02\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2fa145701bfde83c50d064c94ae1e0a3512c4004245c5447b00078e2971c30db","schema":"livegraph.boolean-propagation.v1","spec_hash":"b029860d1909552961eb9364f1dcbb3c9b592b63fd8672c15f42f0825fece9eb","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":13,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-c6ed32cd05f4e11b2ac5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v12\n- v06 --refutes--> v13\n- v07 --supports--> v11\n- v12 --supports--> v02\n- v11 --refutes--> v0d\n- v01 --refutes--> v00\n- v0b --supports--> v03\n- v0d --supports--> v0e\n- v10 --supports--> v06\n- v04 --supports--> v0f\n- v12 --refutes--> v04\n- v04 --supports--> v0b\n- v13 --supports--> v07\n- v09 --refutes--> v10\n- v0e --refutes--> v05\n- v10 --supports--> v0c\n- v01 --supports--> v09\n- v0e --supports--> v08\n- v0a --refutes--> v01\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ee07f0804f623bf0505816c04470f47fa0ccfe7fe1532c63d77051ddf83bd3b6","schema":"livegraph.boolean-propagation.v1","spec_hash":"c6ed32cd05f4e11b2ac5e8bdf1e278f99e3168018a6b5781c3007ec0544f1c86","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-60c21b9c33bb614c5647","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v0b\n- v07 --refutes--> v03\n- v0a --supports--> v01\n- v0b --refutes--> v13\n- v12 --supports--> v06\n- v0d --supports--> v08\n- v05 --refutes--> v04\n- v13 --supports--> v07\n- v07 --refutes--> v0e\n- v09 --supports--> v0f\n- v11 --supports--> v0a\n- v08 --refutes--> v11\n- v0f --supports--> v0d\n- v04 --supports--> v10\n- v07 --supports--> v00\n- v06 --refutes--> v05\n- v04 --supports--> v09\n- v12 --refutes--> v0c\n- v12 --supports--> v02\n\n[VERIFIED ROOT]\nv12 = FALSE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"72d825e62602fd18bbb7e72e45247e8c9769c98a3f50da2cea4c61329c0613ab","schema":"livegraph.boolean-propagation.v1","spec_hash":"60c21b9c33bb614c56473c18738c32b3ddb4eab2d8b8a3f2f6fa25e8d2ce72e4","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v12\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-8e9d8b0d98b82e6e2864","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v03\n- v0a --refutes--> v04\n- v10 --refutes--> v02\n- v0b --refutes--> v11\n- v0f --supports--> v05\n- v04 --supports--> v0c\n- v07 --supports--> v08\n- v04 --supports--> v09\n- v07 --supports--> v01\n- v11 --supports--> v07\n- v07 --refutes--> v0e\n- v0d --supports--> v10\n- v12 --refutes--> v0f\n- v0e --refutes--> v0d\n- v03 --refutes--> v0a\n- v07 --supports--> v00\n- v03 --refutes--> v06\n- v02 --supports--> v12\n- v0f --refutes--> v13\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b9f1e1be9a8407250fa9b90956e158e3f52136e3f0210d1a637587ef4b3ee1d2","schema":"livegraph.boolean-propagation.v1","spec_hash":"8e9d8b0d98b82e6e28646d37f46c0c7108ac0080560546ce45141ca1545f43b0","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-43862477ad8fd674ee6e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v11\n- v02 --supports--> v13\n- v07 --supports--> v0d\n- v04 --supports--> v09\n- v0d --supports--> v02\n- v0d --refutes--> v08\n- v02 --supports--> v10\n- v0b --refutes--> v01\n- v0c --refutes--> v07\n- v12 --supports--> v0f\n- v0b --supports--> v05\n- v04 --refutes--> v0b\n- v11 --refutes--> v06\n- v05 --refutes--> v12\n- v04 --supports--> v0e\n- v06 --refutes--> v00\n- v10 --refutes--> v04\n- v00 --supports--> v0c\n- v0d --refutes--> v0a\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"674d7d1d00ac17b8bae499cabe5fafcd69603f9cac074ded994254bbb924908e","schema":"livegraph.boolean-propagation.v1","spec_hash":"43862477ad8fd674ee6e9eb375170cc64e04ed55f65f6cc909775e79022b1e35","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-fa7313b3e8ef9db19e23","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v09\n- v07 --supports--> v0c\n- v10 --supports--> v07\n- v0e --refutes--> v08\n- v11 --refutes--> v10\n- v04 --refutes--> v0b\n- v06 --supports--> v04\n- v11 --refutes--> v05\n- v09 --supports--> v11\n- v07 --supports--> v0d\n- v0b --supports--> v03\n- v13 --refutes--> v02\n- v12 --supports--> v13\n- v01 --supports--> v12\n- v03 --supports--> v01\n- v04 --supports--> v00\n- v07 --refutes--> v0f\n- v01 --supports--> v0a\n- v02 --supports--> v0e\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a0caeb0eb8ac15b6a2fb9a819be29359e7cf92c7d3bd33cda30df30e9fbd94c4","schema":"livegraph.boolean-propagation.v1","spec_hash":"fa7313b3e8ef9db19e232ff92d52d739d6587f5a54ebbf95d83688ddc72ca5fb","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-086f3e5424eb7e5096ca","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v09\n- v02 --supports--> v0d\n- v09 --refutes--> v00\n- v0d --supports--> v0a\n- v05 --supports--> v08\n- v03 --refutes--> v11\n- v11 --refutes--> v10\n- v08 --supports--> v04\n- v10 --refutes--> v07\n- v04 --supports--> v0b\n- v10 --supports--> v0f\n- v0d --supports--> v0e\n- v0b --refutes--> v0c\n- v0f --refutes--> v05\n- v0c --refutes--> v13\n- v0a --supports--> v12\n- v0c --refutes--> v02\n- v05 --refutes--> v01\n- v02 --supports--> v06\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a4772e416b7aecff645edca5b3ec860c3ca99b1d2edbd9eaed027f6d4234834f","schema":"livegraph.boolean-propagation.v1","spec_hash":"086f3e5424eb7e5096ca7b4dddae6b1ae003d101d20c4ca936223c23a3bceed0","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":13,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-062d4643677cd0153364","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v06\n- v02 --refutes--> v08\n- v00 --supports--> v10\n- v00 --supports--> v07\n- v0d --refutes--> v05\n- v04 --supports--> v11\n- v06 --refutes--> v02\n- v0c --refutes--> v01\n- v04 --supports--> v0f\n- v05 --supports--> v09\n- v05 --supports--> v0e\n- v0e --supports--> v0b\n- v10 --refutes--> v0a\n- v0e --refutes--> v03\n- v08 --supports--> v0c\n- v0e --supports--> v12\n- v11 --refutes--> v0d\n- v0c --supports--> v13\n- v01 --supports--> v00\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ae7df1c1a4d38f015fb4423753ccab1440221358053b6fb4b37804ea0d504d47","schema":"livegraph.boolean-propagation.v1","spec_hash":"062d4643677cd01533643539616c54f5097c1b11715442e5d975a5f29a1b5461","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":13,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-07df937b41d9661666f8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v13\n- v11 --supports--> v07\n- v0b --supports--> v14\n- v14 --refutes--> v03\n- v13 --supports--> v0c\n- v12 --supports--> v08\n- v13 --refutes--> v09\n- v15 --supports--> v12\n- v13 --refutes--> v15\n- v07 --refutes--> v04\n- v01 --supports--> v02\n- v12 --supports--> v0f\n- v02 --supports--> v10\n- v0e --supports--> v11\n- v00 --refutes--> v01\n- v0e --supports--> v0a\n- v14 --supports--> v06\n- v0d --refutes--> v0e\n- v00 --supports--> v05\n- v0f --supports--> v0b\n- v06 --supports--> v00\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"03729fcd2011a512564cdb8187e622f7e8b054f969f9f450da3e359496128184","schema":"livegraph.boolean-propagation.v1","spec_hash":"07df937b41d9661666f835024535d601678c6bcf34117a8c28430ceeb0548c80","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-38dca10ec5e1d7e50911","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --refutes--> v0b\n- v0f --supports--> v13\n- v12 --refutes--> v07\n- v12 --refutes--> v09\n- v09 --supports--> v10\n- v13 --supports--> v00\n- v0e --supports--> v08\n- v13 --supports--> v0d\n- v08 --supports--> v06\n- v15 --refutes--> v12\n- v06 --supports--> v11\n- v12 --supports--> v02\n- v0d --refutes--> v0a\n- v0d --supports--> v15\n- v05 --supports--> v01\n- v06 --supports--> v03\n- v01 --supports--> v0f\n- v14 --supports--> v0c\n- v04 --refutes--> v14\n- v09 --supports--> v04\n- v03 --supports--> v05\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"575506e2fd7245205ad8027db21dd8b2ff5a03aaaf1f7074c4074b2a8533dc86","schema":"livegraph.boolean-propagation.v1","spec_hash":"38dca10ec5e1d7e50911db594a822f04f2de755bea966f9db46829ff7c8f6422","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-e870258610d65d9c30c3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --refutes--> v04\n- v02 --refutes--> v11\n- v04 --refutes--> v01\n- v01 --supports--> v09\n- v0a --refutes--> v03\n- v01 --supports--> v0e\n- v0f --refutes--> v08\n- v06 --refutes--> v10\n- v0e --refutes--> v15\n- v0d --supports--> v00\n- v10 --supports--> v02\n- v07 --refutes--> v0f\n- v06 --refutes--> v0c\n- v02 --refutes--> v12\n- v01 --supports--> v0b\n- v14 --supports--> v07\n- v0a --refutes--> v14\n- v0e --supports--> v05\n- v02 --supports--> v0d\n- v00 --refutes--> v0a\n- v15 --refutes--> v06\n\n[VERIFIED ROOT]\nv13 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"340870dff43e0e910bbb964fc17291245cb8ff9f5dabb7e03a5e6dbec1930370","schema":"livegraph.boolean-propagation.v1","spec_hash":"e870258610d65d9c30c398eb84f62f4b7349bf5279ee9f5698d40a9fbe368eb0","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-01029e5c553939b2655c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v02\n- v15 --refutes--> v0a\n- v0f --refutes--> v04\n- v13 --supports--> v0d\n- v11 --refutes--> v15\n- v02 --refutes--> v07\n- v0e --refutes--> v0b\n- v03 --refutes--> v11\n- v00 --supports--> v08\n- v07 --refutes--> v0e\n- v06 --supports--> v0f\n- v09 --supports--> v01\n- v13 --refutes--> v06\n- v0b --supports--> v05\n- v0b --refutes--> v03\n- v0f --refutes--> v10\n- v0f --refutes--> v0c\n- v13 --supports--> v12\n- v00 --supports--> v14\n- v0c --refutes--> v09\n- v0a --refutes--> v13\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e571858307b087ffaff0b902749f0713663bd86c501b417475ad370768c5180c","schema":"livegraph.boolean-propagation.v1","spec_hash":"01029e5c553939b2655c7c6bbe1b77c98c0addbea0837ace88cc66b1f4044a8b","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-477cfcfe4cc8cdd360d5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v10\n- v01 --supports--> v13\n- v09 --refutes--> v0f\n- v14 --supports--> v01\n- v06 --refutes--> v09\n- v0b --refutes--> v0e\n- v01 --refutes--> v08\n- v12 --refutes--> v00\n- v01 --refutes--> v0d\n- v07 --supports--> v12\n- v03 --refutes--> v14\n- v10 --supports--> v07\n- v08 --refutes--> v0a\n- v00 --refutes--> v05\n- v0f --supports--> v0b\n- v09 --refutes--> v02\n- v01 --refutes--> v15\n- v08 --supports--> v04\n- v0e --refutes--> v03\n- v09 --refutes--> v0c\n- v11 --refutes--> v06\n\n[VERIFIED ROOT]\nv11 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fd2f700ceb683b6da26083c74dfa8549245aff47fa209aad966989d82dc53c39","schema":"livegraph.boolean-propagation.v1","spec_hash":"477cfcfe4cc8cdd360d5f3574ebfe566186ee115179c0c7898d19d46301dab02","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-3cf22744ec50555bb512","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v12\n- v0a --refutes--> v06\n- v0f --refutes--> v03\n- v09 --supports--> v07\n- v14 --refutes--> v00\n- v00 --supports--> v01\n- v0f --refutes--> v0d\n- v15 --refutes--> v14\n- v02 --supports--> v13\n- v0b --refutes--> v0e\n- v04 --refutes--> v0c\n- v03 --supports--> v15\n- v12 --refutes--> v02\n- v00 --supports--> v09\n- v03 --supports--> v08\n- v0e --refutes--> v04\n- v0b --supports--> v11\n- v09 --refutes--> v05\n- v04 --supports--> v10\n- v09 --refutes--> v0b\n- v06 --refutes--> v0f\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e16b4f86c83acfcd3a8daa7bcba53778652228b78d243a54b23b8c547b213563","schema":"livegraph.boolean-propagation.v1","spec_hash":"3cf22744ec50555bb5121469ff5925c8783e6e6eb25a6aeca1568b87955ee0ad","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-5c1dcd208cd02ad84193","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --supports--> v11\n- v14 --supports--> v0d\n- v0c --supports--> v0b\n- v00 --supports--> v14\n- v05 --refutes--> v04\n- v10 --refutes--> v07\n- v02 --refutes--> v01\n- v05 --refutes--> v03\n- v11 --supports--> v09\n- v12 --supports--> v0a\n- v11 --supports--> v0f\n- v0e --supports--> v05\n- v13 --refutes--> v10\n- v15 --supports--> v00\n- v02 --refutes--> v06\n- v10 --supports--> v0c\n- v03 --supports--> v02\n- v0b --refutes--> v15\n- v12 --supports--> v0e\n- v06 --refutes--> v13\n- v03 --supports--> v08\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"35e7ce6db0fd686d9b29a7191ac57412690062a9c161701706ebb9e725c040c9","schema":"livegraph.boolean-propagation.v1","spec_hash":"5c1dcd208cd02ad84193bfe91896965f5f9b2d6fb6e07487a42b1166611b9eca","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-0d484a7a0c223579d2b4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v0d\n- v0a --supports--> v15\n- v0d --supports--> v10\n- v01 --supports--> v13\n- v0d --supports--> v04\n- v15 --supports--> v12\n- v02 --refutes--> v05\n- v11 --refutes--> v07\n- v00 --refutes--> v09\n- v14 --supports--> v11\n- v0e --refutes--> v14\n- v12 --supports--> v01\n- v0e --refutes--> v03\n- v0c --supports--> v0f\n- v00 --refutes--> v0c\n- v08 --refutes--> v0b\n- v10 --refutes--> v0a\n- v08 --refutes--> v02\n- v12 --supports--> v06\n- v02 --supports--> v00\n- v11 --refutes--> v08\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"71a9bc65e33e5df556298d08bfbfee1f40f58a7301d8d1a2cb3777114cccc1a9","schema":"livegraph.boolean-propagation.v1","spec_hash":"0d484a7a0c223579d2b42d90ac641c29dcfe77d8e046c5dd804b8b0dea75fbbf","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-747fe4c7049c30f26793","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v0d\n- v07 --refutes--> v0e\n- v02 --refutes--> v08\n- v07 --supports--> v06\n- v14 --supports--> v09\n- v05 --refutes--> v0c\n- v06 --refutes--> v11\n- v07 --supports--> v03\n- v14 --refutes--> v0b\n- v04 --refutes--> v0f\n- v05 --refutes--> v10\n- v0b --supports--> v05\n- v0c --refutes--> v01\n- v08 --refutes--> v00\n- v0d --supports--> v14\n- v01 --supports--> v04\n- v14 --supports--> v12\n- v0f --supports--> v02\n- v11 --supports--> v0a\n- v05 --supports--> v15\n- v08 --refutes--> v13\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"240771a72ac11ccff5422e19d1cd77ff960e33a191834f3c532c46de96b4de65","schema":"livegraph.boolean-propagation.v1","spec_hash":"747fe4c7049c30f26793c14353ce821d571632a8e67f13a8ef6aed1fcaf3710c","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-8ff91a2c40a0fdd54899","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v00\n- v0d --refutes--> v11\n- v12 --refutes--> v14\n- v15 --supports--> v04\n- v04 --refutes--> v03\n- v12 --refutes--> v0e\n- v09 --refutes--> v10\n- v11 --supports--> v01\n- v05 --refutes--> v09\n- v09 --refutes--> v02\n- v14 --supports--> v15\n- v12 --supports--> v0f\n- v06 --refutes--> v0a\n- v08 --supports--> v0c\n- v08 --supports--> v07\n- v0a --refutes--> v13\n- v01 --refutes--> v08\n- v0a --refutes--> v0d\n- v14 --refutes--> v0b\n- v03 --supports--> v05\n- v0c --refutes--> v12\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ac36a803b0e3016f8e338381c1029a7f3bd0b61c6e7ff47dbc5db379317c5887","schema":"livegraph.boolean-propagation.v1","spec_hash":"8ff91a2c40a0fdd54899d4f17ca3bc91c255c8e21d200055b4e967f9d028f3a9","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-855f25c9cd9e4da089ba","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v13\n- v03 --supports--> v07\n- v14 --refutes--> v11\n- v13 --refutes--> v0c\n- v0c --refutes--> v15\n- v00 --refutes--> v01\n- v14 --refutes--> v09\n- v00 --refutes--> v0f\n- v15 --supports--> v03\n- v10 --supports--> v14\n- v07 --refutes--> v05\n- v14 --supports--> v0a\n- v04 --supports--> v10\n- v0b --refutes--> v06\n- v05 --supports--> v04\n- v05 --supports--> v0d\n- v01 --supports--> v0e\n- v0b --supports--> v00\n- v12 --refutes--> v0b\n- v13 --refutes--> v08\n- v15 --refutes--> v02\n\n[VERIFIED ROOT]\nv12 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bb66dfac4899b80bb124e2b8b0f2e80e78e8354c26bc895b86f044667dc2317c","schema":"livegraph.boolean-propagation.v1","spec_hash":"855f25c9cd9e4da089ba1fca6be1ceab063d8a0d55d8ca1e7ba8f142404d05e7","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v12\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-c1ae03bcdfff4df0ac6c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v0d\n- v10 --refutes--> v0b\n- v14 --refutes--> v08\n- v0d --supports--> v09\n- v00 --refutes--> v0e\n- v00 --supports--> v13\n- v15 --refutes--> v0c\n- v06 --refutes--> v11\n- v0c --refutes--> v0f\n- v14 --refutes--> v03\n- v12 --refutes--> v10\n- v11 --refutes--> v00\n- v15 --supports--> v07\n- v15 --supports--> v01\n- v02 --supports--> v15\n- v0d --supports--> v05\n- v08 --supports--> v06\n- v08 --refutes--> v04\n- v0f --refutes--> v12\n- v05 --supports--> v14\n- v10 --supports--> v0a\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b11e51e93d5aa41a5e5de58411eb03db63b632805dbc020553f9c407c698336f","schema":"livegraph.boolean-propagation.v1","spec_hash":"c1ae03bcdfff4df0ac6c43b95cc9b910f5bf35d2478a6a25de715ff0a668cd22","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-599ad3f57f6fb2a1bbf0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v0f\n- v0d --supports--> v05\n- v09 --refutes--> v13\n- v08 --supports--> v00\n- v09 --refutes--> v14\n- v12 --supports--> v10\n- v04 --supports--> v0a\n- v0a --refutes--> v08\n- v03 --refutes--> v0c\n- v03 --refutes--> v0e\n- v08 --supports--> v07\n- v05 --refutes--> v06\n- v05 --refutes--> v01\n- v04 --supports--> v02\n- v10 --refutes--> v0b\n- v11 --refutes--> v04\n- v0e --supports--> v11\n- v07 --refutes--> v09\n- v13 --supports--> v15\n- v10 --supports--> v0d\n- v13 --supports--> v12\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"aab97e532bcf68528721179f489b6eaf099f174ee5a4591a3d18ebd44ca222ed","schema":"livegraph.boolean-propagation.v1","spec_hash":"599ad3f57f6fb2a1bbf02e306415e8ba076d64d05b6a32becf17b22f92732243","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":14,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-5d98ad94123345de6a1a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v12\n- v0c --refutes--> v05\n- v0f --refutes--> v06\n- v0c --supports--> v11\n- v0b --refutes--> v08\n- v0a --supports--> v14\n- v14 --refutes--> v0e\n- v02 --refutes--> v0b\n- v11 --refutes--> v15\n- v12 --supports--> v01\n- v15 --supports--> v0a\n- v0b --refutes--> v03\n- v14 --refutes--> v13\n- v03 --supports--> v0d\n- v07 --supports--> v09\n- v07 --supports--> v00\n- v0d --refutes--> v04\n- v0d --refutes--> v10\n- v12 --refutes--> v07\n- v04 --refutes--> v0c\n- v00 --supports--> v02\n\n[VERIFIED ROOT]\nv0f = FALSE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1cfec3d56881cc1166222e8d3be5900da05a356735703dc4ead30129b94c2c9b","schema":"livegraph.boolean-propagation.v1","spec_hash":"5d98ad94123345de6a1ad757438d5ae136893a89efa2e05992912b4422d080d7","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-0e65f2266dbf15935eb8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v06\n- v10 --supports--> v14\n- v13 --supports--> v09\n- v04 --supports--> v13\n- v02 --refutes--> v01\n- v13 --supports--> v15\n- v12 --supports--> v10\n- v0b --refutes--> v11\n- v11 --supports--> v07\n- v0f --supports--> v04\n- v0a --supports--> v00\n- v03 --refutes--> v0d\n- v08 --refutes--> v0f\n- v0a --supports--> v0e\n- v01 --supports--> v08\n- v03 --refutes--> v12\n- v04 --refutes--> v05\n- v07 --refutes--> v0a\n- v14 --refutes--> v02\n- v13 --refutes--> v0b\n- v04 --supports--> v0c\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5d877b5ca6c62db5ab376e33059268e35e807b72b1ffdf8a4b1997847533fb6b","schema":"livegraph.boolean-propagation.v1","spec_hash":"0e65f2266dbf15935eb83389ab0560e7346459eb86901105fdb9ebf7558204c8","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-81850d31748aa21b32c8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v0e\n- v13 --refutes--> v14\n- v0a --supports--> v13\n- v01 --supports--> v00\n- v15 --supports--> v06\n- v0e --refutes--> v03\n- v0d --refutes--> v15\n- v0f --supports--> v02\n- v02 --refutes--> v07\n- v03 --supports--> v0a\n- v0a --refutes--> v05\n- v0b --supports--> v0d\n- v01 --refutes--> v12\n- v13 --supports--> v0f\n- v0f --supports--> v08\n- v01 --refutes--> v10\n- v07 --supports--> v01\n- v04 --refutes--> v0b\n- v0d --refutes--> v0c\n- v13 --supports--> v09\n- v11 --supports--> v04\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e5ca82557cbdf8fb621d948b90e6bfbd322c8a1792ea5b91a19dba34d6e8f1c0","schema":"livegraph.boolean-propagation.v1","spec_hash":"81850d31748aa21b32c82ad48416ce9e0c0083827af63b3d4046849d8ea8c5f0","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-3c01c680092ac370100a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v11\n- v0f --supports--> v0c\n- v10 --refutes--> v00\n- v14 --supports--> v0e\n- v0f --supports--> v06\n- v14 --refutes--> v0d\n- v03 --refutes--> v15\n- v10 --supports--> v03\n- v03 --refutes--> v04\n- v15 --supports--> v0f\n- v11 --refutes--> v01\n- v13 --supports--> v05\n- v08 --refutes--> v12\n- v07 --refutes--> v13\n- v01 --refutes--> v14\n- v13 --supports--> v0b\n- v0a --refutes--> v10\n- v03 --supports--> v09\n- v12 --supports--> v0a\n- v10 --supports--> v02\n- v0d --supports--> v07\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9672300dfc5b60b1195e7cb57006ee87efafb30ada1386d9ad63fd297ad79240","schema":"livegraph.boolean-propagation.v1","spec_hash":"3c01c680092ac370100a13ec80022542d88321413d1e0e82f3e437b1bb48a8b9","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-a1d74bdb5d0a8f37639f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v0b\n- v03 --supports--> v13\n- v04 --refutes--> v15\n- v11 --supports--> v12\n- v0e --refutes--> v14\n- v05 --supports--> v11\n- v14 --supports--> v05\n- v13 --refutes--> v0a\n- v11 --refutes--> v10\n- v08 --refutes--> v0d\n- v0c --supports--> v06\n- v00 --supports--> v0e\n- v0c --refutes--> v08\n- v02 --refutes--> v00\n- v0d --refutes--> v09\n- v13 --refutes--> v0c\n- v0d --refutes--> v04\n- v15 --refutes--> v07\n- v10 --refutes--> v03\n- v13 --refutes--> v0f\n- v04 --supports--> v01\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c988a97fd81b685dd841e262a2422311a43d6a45abba405a407172fd5354bf93","schema":"livegraph.boolean-propagation.v1","spec_hash":"a1d74bdb5d0a8f37639f2c08255644afb5dbdedaed5fab6efe8559c355e07c65","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-c4a4e1f9d54d87f7a923","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v11\n- v0f --supports--> v01\n- v00 --refutes--> v0e\n- v0b --supports--> v12\n- v0d --supports--> v13\n- v0d --supports--> v15\n- v09 --supports--> v03\n- v02 --refutes--> v14\n- v0e --supports--> v07\n- v14 --refutes--> v09\n- v07 --supports--> v0b\n- v0b --refutes--> v10\n- v01 --refutes--> v06\n- v0c --refutes--> v04\n- v03 --supports--> v0f\n- v0d --refutes--> v02\n- v06 --supports--> v05\n- v0e --supports--> v08\n- v12 --refutes--> v0d\n- v0d --refutes--> v0a\n- v06 --refutes--> v0c\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5ad02b380af6d6cd0ad28eb6276c2fddc0edd996f0a613c2d920d5647ad23981","schema":"livegraph.boolean-propagation.v1","spec_hash":"c4a4e1f9d54d87f7a923e8138a00ec9cf30d857bb224bd9fb6962ed11341f720","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-dcb7894616b2439ae72c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v07\n- v08 --refutes--> v02\n- v01 --supports--> v00\n- v0a --supports--> v06\n- v13 --refutes--> v12\n- v0c --refutes--> v13\n- v0e --supports--> v14\n- v10 --refutes--> v0e\n- v0c --refutes--> v09\n- v0e --supports--> v11\n- v13 --refutes--> v0b\n- v12 --supports--> v05\n- v11 --supports--> v08\n- v00 --refutes--> v10\n- v0a --refutes--> v0f\n- v08 --refutes--> v03\n- v0d --refutes--> v01\n- v05 --refutes--> v15\n- v15 --supports--> v0a\n- v03 --refutes--> v0c\n- v05 --supports--> v04\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"975990917fa6e11ead7ab1d9386e710d0f80a618c69318bc4fa6b3a04132e849","schema":"livegraph.boolean-propagation.v1","spec_hash":"dcb7894616b2439ae72c9d9e1ff269df5a8d9c78b5994af94b8d7bb10f630fb3","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-88ecd50b3ef594f37df5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v01\n- v0f --supports--> v0e\n- v13 --supports--> v15\n- v00 --refutes--> v05\n- v05 --refutes--> v0d\n- v0c --supports--> v07\n- v09 --supports--> v03\n- v11 --refutes--> v0b\n- v11 --refutes--> v0c\n- v0f --supports--> v02\n- v09 --refutes--> v12\n- v03 --supports--> v14\n- v0f --refutes--> v11\n- v01 --refutes--> v08\n- v13 --supports--> v09\n- v05 --supports--> v13\n- v08 --refutes--> v0a\n- v0a --supports--> v06\n- v14 --supports--> v10\n- v07 --refutes--> v00\n- v14 --supports--> v04\n\n[VERIFIED ROOT]\nv0f = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"67eaf4734e1a386b61c0889cc70dc32af7cec2b4ded1d3bfd6d729d2ab0f19d5","schema":"livegraph.boolean-propagation.v1","spec_hash":"88ecd50b3ef594f37df59ca723c37bc7dc158b71a173bbd20c8ca88f33808bf5","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-71bf9bf1685734cd4d79","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v0b\n- v07 --supports--> v11\n- v04 --refutes--> v08\n- v02 --refutes--> v0d\n- v15 --supports--> v02\n- v0c --refutes--> v01\n- v10 --supports--> v0e\n- v0f --refutes--> v14\n- v10 --supports--> v0f\n- v0c --supports--> v05\n- v0d --supports--> v00\n- v14 --supports--> v15\n- v0d --supports--> v09\n- v08 --supports--> v10\n- v12 --refutes--> v0c\n- v10 --refutes--> v0a\n- v06 --supports--> v07\n- v0f --supports--> v03\n- v09 --refutes--> v12\n- v0d --supports--> v13\n- v01 --supports--> v06\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d00962c2ad3ff3c482b5b4d8dd5ed990892b7e016e6869548af6c55a5a0d5c53","schema":"livegraph.boolean-propagation.v1","spec_hash":"71bf9bf1685734cd4d790c3e3eda00d046cc8573e1e62aafd13991a67971f13f","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-8b9fb53fcd4a299a7e7e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v08\n- v0f --supports--> v0e\n- v11 --supports--> v0d\n- v12 --supports--> v04\n- v01 --supports--> v00\n- v06 --supports--> v15\n- v02 --refutes--> v0a\n- v10 --supports--> v12\n- v13 --refutes--> v0f\n- v0b --supports--> v06\n- v04 --refutes--> v01\n- v0f --refutes--> v10\n- v0d --refutes--> v13\n- v00 --refutes--> v03\n- v04 --supports--> v0c\n- v02 --supports--> v05\n- v0a --supports--> v11\n- v03 --refutes--> v0b\n- v04 --supports--> v14\n- v0b --supports--> v07\n- v03 --refutes--> v09\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv15\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9c83730647e9506d72dfad262ceb71f18eba88884664261e0590bdb644696ccf","schema":"livegraph.boolean-propagation.v1","spec_hash":"8b9fb53fcd4a299a7e7e6e97a9aed10db9f94b7fe368e733b4f4749d8859bd31","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":14,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-80741ccf05a85ba72b8c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v10\n- v03 --supports--> v0e\n- v08 --supports--> v11\n- v0e --refutes--> v0c\n- v01 --supports--> v0d\n- v0d --supports--> v06\n- v04 --supports--> v03\n- v10 --supports--> v0f\n- v15 --refutes--> v08\n- v09 --refutes--> v00\n- v15 --refutes--> v14\n- v05 --supports--> v15\n- v06 --supports--> v12\n- v06 --refutes--> v0a\n- v06 --refutes--> v09\n- v00 --supports--> v04\n- v07 --refutes--> v13\n- v0e --supports--> v07\n- v0e --supports--> v0b\n- v0f --refutes--> v05\n- v04 --refutes--> v02\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f66c6ed1821c9b96047fabe7826a6c6186aaf11d4ea90e56977be7ef7724c67b","schema":"livegraph.boolean-propagation.v1","spec_hash":"80741ccf05a85ba72b8c6d933c1a0946df39750099e3bd3f09ae79f63b48cee4","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-fe98c1910364edae54ed","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v00\n- v0a --refutes--> v0d\n- v0e --supports--> v01\n- v13 --supports--> v0e\n- v0a --refutes--> v12\n- v07 --refutes--> v02\n- v07 --refutes--> v04\n- v10 --supports--> v07\n- v05 --supports--> v11\n- v08 --refutes--> v09\n- v03 --refutes--> v0b\n- v10 --supports--> v06\n- v11 --refutes--> v13\n- v0e --supports--> v0a\n- v0b --supports--> v10\n- v11 --supports--> v15\n- v09 --supports--> v14\n- v07 --supports--> v0f\n- v0c --refutes--> v08\n- v00 --refutes--> v0c\n- v09 --supports--> v05\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3bfd40f35aa4dce35bfccce9843847980df7f939d8f6cb1e847a9a7d90cca88b","schema":"livegraph.boolean-propagation.v1","spec_hash":"fe98c1910364edae54ed5958fb27c50bebd1499318d6fca14e18a4e1fbebd062","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":14,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-bf6050d2a0fcb7326032","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v0b\n- v09 --refutes--> v10\n- v09 --supports--> v02\n- v04 --refutes--> v06\n- v11 --supports--> v0f\n- v03 --refutes--> v12\n- v07 --refutes--> v04\n- v10 --refutes--> v03\n- v15 --supports--> v0a\n- v04 --supports--> v01\n- v12 --supports--> v0e\n- v0b --supports--> v13\n- v15 --supports--> v00\n- v07 --supports--> v0c\n- v14 --supports--> v07\n- v0d --supports--> v09\n- v04 --supports--> v08\n- v06 --refutes--> v0d\n- v13 --refutes--> v15\n- v12 --supports--> v11\n- v0d --refutes--> v05\n\n[VERIFIED ROOT]\nv14 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3ec1ecba7d718c82da60f61acda30b5ed59b1373215b3da45caadb1e26a9b765","schema":"livegraph.boolean-propagation.v1","spec_hash":"bf6050d2a0fcb73260327476f7b93582620c863cce48c7962cbb1dcc9710d3ea","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v14\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-f74ae94b8072575f7574","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v08\n- v0e --supports--> v11\n- v0c --supports--> v04\n- v04 --refutes--> v00\n- v08 --refutes--> v0f\n- v0a --supports--> v14\n- v09 --refutes--> v13\n- v04 --supports--> v02\n- v12 --supports--> v0a\n- v12 --supports--> v0b\n- v01 --supports--> v05\n- v0f --supports--> v10\n- v02 --supports--> v03\n- v10 --supports--> v12\n- v09 --supports--> v0e\n- v01 --supports--> v15\n- v14 --refutes--> v0c\n- v01 --refutes--> v06\n- v02 --supports--> v09\n- v14 --supports--> v0d\n- v06 --supports--> v07\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4cf8d4ed260b6292d06b8c91fa5db7173b798ed47e1c1a258fdd47193ee4481c","schema":"livegraph.boolean-propagation.v1","spec_hash":"f74ae94b8072575f75747c0b3ad5d8a3e7e0b81722af54247d1e7bb74507dc83","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-2b4d69b467545f9ac2f3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v09\n- v0c --supports--> v0e\n- v00 --supports--> v0b\n- v0e --supports--> v08\n- v0b --refutes--> v0c\n- v13 --refutes--> v10\n- v07 --supports--> v04\n- v13 --refutes--> v03\n- v12 --supports--> v0a\n- v02 --refutes--> v11\n- v12 --supports--> v07\n- v15 --supports--> v05\n- v04 --refutes--> v00\n- v02 --supports--> v13\n- v08 --supports--> v15\n- v09 --supports--> v02\n- v03 --refutes--> v14\n- v12 --refutes--> v06\n- v15 --refutes--> v0d\n- v0e --supports--> v01\n- v09 --refutes--> v0f\n\n[VERIFIED ROOT]\nv12 = FALSE\n\n[TARGET]\nv14\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4ae7310bfc2685cbc3853fa431948f0fefe37b7ee6d8e1bf5a55ef3995d53f3b","schema":"livegraph.boolean-propagation.v1","spec_hash":"2b4d69b467545f9ac2f36e55b7974b9362125aa91e9ef8e0f90a6c7ca7b78941","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v12\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-acc20bd63350cab64180","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v0e\n- v00 --supports--> v10\n- v09 --refutes--> v04\n- v05 --supports--> v12\n- v0d --supports--> v05\n- v13 --refutes--> v07\n- v07 --refutes--> v02\n- v07 --supports--> v0f\n- v0c --refutes--> v03\n- v07 --refutes--> v11\n- v06 --supports--> v0a\n- v12 --supports--> v15\n- v04 --supports--> v0c\n- v13 --supports--> v14\n- v06 --refutes--> v0b\n- v00 --supports--> v0d\n- v08 --refutes--> v09\n- v0a --refutes--> v01\n- v03 --supports--> v06\n- v0a --refutes--> v13\n- v15 --refutes--> v08\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c30447d6fc75d78319e7030c9b158b77b2bce30adcf670dcc6fda0ce9e87c0c3","schema":"livegraph.boolean-propagation.v1","spec_hash":"acc20bd63350cab64180236700093d82aafb520e8e58d00d31509b9330596064","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-5b1898c75ef2be1e7b00","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v14\n- v0f --refutes--> v07\n- v15 --supports--> v03\n- v07 --refutes--> v0e\n- v11 --supports--> v08\n- v0a --refutes--> v0f\n- v07 --supports--> v01\n- v05 --supports--> v06\n- v10 --supports--> v15\n- v03 --supports--> v0d\n- v09 --refutes--> v10\n- v09 --supports--> v04\n- v09 --supports--> v0b\n- v07 --refutes--> v02\n- v03 --supports--> v11\n- v06 --refutes--> v0a\n- v08 --supports--> v0c\n- v12 --supports--> v00\n- v0e --supports--> v09\n- v00 --refutes--> v05\n- v03 --supports--> v13\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"97fd88525e54f58388ed5e9da6336389a6e6e725c0b02ea60e9bdf25d990eab1","schema":"livegraph.boolean-propagation.v1","spec_hash":"5b1898c75ef2be1e7b009ebd7e769353808a008a442e9630708431ed38fc8e54","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-fd47900d31c926b02210","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --supports--> v11\n- v00 --supports--> v07\n- v0d --refutes--> v02\n- v10 --supports--> v0c\n- v07 --refutes--> v08\n- v10 --refutes--> v03\n- v11 --supports--> v01\n- v10 --supports--> v0a\n- v10 --supports--> v13\n- v12 --refutes--> v0f\n- v0b --supports--> v00\n- v0f --refutes--> v0b\n- v0e --supports--> v04\n- v09 --supports--> v0e\n- v13 --supports--> v05\n- v0e --supports--> v14\n- v08 --supports--> v0d\n- v06 --refutes--> v09\n- v11 --refutes--> v12\n- v0d --refutes--> v15\n- v15 --supports--> v06\n\n[VERIFIED ROOT]\nv10 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f28b50edad19575a8d08f779604e011e812b93e8cc244852669c08ec792c2a18","schema":"livegraph.boolean-propagation.v1","spec_hash":"fd47900d31c926b022102fb97600c61b19fa29b2cfc6f9f35428c9cf17c33b09","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-9924d827ff2752c8b0e9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v0a\n- v0d --refutes--> v12\n- v15 --refutes--> v07\n- v0a --supports--> v13\n- v03 --refutes--> v02\n- v0e --refutes--> v06\n- v0b --refutes--> v01\n- v07 --supports--> v0e\n- v05 --supports--> v15\n- v05 --supports--> v10\n- v15 --refutes--> v0f\n- v09 --supports--> v03\n- v0d --refutes--> v05\n- v0a --supports--> v00\n- v08 --supports--> v0c\n- v00 --refutes--> v08\n- v05 --supports--> v11\n- v06 --refutes--> v0b\n- v0c --refutes--> v0d\n- v0b --refutes--> v04\n- v14 --refutes--> v09\n\n[VERIFIED ROOT]\nv14 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a637a1b701272a916ecf046d3d958eb30bef1eb5adf3fb2b00afb36b91c5cbef","schema":"livegraph.boolean-propagation.v1","spec_hash":"9924d827ff2752c8b0e981813a4e9cad76c0f84fd34b941367ba1a1313a0a503","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v14\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-6bc1a629beafbbb2ee8c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v01\n- v06 --refutes--> v0a\n- v13 --supports--> v0b\n- v14 --supports--> v03\n- v14 --supports--> v13\n- v07 --supports--> v05\n- v15 --supports--> v0c\n- v02 --refutes--> v14\n- v0b --refutes--> v07\n- v0e --supports--> v15\n- v14 --refutes--> v12\n- v0a --refutes--> v09\n- v05 --supports--> v08\n- v0c --refutes--> v02\n- v10 --refutes--> v11\n- v11 --refutes--> v06\n- v14 --supports--> v00\n- v0f --supports--> v0e\n- v0a --supports--> v0f\n- v05 --refutes--> v04\n- v05 --supports--> v0d\n\n[VERIFIED ROOT]\nv10 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0fc54aff93c5fe5450a73d60cac8329ae8f0c40da6e7de2123125be81a46a366","schema":"livegraph.boolean-propagation.v1","spec_hash":"6bc1a629beafbbb2ee8c0b56ed8382fad31fe2bcb1d30a6d32cd26f56ea57f3b","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-7da14cd3e701747e537c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v07\n- v0b --refutes--> v14\n- v01 --refutes--> v03\n- v0e --supports--> v06\n- v13 --refutes--> v0b\n- v04 --supports--> v15\n- v07 --refutes--> v10\n- v04 --supports--> v0d\n- v0b --refutes--> v0e\n- v02 --supports--> v08\n- v09 --supports--> v0a\n- v09 --supports--> v02\n- v09 --refutes--> v11\n- v0c --supports--> v13\n- v08 --supports--> v04\n- v06 --supports--> v00\n- v06 --supports--> v09\n- v10 --supports--> v0c\n- v12 --supports--> v01\n- v10 --supports--> v0f\n- v09 --refutes--> v05\n\n[VERIFIED ROOT]\nv12 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"91e6b0466aa2e662e9c427a2a72032bd3e8d1be4c394f4377f5176f9a87210a7","schema":"livegraph.boolean-propagation.v1","spec_hash":"7da14cd3e701747e537cc8d05bcaad098b4723e5ec488822e9030eadad1a8208","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":14,\"root\":\"v12\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-0913e5b71d0c825d75be","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v10\n- v0e --refutes--> v0c\n- v13 --supports--> v04\n- v04 --supports--> v14\n- v08 --refutes--> v06\n- v0b --refutes--> v13\n- v0e --refutes--> v07\n- v02 --refutes--> v0e\n- v11 --supports--> v15\n- v08 --refutes--> v12\n- v14 --refutes--> v11\n- v03 --supports--> v01\n- v10 --refutes--> v03\n- v0d --supports--> v0a\n- v12 --supports--> v05\n- v0d --supports--> v02\n- v0e --supports--> v0b\n- v01 --refutes--> v0f\n- v0d --supports--> v09\n- v12 --refutes--> v00\n- v0f --refutes--> v08\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3b610cf180da681b63033272a2a678bf6fa63402dc96c7fbe6436f1667595e20","schema":"livegraph.boolean-propagation.v1","spec_hash":"0913e5b71d0c825d75beb836d45b8e9bf1ee80f76773ed07e47bf8fd904caebd","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-de0366e72c99b18c2f79","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v11\n- v0b --supports--> v12\n- v03 --refutes--> v0c\n- v08 --refutes--> v0e\n- v0d --supports--> v08\n- v07 --supports--> v15\n- v14 --refutes--> v03\n- v07 --supports--> v01\n- v05 --refutes--> v13\n- v00 --supports--> v0a\n- v07 --supports--> v09\n- v12 --supports--> v04\n- v11 --supports--> v0d\n- v01 --supports--> v0f\n- v03 --refutes--> v05\n- v06 --refutes--> v00\n- v0f --supports--> v06\n- v08 --supports--> v14\n- v01 --supports--> v10\n- v08 --refutes--> v02\n- v05 --supports--> v07\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a2d20d361ff4df33954594c30735db0ee2c4132d177bef5b3e2b87b65000d616","schema":"livegraph.boolean-propagation.v1","spec_hash":"de0366e72c99b18c2f79d414c42c426fb8fe14bfdc60868e5e7359efdd357aed","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-538c6326bfd48e8e2712","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v03\n- v0d --supports--> v0f\n- v05 --refutes--> v14\n- v14 --refutes--> v15\n- v15 --refutes--> v0e\n- v01 --supports--> v02\n- v15 --refutes--> v04\n- v0b --refutes--> v00\n- v0c --refutes--> v05\n- v10 --refutes--> v06\n- v01 --refutes--> v11\n- v11 --refutes--> v10\n- v04 --supports--> v0d\n- v10 --supports--> v07\n- v06 --supports--> v0b\n- v09 --supports--> v0a\n- v09 --supports--> v12\n- v04 --supports--> v08\n- v15 --supports--> v13\n- v00 --refutes--> v0c\n- v0f --supports--> v09\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b7bd02fec1e523010651e320c414a7f7625ef1e312fe618b0d244a08ef400be7","schema":"livegraph.boolean-propagation.v1","spec_hash":"538c6326bfd48e8e2712afcb16946e9229154773d9635ae8b9fded007ad7f2aa","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-195a1ceb62f85950cc0c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v0a\n- v0b --refutes--> v02\n- v0c --supports--> v14\n- v0f --refutes--> v03\n- v04 --supports--> v00\n- v0c --refutes--> v07\n- v0b --refutes--> v12\n- v13 --supports--> v05\n- v0a --supports--> v08\n- v15 --supports--> v0c\n- v0a --refutes--> v10\n- v09 --supports--> v0e\n- v00 --refutes--> v0b\n- v07 --supports--> v06\n- v09 --supports--> v0d\n- v02 --supports--> v0f\n- v0f --supports--> v13\n- v10 --supports--> v01\n- v0d --refutes--> v15\n- v10 --supports--> v11\n- v07 --refutes--> v04\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e61b67848918a25381592dda6afdf7c9a5225d2d722c22a360a89daf719d7198","schema":"livegraph.boolean-propagation.v1","spec_hash":"195a1ceb62f85950cc0c93c015ce99aaa8a9b4c252c89ad5e4ba5e857a502488","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-12a5bb6aa3e925565a4b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v0a\n- v14 --supports--> v09\n- v08 --supports--> v02\n- v06 --supports--> v04\n- v15 --refutes--> v00\n- v00 --supports--> v12\n- v0e --refutes--> v13\n- v05 --supports--> v08\n- v0e --refutes--> v03\n- v05 --refutes--> v07\n- v03 --refutes--> v0b\n- v0e --refutes--> v11\n- v02 --refutes--> v14\n- v14 --refutes--> v0c\n- v06 --refutes--> v0f\n- v0c --supports--> v01\n- v01 --refutes--> v0d\n- v0b --refutes--> v15\n- v00 --refutes--> v05\n- v0d --refutes--> v06\n- v06 --supports--> v10\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"21f4d6accc86c2395948f26a9a39bf869e35604654298c2f3b7a715d381eef36","schema":"livegraph.boolean-propagation.v1","spec_hash":"12a5bb6aa3e925565a4b751b3a7f4abbb5f87b3c72fd78f201e90ccb0e3da037","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-1a47c1fe75fdfa8a5588","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v0e\n- v05 --refutes--> v09\n- v10 --supports--> v15\n- v09 --refutes--> v0f\n- v01 --supports--> v04\n- v14 --refutes--> v0b\n- v13 --refutes--> v0d\n- v00 --supports--> v13\n- v10 --supports--> v03\n- v0d --supports--> v0a\n- v08 --refutes--> v02\n- v0c --supports--> v00\n- v02 --supports--> v0c\n- v0d --refutes--> v11\n- v0b --refutes--> v10\n- v00 --refutes--> v12\n- v0b --supports--> v06\n- v04 --supports--> v05\n- v0d --supports--> v07\n- v0f --refutes--> v14\n- v10 --supports--> v08\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ae5c25fa192909dd683a4e9d526d91a78edcafe7f2b373e466213280f99a7890","schema":"livegraph.boolean-propagation.v1","spec_hash":"1a47c1fe75fdfa8a5588dc43393e0795dc94ce4f41ce8dcb17062e59effac159","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-108e01c2ecd4093a1fed","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v13\n- v00 --refutes--> v12\n- v00 --supports--> v10\n- v00 --supports--> v03\n- v0d --refutes--> v0e\n- v14 --refutes--> v07\n- v14 --refutes--> v08\n- v05 --supports--> v0d\n- v0f --supports--> v06\n- v15 --supports--> v05\n- v09 --refutes--> v0b\n- v01 --supports--> v14\n- v14 --supports--> v0c\n- v0e --supports--> v09\n- v13 --refutes--> v15\n- v0c --supports--> v00\n- v0c --refutes--> v11\n- v0b --refutes--> v01\n- v0d --supports--> v02\n- v0d --refutes--> v04\n- v0a --refutes--> v0f\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"44c879bf800b90bdcdd4816dda156b297dec92fdf3e0399dc73218065508090f","schema":"livegraph.boolean-propagation.v1","spec_hash":"108e01c2ecd4093a1fedce1995982eb8b3b939742498fb97337b8b12def36d83","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-18dc69a1a0a4ad5fe367","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v05\n- v13 --supports--> v15\n- v15 --supports--> v02\n- v10 --refutes--> v08\n- v0d --refutes--> v07\n- v0e --refutes--> v03\n- v15 --refutes--> v0a\n- v0c --refutes--> v12\n- v08 --supports--> v14\n- v00 --supports--> v01\n- v01 --supports--> v0e\n- v0d --supports--> v0b\n- v09 --refutes--> v04\n- v11 --supports--> v0c\n- v0c --supports--> v00\n- v0b --refutes--> v11\n- v0e --supports--> v09\n- v0f --supports--> v13\n- v08 --supports--> v0f\n- v11 --refutes--> v06\n- v02 --supports--> v0d\n\n[VERIFIED ROOT]\nv10 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c1df149d9c2140b9e014b7b7805e87ab0fd4211fbf5046a870b8ed5b9b6f7175","schema":"livegraph.boolean-propagation.v1","spec_hash":"18dc69a1a0a4ad5fe3674c48289ebc1188422c433ae01871bae883d47a9eb8e2","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-ce70a3d9debce4c5d7d6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v07\n- v06 --refutes--> v0d\n- v04 --refutes--> v05\n- v00 --refutes--> v09\n- v0c --supports--> v06\n- v07 --refutes--> v03\n- v04 --refutes--> v11\n- v13 --refutes--> v10\n- v00 --supports--> v0c\n- v0e --supports--> v0f\n- v13 --supports--> v15\n- v04 --refutes--> v01\n- v13 --refutes--> v0b\n- v10 --refutes--> v0e\n- v11 --supports--> v12\n- v12 --supports--> v0a\n- v10 --supports--> v14\n- v02 --refutes--> v00\n- v06 --refutes--> v08\n- v0a --supports--> v02\n- v03 --supports--> v04\n\n[VERIFIED ROOT]\nv13 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"69c223d52ee5fbafe7e70ff45e7657aad425e344dd457eb312e4c8e5a428eecd","schema":"livegraph.boolean-propagation.v1","spec_hash":"ce70a3d9debce4c5d7d6fc1757f6ff6b93d95e2d24593eb22b592c8c0fa36e70","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":14,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-4635290d304fe7aa0a09","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v03\n- v01 --supports--> v02\n- v07 --refutes--> v06\n- v13 --refutes--> v0b\n- v07 --refutes--> v15\n- v10 --supports--> v04\n- v12 --supports--> v11\n- v0c --refutes--> v10\n- v12 --refutes--> v0c\n- v0f --supports--> v13\n- v04 --supports--> v08\n- v0d --refutes--> v0e\n- v0d --supports--> v09\n- v05 --supports--> v14\n- v05 --supports--> v07\n- v03 --refutes--> v05\n- v0c --supports--> v00\n- v0a --refutes--> v0d\n- v15 --supports--> v0f\n- v01 --refutes--> v0a\n- v0b --refutes--> v12\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f41821a4012880a213e605c26b3a5bc5c9faf177e62211fcf2f6da0f6ce465d4","schema":"livegraph.boolean-propagation.v1","spec_hash":"4635290d304fe7aa0a09e3ec2b405b4a7df22708b4dbdd13b5934ca479cd2dff","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-70d7b976b9d531c0b875","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --supports--> v05\n- v06 --refutes--> v0e\n- v14 --supports--> v12\n- v05 --refutes--> v01\n- v0a --supports--> v00\n- v01 --supports--> v0b\n- v03 --supports--> v0a\n- v14 --supports--> v11\n- v07 --supports--> v06\n- v06 --supports--> v10\n- v0b --refutes--> v07\n- v13 --supports--> v03\n- v02 --supports--> v13\n- v00 --supports--> v0d\n- v0f --supports--> v0c\n- v0f --supports--> v14\n- v0f --supports--> v04\n- v0d --supports--> v0f\n- v0e --refutes--> v09\n- v0f --supports--> v08\n- v14 --supports--> v15\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"875dd6da5fc9d802eb7dc905a8b652b300b6027adc295a34fe4bca68e815bb1e","schema":"livegraph.boolean-propagation.v1","spec_hash":"70d7b976b9d531c0b87513ef3383d3eb468d79e1536fb81f824ccacaa343f4c6","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-51fa375646f61204fc49","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v0a\n- v04 --refutes--> v0f\n- v14 --refutes--> v0b\n- v06 --refutes--> v00\n- v0a --supports--> v09\n- v05 --refutes--> v04\n- v0f --supports--> v10\n- v08 --supports--> v11\n- v02 --refutes--> v14\n- v0f --supports--> v0c\n- v0f --refutes--> v03\n- v0a --supports--> v12\n- v00 --refutes--> v0d\n- v11 --refutes--> v13\n- v0a --supports--> v0e\n- v09 --supports--> v08\n- v0d --refutes--> v01\n- v08 --supports--> v07\n- v10 --supports--> v06\n- v14 --refutes--> v05\n- v0a --supports--> v15\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a962502a3fbd1eec434c6cc48981a21825a91fd27e8cca7d5e0253c8f466eec0","schema":"livegraph.boolean-propagation.v1","spec_hash":"51fa375646f61204fc4946fa70762cfff739c4fff0b611af3c0d77db9ebb5c52","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-35af33fbf4b0885def97","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v08\n- v0e --refutes--> v03\n- v09 --refutes--> v07\n- v15 --refutes--> v02\n- v09 --supports--> v0f\n- v0b --refutes--> v09\n- v10 --refutes--> v15\n- v14 --refutes--> v04\n- v04 --refutes--> v0b\n- v03 --refutes--> v0a\n- v10 --refutes--> v06\n- v08 --refutes--> v12\n- v0b --refutes--> v0d\n- v03 --supports--> v00\n- v0c --supports--> v01\n- v02 --refutes--> v14\n- v03 --supports--> v10\n- v08 --refutes--> v11\n- v05 --refutes--> v13\n- v07 --supports--> v0c\n- v01 --refutes--> v05\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3b6da9c3fd59b67bed05ff6aa04d3db25151c1b63c0d35a049c88fd410412946","schema":"livegraph.boolean-propagation.v1","spec_hash":"35af33fbf4b0885def97878a538909207c0f748ff74a5472709b4b25a8d84b93","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-dbfe856b819db945511a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v04\n- v08 --refutes--> v13\n- v0e --refutes--> v08\n- v0d --refutes--> v02\n- v12 --refutes--> v0e\n- v0e --refutes--> v14\n- v05 --refutes--> v15\n- v10 --supports--> v12\n- v0f --refutes--> v05\n- v11 --supports--> v03\n- v00 --supports--> v01\n- v13 --refutes--> v0c\n- v0b --supports--> v11\n- v0d --supports--> v06\n- v07 --supports--> v09\n- v07 --supports--> v0f\n- v03 --supports--> v0d\n- v05 --supports--> v10\n- v0e --supports--> v0a\n- v02 --supports--> v07\n- v13 --refutes--> v00\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ae54bf0d1c7fdba3de5c933b59e0e7cc80a48af4726d1d55fb4ad488f59d9301","schema":"livegraph.boolean-propagation.v1","spec_hash":"dbfe856b819db945511ad2b05218823e0db7028bb04773d3568230c55fb0dff1","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":14,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-bf1b96fddefacf531c49","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v03\n- v12 --supports--> v0a\n- v14 --supports--> v08\n- v0b --refutes--> v04\n- v06 --supports--> v11\n- v0c --supports--> v12\n- v06 --refutes--> v10\n- v0a --supports--> v02\n- v08 --refutes--> v0b\n- v0b --refutes--> v15\n- v01 --refutes--> v0e\n- v06 --refutes--> v00\n- v01 --refutes--> v0f\n- v06 --supports--> v13\n- v03 --supports--> v14\n- v07 --supports--> v05\n- v01 --supports--> v0d\n- v08 --supports--> v09\n- v15 --supports--> v0c\n- v02 --supports--> v06\n- v0d --refutes--> v07\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0a52c5badea6fd664d41a3db513ec211c7061656fcd5e8de9b9087888ebbb78c","schema":"livegraph.boolean-propagation.v1","spec_hash":"bf1b96fddefacf531c492f6ced0e6430c3055c98ad38a1aa950775a5a64782a2","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-da39a5f4f92e80305dfc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v0b\n- v10 --supports--> v04\n- v03 --refutes--> v12\n- v03 --supports--> v08\n- v14 --refutes--> v13\n- v14 --supports--> v0d\n- v12 --refutes--> v05\n- v0f --supports--> v0c\n- v12 --refutes--> v07\n- v09 --supports--> v10\n- v00 --supports--> v0a\n- v0c --refutes--> v15\n- v15 --supports--> v11\n- v15 --supports--> v06\n- v13 --supports--> v00\n- v04 --refutes--> v01\n- v07 --refutes--> v0f\n- v01 --refutes--> v0e\n- v0a --supports--> v03\n- v10 --refutes--> v02\n- v01 --supports--> v14\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"24f811778e4ceba3c73fb1971abac2a13c8b5c923a77fd5794bd23b2550717aa","schema":"livegraph.boolean-propagation.v1","spec_hash":"da39a5f4f92e80305dfc5723808a42c7abc3b3edb512efa3db6915a39bc54687","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-6ae95e7a57a9def62069","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v09\n- v0a --refutes--> v04\n- v0d --supports--> v0b\n- v14 --supports--> v12\n- v0b --refutes--> v06\n- v00 --supports--> v14\n- v08 --refutes--> v00\n- v11 --refutes--> v05\n- v0f --refutes--> v08\n- v11 --supports--> v0d\n- v0a --supports--> v10\n- v04 --refutes--> v02\n- v13 --refutes--> v0f\n- v08 --refutes--> v0c\n- v0d --supports--> v0e\n- v04 --refutes--> v07\n- v12 --supports--> v15\n- v07 --supports--> v03\n- v01 --refutes--> v0a\n- v15 --refutes--> v11\n- v0b --supports--> v01\n\n[VERIFIED ROOT]\nv13 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0913e0944dfb1c21b85744ce6e79203a6d3719f10b818657646d21e67e160aa5","schema":"livegraph.boolean-propagation.v1","spec_hash":"6ae95e7a57a9def620696a9571e5142361ceb9c21424a38afc1c8a8efbf5a31c","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-a5e763974c49984cbafb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v0d\n- v04 --refutes--> v02\n- v0e --refutes--> v06\n- v01 --supports--> v03\n- v01 --supports--> v12\n- v14 --refutes--> v15\n- v11 --supports--> v0b\n- v0c --supports--> v00\n- v0b --supports--> v01\n- v0d --refutes--> v14\n- v10 --supports--> v05\n- v12 --refutes--> v0e\n- v0b --supports--> v09\n- v0d --supports--> v08\n- v01 --refutes--> v0a\n- v04 --refutes--> v07\n- v07 --supports--> v11\n- v06 --supports--> v0c\n- v14 --refutes--> v0f\n- v06 --refutes--> v13\n- v05 --supports--> v04\n\n[VERIFIED ROOT]\nv10 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c22dc79229f9b164f0cd289cd229520b65a644246d4887201c7ab8d274127193","schema":"livegraph.boolean-propagation.v1","spec_hash":"a5e763974c49984cbafbf219732c178a3abcefb74d4fdb6287099699ddc6c6e9","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-9aaaac5746d410ff2185","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v03\n- v09 --supports--> v04\n- v0c --refutes--> v07\n- v07 --refutes--> v05\n- v11 --supports--> v0e\n- v06 --supports--> v0f\n- v0f --refutes--> v01\n- v02 --supports--> v13\n- v10 --supports--> v08\n- v0b --refutes--> v10\n- v05 --refutes--> v0b\n- v0b --refutes--> v0a\n- v11 --supports--> v09\n- v07 --supports--> v00\n- v10 --refutes--> v12\n- v0d --refutes--> v02\n- v04 --supports--> v0c\n- v14 --supports--> v0d\n- v08 --supports--> v14\n- v02 --supports--> v06\n- v0c --refutes--> v15\n\n[VERIFIED ROOT]\nv11 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e08734500c4beca4dcc6de6f59c7d1e63723c10898746099d1e65ccd3f36baf9","schema":"livegraph.boolean-propagation.v1","spec_hash":"9aaaac5746d410ff21857300b2291ecac6c5eb73ed21f633bf00296b59a2d969","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-6c2e96aee2d766ecc019","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v10\n- v0a --supports--> v14\n- v09 --supports--> v15\n- v06 --refutes--> v04\n- v15 --supports--> v0c\n- v04 --refutes--> v0f\n- v10 --supports--> v08\n- v0a --supports--> v05\n- v08 --refutes--> v11\n- v01 --supports--> v0e\n- v11 --supports--> v00\n- v09 --supports--> v03\n- v15 --supports--> v12\n- v00 --supports--> v09\n- v10 --refutes--> v07\n- v01 --refutes--> v02\n- v0e --supports--> v0d\n- v11 --refutes--> v0b\n- v0c --refutes--> v06\n- v0d --supports--> v0a\n- v01 --supports--> v13\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cd64d74bc3a2990bb03038a653a6061ce7658f373c66b3900130720f94a0aa45","schema":"livegraph.boolean-propagation.v1","spec_hash":"6c2e96aee2d766ecc019caab9dd937d2e959ba42034875f749982fa96291e58d","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-fe1fbd811c139a1015f1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --supports--> v04\n- v0a --refutes--> v0b\n- v15 --refutes--> v0e\n- v07 --supports--> v09\n- v09 --refutes--> v12\n- v04 --supports--> v10\n- v03 --refutes--> v15\n- v13 --refutes--> v0d\n- v01 --refutes--> v00\n- v0b --supports--> v06\n- v09 --supports--> v13\n- v02 --supports--> v0a\n- v10 --supports--> v11\n- v0b --supports--> v01\n- v10 --supports--> v0f\n- v13 --supports--> v14\n- v0e --refutes--> v02\n- v0b --refutes--> v0c\n- v14 --supports--> v08\n- v0f --refutes--> v03\n- v09 --supports--> v05\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"959b687ecc51122c759acc35876c078abce60b53f5ea0d554ddc21d1c11c2f96","schema":"livegraph.boolean-propagation.v1","spec_hash":"fe1fbd811c139a1015f1a2aeb4e5fea80453d7b0da74f3f8cece14d927e6402f","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-774e56a010ea125df575","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v0e\n- v0f --supports--> v05\n- v0b --refutes--> v03\n- v0f --refutes--> v07\n- v06 --refutes--> v01\n- v04 --refutes--> v0b\n- v09 --supports--> v12\n- v10 --refutes--> v0c\n- v0a --refutes--> v10\n- v08 --refutes--> v02\n- v13 --supports--> v0a\n- v10 --supports--> v11\n- v00 --refutes--> v13\n- v10 --refutes--> v14\n- v0d --supports--> v0f\n- v0c --supports--> v08\n- v09 --supports--> v04\n- v03 --refutes--> v0d\n- v09 --refutes--> v15\n- v06 --refutes--> v09\n- v05 --supports--> v00\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6c69efe745d9506094993ec8cf1fa39ff4f416c34cbfa6d6713ec83a0a8298ad","schema":"livegraph.boolean-propagation.v1","spec_hash":"774e56a010ea125df575207fc233785443b5c52a6093eb3db35b45417388b351","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":14,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-d2eb4408ea67c9295017","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v04\n- v13 --refutes--> v10\n- v14 --refutes--> v02\n- v0b --supports--> v13\n- v04 --supports--> v0f\n- v00 --supports--> v06\n- v00 --supports--> v05\n- v11 --supports--> v0e\n- v06 --supports--> v0b\n- v0a --refutes--> v14\n- v0d --refutes--> v12\n- v11 --refutes--> v03\n- v12 --supports--> v11\n- v0f --supports--> v00\n- v14 --refutes--> v01\n- v0a --refutes--> v09\n- v15 --supports--> v0a\n- v15 --refutes--> v07\n- v03 --supports--> v15\n- v13 --refutes--> v0d\n- v06 --refutes--> v0c\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b20a3abe35c9dcc4d68c1ba294427620645afd241fffc712f3e54a33b0501b41","schema":"livegraph.boolean-propagation.v1","spec_hash":"d2eb4408ea67c9295017159ee705f6d20342816d120c62d9c638ad72172f83bf","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-0eb9232bd7c99bbc6c34","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v0d\n- v0b --refutes--> v0f\n- v0a --refutes--> v15\n- v15 --refutes--> v07\n- v10 --refutes--> v0b\n- v03 --refutes--> v0c\n- v05 --supports--> v09\n- v04 --supports--> v00\n- v0d --refutes--> v02\n- v00 --supports--> v14\n- v02 --supports--> v13\n- v0f --refutes--> v11\n- v04 --supports--> v12\n- v0b --refutes--> v01\n- v0b --refutes--> v0e\n- v07 --refutes--> v03\n- v02 --supports--> v04\n- v06 --supports--> v0a\n- v0d --refutes--> v08\n- v0f --supports--> v06\n- v03 --refutes--> v05\n\n[VERIFIED ROOT]\nv10 = FALSE\n\n[TARGET]\nv14\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b5986ff65c0426b768a83df3ef543ecd595664994ca2612990270dac8b93abbb","schema":"livegraph.boolean-propagation.v1","spec_hash":"0eb9232bd7c99bbc6c343c53687464b82d217b13b7d334d228713b050c0a6e0b","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-0cd2549296f7327a850d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v11\n- v05 --supports--> v06\n- v01 --refutes--> v07\n- v12 --refutes--> v0e\n- v02 --refutes--> v05\n- v0e --supports--> v09\n- v04 --supports--> v0a\n- v03 --supports--> v02\n- v0a --refutes--> v03\n- v11 --refutes--> v12\n- v0e --supports--> v14\n- v03 --refutes--> v00\n- v05 --supports--> v0b\n- v0a --supports--> v10\n- v0d --supports--> v13\n- v0e --refutes--> v04\n- v08 --supports--> v0d\n- v15 --refutes--> v08\n- v07 --refutes--> v0f\n- v01 --refutes--> v0c\n- v05 --supports--> v15\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2bc2452283a730a3400ab5dc735a2adced6d3cbee67a01ad9e17ae43941ea164","schema":"livegraph.boolean-propagation.v1","spec_hash":"0cd2549296f7327a850da099bf9f13c3eb6ee66b77c655b9bcd9344e4a5abd74","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-5eb35014690faa6e22a5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v0c\n- v0c --refutes--> v15\n- v0c --supports--> v08\n- v07 --refutes--> v09\n- v00 --refutes--> v0f\n- v0c --supports--> v13\n- v11 --supports--> v0a\n- v10 --supports--> v03\n- v03 --refutes--> v02\n- v12 --refutes--> v14\n- v03 --supports--> v05\n- v00 --refutes--> v0b\n- v01 --refutes--> v06\n- v0a --supports--> v00\n- v14 --refutes--> v04\n- v03 --supports--> v12\n- v11 --supports--> v0d\n- v08 --refutes--> v10\n- v09 --supports--> v01\n- v0b --refutes--> v07\n- v01 --supports--> v0e\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8727c5e9f1ad1ec79abdb4d76022df5c85891ae25646601730114e481d5a43ad","schema":"livegraph.boolean-propagation.v1","spec_hash":"5eb35014690faa6e22a526d344990ce53362da3be68a35ea1135637a6b5a85ba","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-9165784b13ea844a6bac","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v06\n- v0f --refutes--> v11\n- v08 --refutes--> v0d\n- v15 --refutes--> v12\n- v08 --refutes--> v03\n- v01 --supports--> v00\n- v14 --supports--> v08\n- v11 --supports--> v04\n- v06 --refutes--> v05\n- v15 --supports--> v10\n- v01 --refutes--> v0f\n- v08 --supports--> v01\n- v0b --refutes--> v15\n- v09 --refutes--> v13\n- v0a --refutes--> v0e\n- v0b --refutes--> v02\n- v09 --refutes--> v0a\n- v06 --refutes--> v09\n- v04 --supports--> v0c\n- v0c --supports--> v07\n- v0e --supports--> v14\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"39cf90dbc39babf5b001f60a850b8236d9dc42d0030bdb197481e391c0e75b35","schema":"livegraph.boolean-propagation.v1","spec_hash":"9165784b13ea844a6bac649637407898cbfe09c5de33bcad1470767134d2cf59","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-42dd57496a56ea432545","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v10\n- v0e --supports--> v0c\n- v12 --supports--> v08\n- v14 --refutes--> v01\n- v09 --supports--> v05\n- v15 --supports--> v04\n- v01 --supports--> v09\n- v12 --refutes--> v0b\n- v12 --refutes--> v11\n- v0c --supports--> v14\n- v05 --supports--> v13\n- v0d --supports--> v0e\n- v04 --refutes--> v0d\n- v00 --supports--> v0a\n- v02 --refutes--> v0f\n- v0b --supports--> v15\n- v04 --refutes--> v07\n- v00 --supports--> v02\n- v0f --refutes--> v12\n- v0b --supports--> v06\n- v0c --supports--> v03\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f3901cf216da358509ebfdddd39aaf668eb0fa8da37f0fd51e4d3742789b7429","schema":"livegraph.boolean-propagation.v1","spec_hash":"42dd57496a56ea432545a2150c59963f41c06bffe40951ce92192a0cceccd6af","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-3bf73f1d52e797bdb64f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v0b\n- v08 --refutes--> v0d\n- v00 --refutes--> v10\n- v15 --supports--> v09\n- v15 --supports--> v05\n- v12 --supports--> v03\n- v13 --supports--> v15\n- v0d --supports--> v0f\n- v05 --supports--> v14\n- v08 --supports--> v06\n- v00 --supports--> v04\n- v0a --refutes--> v0c\n- v13 --supports--> v11\n- v0f --refutes--> v00\n- v0e --supports--> v13\n- v02 --supports--> v08\n- v0c --refutes--> v07\n- v15 --refutes--> v01\n- v12 --refutes--> v0e\n- v07 --refutes--> v02\n- v04 --refutes--> v12\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv14\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"54dd673c28a313c18ab04ac493586aa9787615570ae1aaffe144de194d593ea5","schema":"livegraph.boolean-propagation.v1","spec_hash":"3bf73f1d52e797bdb64f3df876b843f2de5d0efce14a3a6a6c4d29f98694f356","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-9161cb82233fcee983f6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v05\n- v14 --supports--> v15\n- v08 --refutes--> v06\n- v0b --supports--> v14\n- v12 --refutes--> v0e\n- v15 --refutes--> v12\n- v00 --refutes--> v0b\n- v05 --refutes--> v01\n- v0c --supports--> v00\n- v12 --supports--> v10\n- v06 --refutes--> v0d\n- v09 --supports--> v02\n- v12 --supports--> v0f\n- v0e --refutes--> v11\n- v05 --refutes--> v09\n- v07 --supports--> v04\n- v06 --supports--> v07\n- v0e --refutes--> v0a\n- v04 --refutes--> v03\n- v0e --supports--> v08\n- v03 --refutes--> v13\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b4d93b016c30366a965aace3ec71ec7d3746cc32779a1dd3a5f2d8b20760469e","schema":"livegraph.boolean-propagation.v1","spec_hash":"9161cb82233fcee983f603173384918d2de2bd66a840a0296565575890efe3e4","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-a0cb69c619d18d6e56f9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v00\n- v11 --supports--> v12\n- v0b --refutes--> v10\n- v03 --supports--> v13\n- v12 --refutes--> v07\n- v09 --supports--> v0b\n- v0b --supports--> v0d\n- v0a --supports--> v14\n- v02 --refutes--> v05\n- v0d --supports--> v11\n- v06 --supports--> v01\n- v13 --supports--> v0c\n- v0e --refutes--> v06\n- v12 --supports--> v08\n- v06 --refutes--> v0f\n- v14 --refutes--> v0e\n- v13 --supports--> v0a\n- v0f --supports--> v02\n- v02 --refutes--> v04\n- v05 --supports--> v09\n- v09 --refutes--> v15\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c7e92dfaa81bbf7d83db1047f91764641d425aa942575191e677e86653fc2736","schema":"livegraph.boolean-propagation.v1","spec_hash":"a0cb69c619d18d6e56f937ab1df024de8171e8a6b165ae8a20b77320e67f4016","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-17670bdc181d6bf72d79","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v0a\n- v0f --refutes--> v00\n- v0e --refutes--> v10\n- v07 --supports--> v04\n- v01 --refutes--> v0d\n- v03 --supports--> v11\n- v10 --supports--> v02\n- v08 --refutes--> v15\n- v04 --supports--> v03\n- v11 --refutes--> v0e\n- v12 --refutes--> v07\n- v10 --supports--> v01\n- v13 --refutes--> v08\n- v0d --refutes--> v14\n- v14 --supports--> v13\n- v14 --supports--> v05\n- v07 --refutes--> v09\n- v00 --refutes--> v12\n- v12 --refutes--> v06\n- v00 --supports--> v0b\n- v07 --refutes--> v0c\n\n[VERIFIED ROOT]\nv0f = FALSE\n\n[TARGET]\nv15\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ed48069266495927e801bef53c7b6ff26fc48a86c6fecb6bdc08468960115e56","schema":"livegraph.boolean-propagation.v1","spec_hash":"17670bdc181d6bf72d79af9fac63d018184a45cfc31b94fe3c94a7aca025f512","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-4729df43ec736347e310","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v0d\n- v08 --refutes--> v0c\n- v09 --supports--> v04\n- v15 --refutes--> v01\n- v00 --refutes--> v05\n- v09 --refutes--> v07\n- v0c --supports--> v0b\n- v03 --supports--> v08\n- v05 --refutes--> v10\n- v05 --refutes--> v0e\n- v0f --refutes--> v15\n- v02 --refutes--> v12\n- v0c --refutes--> v09\n- v02 --supports--> v13\n- v01 --supports--> v06\n- v0f --refutes--> v11\n- v0f --supports--> v14\n- v06 --supports--> v00\n- v0a --supports--> v02\n- v10 --supports--> v0a\n- v04 --refutes--> v0f\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6cb04883e39b6abaae7783820381ea7417d5db6912c309690932cbf44755ff0e","schema":"livegraph.boolean-propagation.v1","spec_hash":"4729df43ec736347e310ad8592ff1d7fa1a3e920f6c5172c9b7ac112ececb723","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-c07d3823381a1ccb4ce3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v00\n- v05 --supports--> v0d\n- v06 --refutes--> v0c\n- v08 --supports--> v06\n- v0c --refutes--> v0e\n- v03 --refutes--> v0f\n- v05 --refutes--> v03\n- v08 --refutes--> v01\n- v0a --supports--> v14\n- v10 --supports--> v13\n- v12 --refutes--> v0b\n- v09 --supports--> v08\n- v13 --supports--> v04\n- v0a --supports--> v09\n- v0e --supports--> v05\n- v12 --supports--> v11\n- v13 --refutes--> v12\n- v0b --refutes--> v0a\n- v02 --supports--> v15\n- v0a --refutes--> v07\n- v15 --refutes--> v10\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"63a984b8701d551e0f0b63bd93c4fba4e3832e4dc819d0ca333cd84d51d3331f","schema":"livegraph.boolean-propagation.v1","spec_hash":"c07d3823381a1ccb4ce3d3bf55018472bb745071eacc0a42597d00a1fcbd04a4","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-5d429bf0dc2cc735eac1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --supports--> v07\n- v15 --supports--> v03\n- v13 --refutes--> v05\n- v00 --supports--> v04\n- v0d --supports--> v11\n- v0f --supports--> v13\n- v0c --supports--> v09\n- v0b --supports--> v08\n- v00 --supports--> v15\n- v01 --supports--> v12\n- v03 --supports--> v06\n- v0a --supports--> v02\n- v0c --supports--> v10\n- v01 --refutes--> v0d\n- v0e --refutes--> v00\n- v05 --refutes--> v0e\n- v0a --refutes--> v0f\n- v00 --supports--> v14\n- v0b --refutes--> v0a\n- v07 --supports--> v0c\n- v0c --supports--> v0b\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"15a162e57def84d4464f42efce10607a5c368d4c28b0d86064ea9b1ca2affe1c","schema":"livegraph.boolean-propagation.v1","spec_hash":"5d429bf0dc2cc735eac10a57ac10f51e2690021ab05e4bc1cfde6a9d6a0e9a18","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-af613d9687b1125fbbfe","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v07\n- v07 --supports--> v0b\n- v08 --supports--> v0d\n- v04 --supports--> v09\n- v06 --supports--> v10\n- v12 --supports--> v01\n- v0a --supports--> v13\n- v14 --supports--> v0f\n- v08 --supports--> v14\n- v04 --supports--> v15\n- v10 --supports--> v0a\n- v06 --refutes--> v00\n- v01 --supports--> v11\n- v13 --refutes--> v12\n- v11 --supports--> v0c\n- v12 --supports--> v05\n- v09 --refutes--> v06\n- v07 --supports--> v03\n- v0c --supports--> v08\n- v11 --refutes--> v0e\n- v0a --refutes--> v02\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"72f7ad42332a9fbb453c237f3c11f5074d0be3b0cd98e147d4a26bf273e120b8","schema":"livegraph.boolean-propagation.v1","spec_hash":"af613d9687b1125fbbfe91882e797991054ced4770d83522f54305eaeca60da4","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-5ee47c66cdc45609dd1d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v13\n- v03 --supports--> v01\n- v0f --refutes--> v0e\n- v0c --refutes--> v02\n- v00 --supports--> v14\n- v0c --refutes--> v06\n- v0d --refutes--> v00\n- v09 --refutes--> v0d\n- v13 --refutes--> v05\n- v11 --refutes--> v07\n- v10 --supports--> v03\n- v0e --supports--> v04\n- v12 --supports--> v11\n- v0e --supports--> v12\n- v0a --supports--> v0f\n- v14 --supports--> v0a\n- v13 --supports--> v15\n- v10 --supports--> v0b\n- v11 --supports--> v08\n- v15 --refutes--> v10\n- v03 --supports--> v09\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cd8617b0b34da967b542e75d7efddbb1139b01005aadbb64d666e2993fbeb14e","schema":"livegraph.boolean-propagation.v1","spec_hash":"5ee47c66cdc45609dd1d19124fb234a8fecdb5e2852e68d48d7e69941b8da0bd","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-398cbc6e95ecd86225c8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v09\n- v0d --refutes--> v04\n- v0b --supports--> v12\n- v13 --supports--> v15\n- v05 --refutes--> v06\n- v01 --supports--> v03\n- v04 --supports--> v11\n- v04 --refutes--> v0f\n- v06 --supports--> v0c\n- v04 --supports--> v0e\n- v0c --refutes--> v01\n- v00 --refutes--> v0d\n- v12 --supports--> v07\n- v0b --refutes--> v14\n- v03 --refutes--> v0b\n- v0f --refutes--> v05\n- v0f --supports--> v08\n- v15 --refutes--> v0a\n- v03 --supports--> v02\n- v0a --refutes--> v00\n- v0c --supports--> v10\n\n[VERIFIED ROOT]\nv13 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bd2427cd368e66766b4f678173ab664dd62849a62d55fbc78b60d3b279d73791","schema":"livegraph.boolean-propagation.v1","spec_hash":"398cbc6e95ecd86225c8fdf019dda66e8ffea07852cfebec9c86ee7e508442f0","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v13\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-b6c6bcbbce257af301cf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v0a\n- v04 --supports--> v08\n- v0a --refutes--> v03\n- v15 --supports--> v07\n- v09 --refutes--> v01\n- v07 --refutes--> v00\n- v0b --supports--> v11\n- v03 --supports--> v02\n- v0b --supports--> v09\n- v07 --supports--> v0d\n- v09 --refutes--> v12\n- v14 --refutes--> v0b\n- v06 --refutes--> v10\n- v02 --supports--> v06\n- v05 --refutes--> v0e\n- v06 --supports--> v15\n- v04 --supports--> v13\n- v12 --supports--> v0c\n- v05 --refutes--> v04\n- v04 --supports--> v14\n- v0d --supports--> v0f\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3827f20610f6ff69bbc69efe94f634f9507c86063c8f1ce1397e125f54d855dd","schema":"livegraph.boolean-propagation.v1","spec_hash":"b6c6bcbbce257af301cfb8aefa4fd4cd8cf158466cd7de5dacae1cc472252f7c","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-b0695eea978f9adbe070","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v0b\n- v04 --supports--> v00\n- v0f --supports--> v08\n- v06 --supports--> v11\n- v0f --supports--> v12\n- v0f --supports--> v09\n- v00 --supports--> v0e\n- v00 --supports--> v15\n- v00 --supports--> v03\n- v14 --refutes--> v05\n- v10 --refutes--> v01\n- v01 --supports--> v04\n- v15 --supports--> v0a\n- v0d --refutes--> v06\n- v12 --supports--> v0d\n- v0a --refutes--> v0f\n- v00 --refutes--> v07\n- v0a --refutes--> v13\n- v05 --refutes--> v10\n- v0a --supports--> v02\n- v0c --supports--> v14\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1c2667083e4648505490bdd18c0b9c492d113fea9293bdb641ad3ec0549277c7","schema":"livegraph.boolean-propagation.v1","spec_hash":"b0695eea978f9adbe070552753a7bdf571808479ef86e86d09d7ae6efda2a78a","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-282302e7fef90ce40981","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v0d\n- v11 --supports--> v13\n- v12 --refutes--> v00\n- v15 --supports--> v04\n- v01 --supports--> v0f\n- v04 --supports--> v02\n- v0b --supports--> v11\n- v12 --refutes--> v07\n- v15 --supports--> v06\n- v03 --refutes--> v01\n- v0c --supports--> v12\n- v0f --supports--> v0b\n- v11 --refutes--> v10\n- v09 --refutes--> v08\n- v11 --refutes--> v09\n- v04 --supports--> v14\n- v0e --refutes--> v0c\n- v03 --supports--> v05\n- v0d --refutes--> v03\n- v08 --supports--> v0e\n- v0f --refutes--> v0a\n\n[VERIFIED ROOT]\nv15 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2c107bf2ce032b2b88e7dfa646cd973b3342fc9ff2c92e2aca0a58c867a89313","schema":"livegraph.boolean-propagation.v1","spec_hash":"282302e7fef90ce4098161a371c1443b156ddbd1c4f81ceda966d8daf4f4f7d0","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v15\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-7b2327be2d77698d2f52","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v0a\n- v0d --supports--> v13\n- v12 --refutes--> v0f\n- v05 --supports--> v03\n- v08 --supports--> v0e\n- v01 --supports--> v00\n- v14 --refutes--> v08\n- v0e --supports--> v01\n- v0f --supports--> v14\n- v05 --refutes--> v02\n- v10 --refutes--> v12\n- v11 --refutes--> v09\n- v03 --supports--> v11\n- v0c --refutes--> v15\n- v01 --refutes--> v04\n- v13 --supports--> v0b\n- v11 --supports--> v0c\n- v0c --refutes--> v0d\n- v14 --refutes--> v07\n- v04 --refutes--> v05\n- v14 --refutes--> v06\n\n[VERIFIED ROOT]\nv10 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"db0e37943dc276512d80fd0ade4e6ca9942f98ecd179f9032ed853d7790ea194","schema":"livegraph.boolean-propagation.v1","spec_hash":"7b2327be2d77698d2f52ad4b66e85835239cc4fdec7493704dc39c26f0054c7f","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-de3a2c90b9fd767083a1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --supports--> v00\n- v00 --supports--> v0b\n- v02 --refutes--> v03\n- v08 --supports--> v0d\n- v06 --refutes--> v13\n- v0d --refutes--> v02\n- v00 --supports--> v11\n- v07 --supports--> v0a\n- v0e --refutes--> v0c\n- v07 --refutes--> v06\n- v11 --supports--> v0f\n- v0d --supports--> v09\n- v04 --refutes--> v05\n- v14 --refutes--> v04\n- v04 --refutes--> v08\n- v0d --supports--> v10\n- v01 --refutes--> v07\n- v03 --refutes--> v0e\n- v0d --supports--> v12\n- v0f --supports--> v01\n- v0e --refutes--> v15\n\n[VERIFIED ROOT]\nv14 = TRUE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c47b2edc583f4be764f8a12c4e90b1ec87eaecf67307d80be124b569c2e63bbf","schema":"livegraph.boolean-propagation.v1","spec_hash":"de3a2c90b9fd767083a187ee968b8101c010a1e948a7839a8f15d1b865ae85e2","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v14\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-4889bdd1e195767ad07c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v00\n- v03 --supports--> v11\n- v04 --refutes--> v0f\n- v0f --refutes--> v0e\n- v0b --refutes--> v13\n- v14 --supports--> v08\n- v02 --refutes--> v05\n- v08 --refutes--> v04\n- v0e --supports--> v12\n- v0c --supports--> v01\n- v13 --supports--> v0c\n- v02 --supports--> v0b\n- v0a --supports--> v06\n- v13 --refutes--> v10\n- v04 --supports--> v09\n- v0d --supports--> v0a\n- v0c --supports--> v03\n- v0d --supports--> v07\n- v00 --refutes--> v02\n- v03 --supports--> v14\n- v0e --supports--> v15\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"608126b933a940c2f02a1004861a20d207f83e37420c64bf9413916dee8e3c76","schema":"livegraph.boolean-propagation.v1","spec_hash":"4889bdd1e195767ad07c5deab4e61a42a4bebb4aea755d4c7d1787b6218d5e15","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-d0f3b5c51a725c05565c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v12\n- v00 --supports--> v13\n- v08 --refutes--> v01\n- v08 --refutes--> v0e\n- v15 --supports--> v14\n- v02 --refutes--> v0c\n- v13 --refutes--> v08\n- v06 --refutes--> v0b\n- v0f --supports--> v00\n- v02 --refutes--> v0f\n- v15 --supports--> v02\n- v09 --supports--> v07\n- v08 --supports--> v10\n- v10 --refutes--> v0a\n- v0a --supports--> v09\n- v11 --supports--> v05\n- v07 --refutes--> v06\n- v12 --refutes--> v15\n- v08 --refutes--> v03\n- v0f --supports--> v0d\n- v07 --supports--> v04\n\n[VERIFIED ROOT]\nv11 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"23de0d667ea45dca44876c14bb2e41c939704f7b94cda34e688884312114e54a","schema":"livegraph.boolean-propagation.v1","spec_hash":"d0f3b5c51a725c05565cc33ff05e8c9362634e229d155bc32716bc674e004795","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-4764cf93a51032297cc7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v07\n- v09 --supports--> v12\n- v0f --supports--> v0b\n- v13 --refutes--> v06\n- v0e --supports--> v0f\n- v0c --refutes--> v0d\n- v01 --refutes--> v08\n- v01 --refutes--> v02\n- v07 --refutes--> v13\n- v0f --refutes--> v09\n- v00 --refutes--> v11\n- v08 --refutes--> v14\n- v13 --refutes--> v01\n- v11 --supports--> v0e\n- v0a --supports--> v15\n- v08 --supports--> v10\n- v07 --supports--> v04\n- v0d --refutes--> v03\n- v10 --supports--> v00\n- v0a --refutes--> v05\n- v0a --supports--> v0c\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9b915537f90b522504812e994e65139e7156b1d825cc8c8bc0587e179ce3f208","schema":"livegraph.boolean-propagation.v1","spec_hash":"4764cf93a51032297cc75db208296a4601cbb35ee5fe18d3b52a242028b220bc","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":14,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-95972b7b2f92cb3f65ef","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --refutes--> v0e\n- v0c --refutes--> v05\n- v08 --supports--> v02\n- v13 --supports--> v01\n- v0d --refutes--> v07\n- v0b --supports--> v09\n- v10 --refutes--> v08\n- v01 --refutes--> v0d\n- v14 --supports--> v0f\n- v0b --supports--> v06\n- v01 --refutes--> v0a\n- v02 --refutes--> v0b\n- v01 --refutes--> v03\n- v0f --refutes--> v04\n- v0f --refutes--> v12\n- v0f --supports--> v00\n- v11 --refutes--> v15\n- v05 --refutes--> v10\n- v12 --refutes--> v11\n- v0d --refutes--> v14\n- v0b --supports--> v13\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f7efed1fa80ccfb4ef3457c601271b5ee1341bf355ff40d897821351b55b19cc","schema":"livegraph.boolean-propagation.v1","spec_hash":"95972b7b2f92cb3f65efab24bd7e0ebe08f642ca0eb4265bae6dbe8602162493","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-c76372ad80be62609210","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v06\n- v0b --refutes--> v07\n- v10 --refutes--> v0a\n- v02 --refutes--> v14\n- v0f --refutes--> v03\n- v08 --refutes--> v0e\n- v04 --refutes--> v13\n- v15 --refutes--> v04\n- v05 --supports--> v02\n- v12 --refutes--> v11\n- v14 --refutes--> v12\n- v10 --refutes--> v15\n- v11 --supports--> v08\n- v08 --refutes--> v00\n- v03 --supports--> v10\n- v13 --refutes--> v0b\n- v0b --refutes--> v0d\n- v03 --supports--> v09\n- v15 --supports--> v0c\n- v13 --supports--> v01\n- v00 --supports--> v0f\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"99b234510a0d53eac0708e72bafee757204192594fba01ac0db6e3ba2bd6e955","schema":"livegraph.boolean-propagation.v1","spec_hash":"c76372ad80be626092108767d3a2f0e3d484c8916082de0a34ae80dbf4994363","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-d6ff46d5b5520ba84354","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v00\n- v0e --supports--> v0a\n- v0a --supports--> v0f\n- v06 --refutes--> v09\n- v15 --supports--> v10\n- v0a --refutes--> v07\n- v03 --refutes--> v01\n- v07 --refutes--> v0b\n- v02 --supports--> v15\n- v15 --supports--> v11\n- v0d --refutes--> v06\n- v12 --supports--> v04\n- v04 --refutes--> v08\n- v0b --supports--> v02\n- v10 --refutes--> v03\n- v04 --refutes--> v0c\n- v04 --supports--> v0d\n- v01 --refutes--> v14\n- v05 --refutes--> v12\n- v12 --supports--> v13\n- v06 --supports--> v0e\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv14\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1dd471b175215845dc8104ff0086bb930fefa3494f46c22a8c4b03ddaca82d68","schema":"livegraph.boolean-propagation.v1","spec_hash":"d6ff46d5b5520ba84354b4d3065b47a5088802748be50f3b008d0cc7c9e26140","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-577e2b7594b22e36912b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v07\n- v0b --refutes--> v0a\n- v03 --supports--> v0f\n- v11 --supports--> v0c\n- v11 --refutes--> v13\n- v10 --refutes--> v11\n- v10 --refutes--> v04\n- v0a --refutes--> v02\n- v0f --refutes--> v0b\n- v07 --refutes--> v06\n- v13 --supports--> v03\n- v10 --refutes--> v12\n- v00 --refutes--> v15\n- v0d --supports--> v05\n- v14 --supports--> v08\n- v13 --refutes--> v0e\n- v08 --refutes--> v01\n- v02 --supports--> v14\n- v06 --refutes--> v0d\n- v08 --supports--> v09\n- v08 --refutes--> v00\n\n[VERIFIED ROOT]\nv10 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"228ceb9d1398c4a6da471dd45cd8774639d6c7092baa2ac26539966afdc3ef53","schema":"livegraph.boolean-propagation.v1","spec_hash":"577e2b7594b22e36912b4e5e3541b7a7ec066d46b6d635ea848651c76da64b13","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-dd694b722ecf3a3b0575","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v08\n- v15 --supports--> v0e\n- v12 --supports--> v06\n- v05 --refutes--> v01\n- v06 --refutes--> v15\n- v00 --refutes--> v04\n- v0e --supports--> v0b\n- v02 --refutes--> v0d\n- v08 --supports--> v03\n- v0d --supports--> v12\n- v11 --supports--> v10\n- v04 --supports--> v07\n- v0d --refutes--> v0c\n- v07 --refutes--> v14\n- v14 --supports--> v05\n- v11 --supports--> v00\n- v00 --refutes--> v0f\n- v02 --refutes--> v13\n- v08 --refutes--> v02\n- v0d --refutes--> v0a\n- v0e --supports--> v09\n\n[VERIFIED ROOT]\nv11 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"56e5c72c23d03ff9b7bfd65f4b1afbd9ab9af47a97ec2715bc2f1ff964e346ff","schema":"livegraph.boolean-propagation.v1","spec_hash":"dd694b722ecf3a3b0575be6c4fa2b6697572015891a627bdb4bbf1a06c1f0426","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-bd6f145f258f253fe413","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v10\n- v08 --supports--> v04\n- v0c --refutes--> v05\n- v0a --refutes--> v0f\n- v0b --refutes--> v01\n- v12 --supports--> v09\n- v09 --supports--> v0a\n- v0a --supports--> v0b\n- v01 --supports--> v13\n- v00 --supports--> v0c\n- v0d --refutes--> v03\n- v0d --supports--> v06\n- v0e --refutes--> v12\n- v10 --refutes--> v15\n- v0c --refutes--> v14\n- v15 --refutes--> v11\n- v04 --refutes--> v07\n- v15 --supports--> v00\n- v12 --supports--> v02\n- v01 --refutes--> v0d\n- v0d --supports--> v08\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7cf8891327c8d427e563d95c455c1bd2591a0f0feef0dc75d0cb41e57df89e1e","schema":"livegraph.boolean-propagation.v1","spec_hash":"bd6f145f258f253fe413338f6f261f89ad66a98c09eb422a770b416f7e025192","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-8e1cd8555c63e20016e8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v10\n- v03 --supports--> v04\n- v12 --refutes--> v00\n- v09 --supports--> v08\n- v10 --refutes--> v0e\n- v05 --supports--> v01\n- v11 --refutes--> v09\n- v12 --refutes--> v02\n- v08 --supports--> v13\n- v08 --refutes--> v14\n- v01 --supports--> v0c\n- v08 --refutes--> v03\n- v0b --supports--> v06\n- v07 --refutes--> v12\n- v00 --refutes--> v05\n- v04 --supports--> v0f\n- v0b --refutes--> v11\n- v01 --refutes--> v0a\n- v10 --supports--> v0b\n- v12 --refutes--> v0d\n- v15 --supports--> v07\n\n[VERIFIED ROOT]\nv15 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dea2224f5d5f53703daf32d51dc9e0e421797d0486009ca300b6db0c99268adb","schema":"livegraph.boolean-propagation.v1","spec_hash":"8e1cd8555c63e20016e8134e2a1f841b0ed36e0daf1d8e5f1a0649e5c61fd6a4","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v15\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-a6b8e3fc3f4da80e0185","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v04\n- v0a --supports--> v05\n- v03 --supports--> v02\n- v11 --supports--> v10\n- v0b --refutes--> v15\n- v0a --supports--> v09\n- v00 --supports--> v0a\n- v15 --supports--> v12\n- v12 --refutes--> v06\n- v0c --refutes--> v0f\n- v06 --refutes--> v0d\n- v05 --supports--> v0c\n- v0f --refutes--> v03\n- v00 --supports--> v01\n- v15 --refutes--> v08\n- v02 --refutes--> v13\n- v13 --supports--> v0b\n- v02 --refutes--> v07\n- v04 --supports--> v14\n- v04 --supports--> v11\n- v11 --supports--> v0e\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9786f10964552e92df6adb1cda72cbe49ff95a9968f4983830130bdf33384127","schema":"livegraph.boolean-propagation.v1","spec_hash":"a6b8e3fc3f4da80e0185b529e2726f13a8d3b7a74473ed13e9e3ba8724f8f27a","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-0b1cb9927dcb41ccaeea","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v00\n- v01 --supports--> v05\n- v08 --refutes--> v15\n- v10 --refutes--> v14\n- v0f --supports--> v13\n- v0c --supports--> v08\n- v10 --supports--> v03\n- v12 --refutes--> v0e\n- v0c --supports--> v0d\n- v09 --supports--> v11\n- v0e --refutes--> v06\n- v15 --supports--> v0a\n- v05 --supports--> v09\n- v0e --refutes--> v0b\n- v07 --supports--> v02\n- v01 --supports--> v04\n- v10 --refutes--> v0f\n- v13 --supports--> v07\n- v02 --refutes--> v01\n- v11 --refutes--> v0c\n- v15 --supports--> v12\n\n[VERIFIED ROOT]\nv10 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"860e0016a97dccf818aea71aec9a9087c04d91fd4aabdb836221415e4f84e7d1","schema":"livegraph.boolean-propagation.v1","spec_hash":"0b1cb9927dcb41ccaeeae4bf573b21583d4a83dbb7d10f583624ada9edbdb25b","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-773aeda98a64cc2f9e6f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v00\n- v12 --refutes--> v10\n- v01 --supports--> v0f\n- v0a --supports--> v03\n- v00 --refutes--> v0e\n- v0e --supports--> v14\n- v06 --supports--> v09\n- v0e --refutes--> v04\n- v0d --supports--> v08\n- v03 --refutes--> v01\n- v06 --supports--> v05\n- v02 --supports--> v0c\n- v12 --supports--> v0a\n- v11 --supports--> v0b\n- v14 --supports--> v11\n- v06 --refutes--> v07\n- v15 --supports--> v13\n- v0b --supports--> v0d\n- v01 --supports--> v06\n- v0d --supports--> v02\n- v0c --refutes--> v15\n\n[VERIFIED ROOT]\nv12 = FALSE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"734abe86a85a0babeb5db19fb6dd1d32f0d23cb597437e385f43e738b3f3ff5e","schema":"livegraph.boolean-propagation.v1","spec_hash":"773aeda98a64cc2f9e6f20a8478d055634d5a40fb45e7f3cc9ae4821377ce2d3","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v12\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-bffdb9f8070baf34d48b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v05\n- v04 --supports--> v02\n- v05 --supports--> v0b\n- v15 --supports--> v0e\n- v13 --refutes--> v06\n- v01 --supports--> v14\n- v09 --refutes--> v08\n- v04 --supports--> v07\n- v0a --supports--> v15\n- v10 --refutes--> v0c\n- v12 --refutes--> v03\n- v0b --refutes--> v04\n- v08 --supports--> v12\n- v03 --refutes--> v13\n- v14 --refutes--> v00\n- v09 --refutes--> v0d\n- v09 --refutes--> v11\n- v0e --refutes--> v09\n- v13 --supports--> v01\n- v04 --refutes--> v0a\n- v05 --refutes--> v0f\n\n[VERIFIED ROOT]\nv10 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c042726efcf74a564c3fdbd64e0f30f1a7d869f0d385c6448faccc8100cd6e6c","schema":"livegraph.boolean-propagation.v1","spec_hash":"bffdb9f8070baf34d48b0db10a7ac403777e33cd0d0ec72a85556a643e1e3dbe","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-79f844728a0f1f56c0eb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v0d\n- v00 --supports--> v05\n- v04 --supports--> v0e\n- v12 --supports--> v03\n- v07 --supports--> v0f\n- v0e --supports--> v10\n- v09 --refutes--> v0a\n- v0d --refutes--> v01\n- v11 --supports--> v04\n- v09 --refutes--> v11\n- v08 --supports--> v12\n- v0b --supports--> v07\n- v10 --refutes--> v0c\n- v0b --refutes--> v06\n- v00 --supports--> v0b\n- v10 --refutes--> v13\n- v0d --supports--> v02\n- v0f --refutes--> v15\n- v01 --refutes--> v14\n- v0f --refutes--> v09\n- v01 --refutes--> v08\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"20f68fd86e0e037b0c5538bb62a8ff4d6ed6406f3eb8a625c0ff5a29ab72244b","schema":"livegraph.boolean-propagation.v1","spec_hash":"79f844728a0f1f56c0ebc53c0152eaf5939af9d727501b5dd5cb4cfda6696e7d","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":14,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-cb2def5e945e5a2c52bd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --supports--> v0d\n- v0c --refutes--> v03\n- v05 --refutes--> v0c\n- v01 --refutes--> v05\n- v08 --refutes--> v13\n- v0f --supports--> v01\n- v02 --supports--> v0f\n- v0b --supports--> v02\n- v0d --supports--> v15\n- v13 --supports--> v00\n- v0b --refutes--> v10\n- v07 --supports--> v12\n- v08 --supports--> v0a\n- v13 --supports--> v06\n- v06 --supports--> v0b\n- v12 --refutes--> v0e\n- v06 --refutes--> v04\n- v07 --refutes--> v11\n- v12 --supports--> v08\n- v15 --supports--> v09\n- v15 --refutes--> v07\n\n[VERIFIED ROOT]\nv14 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"53179b37f699669c047ae7941c66f117726bd6bdaffa98eb98c757a9838f2f12","schema":"livegraph.boolean-propagation.v1","spec_hash":"cb2def5e945e5a2c52bd7a760a67c8b7b4fec8c6561c6957d7db42491a81e48b","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v14\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-75db47ce5c7a468c23c2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v03\n- v14 --refutes--> v00\n- v03 --refutes--> v05\n- v11 --supports--> v0a\n- v08 --refutes--> v0f\n- v14 --refutes--> v0e\n- v12 --supports--> v06\n- v15 --refutes--> v0d\n- v02 --supports--> v10\n- v08 --supports--> v07\n- v0b --refutes--> v12\n- v09 --refutes--> v0b\n- v0d --supports--> v11\n- v0b --refutes--> v0c\n- v0a --supports--> v09\n- v13 --supports--> v15\n- v02 --supports--> v14\n- v12 --refutes--> v01\n- v03 --supports--> v02\n- v14 --supports--> v08\n- v07 --refutes--> v04\n\n[VERIFIED ROOT]\nv13 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f767e6b7485680c2ac412b77c10357489ce9b935a40059be5c1138e2b0d57822","schema":"livegraph.boolean-propagation.v1","spec_hash":"75db47ce5c7a468c23c2b1d1b38f0506cd912e9d3929001b393782defffa3229","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v13\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-b9f22474f86d4672dd0b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v0d\n- v09 --refutes--> v03\n- v00 --refutes--> v01\n- v10 --supports--> v0e\n- v09 --refutes--> v0b\n- v07 --supports--> v08\n- v10 --refutes--> v04\n- v05 --refutes--> v0f\n- v01 --refutes--> v05\n- v0f --supports--> v02\n- v15 --refutes--> v00\n- v0b --refutes--> v12\n- v12 --refutes--> v10\n- v0c --refutes--> v15\n- v02 --refutes--> v07\n- v0f --refutes--> v06\n- v0c --refutes--> v14\n- v07 --refutes--> v09\n- v0b --refutes--> v0a\n- v02 --supports--> v13\n- v11 --supports--> v0c\n\n[VERIFIED ROOT]\nv11 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"38b0d98f7331402d6f11384f70b5b1494c6555f9d01bb9c05706752614290a55","schema":"livegraph.boolean-propagation.v1","spec_hash":"b9f22474f86d4672dd0b9c30340754164e407fa94c9ebc68c6bfe3956d16d423","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-270db197aab0dd86f382","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v04\n- v04 --supports--> v00\n- v06 --refutes--> v08\n- v04 --supports--> v14\n- v08 --refutes--> v0a\n- v03 --refutes--> v10\n- v10 --refutes--> v0b\n- v03 --supports--> v15\n- v12 --refutes--> v05\n- v0b --supports--> v09\n- v0e --refutes--> v01\n- v02 --refutes--> v13\n- v08 --refutes--> v03\n- v11 --refutes--> v06\n- v12 --supports--> v0f\n- v0f --supports--> v02\n- v12 --refutes--> v07\n- v02 --refutes--> v0c\n- v0e --refutes--> v12\n- v14 --supports--> v0e\n- v09 --refutes--> v0d\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"23caa595ad1fed978734efbbec56dc03323b455480c47cf43b1408c726b7a62a","schema":"livegraph.boolean-propagation.v1","spec_hash":"270db197aab0dd86f382217925071c4f41c0ee385d6d5735a125b25dea9e7a98","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-410dca414df8d47c7d91","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v0d\n- v07 --refutes--> v15\n- v01 --supports--> v0a\n- v10 --supports--> v06\n- v03 --refutes--> v14\n- v14 --refutes--> v11\n- v01 --refutes--> v13\n- v08 --supports--> v04\n- v0e --refutes--> v09\n- v0d --refutes--> v03\n- v0e --supports--> v00\n- v10 --supports--> v01\n- v0f --refutes--> v05\n- v0a --refutes--> v12\n- v11 --supports--> v08\n- v12 --refutes--> v0f\n- v04 --supports--> v0e\n- v14 --refutes--> v0b\n- v10 --supports--> v0c\n- v11 --supports--> v02\n- v07 --supports--> v10\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c060c502f53970dab019b92747f3e5189aaf7f268b438ca6eb591edcbfb3393c","schema":"livegraph.boolean-propagation.v1","spec_hash":"410dca414df8d47c7d91f3394797412f1d37fe62c70df998091fb803aa58d653","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-5610b2053d6f78b93ee7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v0e\n- v10 --supports--> v12\n- v0e --refutes--> v07\n- v04 --supports--> v02\n- v0b --refutes--> v14\n- v11 --supports--> v04\n- v09 --refutes--> v0d\n- v0e --supports--> v10\n- v13 --supports--> v11\n- v0d --supports--> v05\n- v0c --supports--> v13\n- v0f --supports--> v01\n- v02 --supports--> v0b\n- v10 --supports--> v03\n- v11 --refutes--> v06\n- v15 --refutes--> v09\n- v14 --refutes--> v15\n- v01 --supports--> v0a\n- v0f --refutes--> v00\n- v0c --supports--> v08\n- v01 --refutes--> v0c\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4a72a019db9f59fcc99a8d9eab54b80da796ad250b4f288e0b5aa6bce2157e4e","schema":"livegraph.boolean-propagation.v1","spec_hash":"5610b2053d6f78b93ee7598b0a4ad7a92f9b8773c1b45f126bde468a4cf796f2","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-dc331f218eb42d7e6ec2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --refutes--> v0a\n- v10 --refutes--> v0d\n- v12 --refutes--> v10\n- v15 --refutes--> v00\n- v0b --refutes--> v06\n- v0e --refutes--> v0f\n- v0e --refutes--> v0b\n- v0d --supports--> v0c\n- v13 --refutes--> v07\n- v02 --supports--> v12\n- v09 --supports--> v05\n- v0a --refutes--> v11\n- v09 --refutes--> v08\n- v0a --refutes--> v13\n- v04 --refutes--> v15\n- v13 --refutes--> v0e\n- v05 --refutes--> v01\n- v0c --refutes--> v04\n- v05 --refutes--> v14\n- v01 --refutes--> v02\n- v0b --supports--> v03\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a756dd3dfe22cdd9f75365213e0acdab9c081ac96bc5b994a22e74cc7c27c2a9","schema":"livegraph.boolean-propagation.v1","spec_hash":"dc331f218eb42d7e6ec20758ed56fe47690a3ddb0c269cd8e85bd4615694d98d","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-a4e8daffd13456674799","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v13\n- v12 --refutes--> v0b\n- v12 --refutes--> v02\n- v0f --supports--> v01\n- v07 --supports--> v0f\n- v08 --supports--> v05\n- v0c --refutes--> v15\n- v14 --refutes--> v00\n- v0d --supports--> v12\n- v02 --supports--> v03\n- v0e --refutes--> v10\n- v01 --supports--> v0d\n- v10 --refutes--> v04\n- v15 --refutes--> v06\n- v14 --supports--> v0a\n- v05 --supports--> v07\n- v10 --supports--> v09\n- v14 --supports--> v0c\n- v03 --supports--> v11\n- v15 --refutes--> v08\n- v03 --supports--> v0e\n\n[VERIFIED ROOT]\nv14 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d4861fabb867d316887daf337b5c22a3bb2ed3146d226d7009b3ffd889cd9945","schema":"livegraph.boolean-propagation.v1","spec_hash":"a4e8daffd13456674799dfc44a801c718670e33fd81084fb8f4b61ab22bf7de5","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v14\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-d19e87beaab7da5890d3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v06\n- v12 --refutes--> v0d\n- v14 --refutes--> v10\n- v08 --refutes--> v0a\n- v0e --supports--> v13\n- v14 --refutes--> v02\n- v05 --supports--> v14\n- v06 --supports--> v0c\n- v03 --supports--> v04\n- v04 --refutes--> v05\n- v0e --refutes--> v12\n- v09 --refutes--> v0f\n- v05 --refutes--> v07\n- v10 --refutes--> v15\n- v05 --supports--> v11\n- v15 --refutes--> v08\n- v01 --supports--> v03\n- v00 --refutes--> v01\n- v06 --refutes--> v0e\n- v0f --supports--> v00\n- v0f --supports--> v0b\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"143d4d9ff9da09c5c13783df958c39725ed56db1324d84d292a052f5f82ae2a4","schema":"livegraph.boolean-propagation.v1","spec_hash":"d19e87beaab7da5890d3b33b99660320055a44e8e26d4e4b8eff503883fbbeff","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-bfb51bc9f936ed773ea4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v15\n- v02 --supports--> v14\n- v0f --supports--> v12\n- v10 --supports--> v00\n- v11 --refutes--> v10\n- v02 --refutes--> v0b\n- v0a --refutes--> v06\n- v15 --supports--> v02\n- v09 --supports--> v0c\n- v0c --refutes--> v0a\n- v10 --supports--> v03\n- v09 --supports--> v0e\n- v0f --refutes--> v09\n- v09 --refutes--> v04\n- v0d --refutes--> v08\n- v0a --refutes--> v05\n- v0d --refutes--> v13\n- v05 --refutes--> v11\n- v01 --supports--> v0d\n- v07 --supports--> v01\n- v13 --supports--> v0f\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a21e86b61b072eab299df031b5e06b18e9c66a9090faff953d40f619e8a9e252","schema":"livegraph.boolean-propagation.v1","spec_hash":"bfb51bc9f936ed773ea4000500c9be4630231c69ad1165f93145285eff987509","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-b09802e678cc25c08d6e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v11\n- v03 --refutes--> v0f\n- v0f --refutes--> v07\n- v10 --supports--> v0c\n- v00 --supports--> v15\n- v03 --supports--> v12\n- v0b --supports--> v10\n- v07 --refutes--> v06\n- v14 --refutes--> v00\n- v09 --refutes--> v0b\n- v0f --supports--> v0d\n- v04 --supports--> v0e\n- v15 --supports--> v13\n- v0c --refutes--> v03\n- v15 --supports--> v09\n- v10 --supports--> v01\n- v0b --refutes--> v02\n- v0a --refutes--> v08\n- v0a --refutes--> v04\n- v11 --supports--> v0a\n- v05 --refutes--> v14\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d682ddb7da1bf94fffc080ca5be4732a3beabfd8962b5665d4ef19362c0ad9cd","schema":"livegraph.boolean-propagation.v1","spec_hash":"b09802e678cc25c08d6edecab21526c6b4a5ab5c25219f34eae19bca84267748","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-50043444ae22c161285e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --refutes--> v01\n- v13 --refutes--> v05\n- v08 --supports--> v0a\n- v0a --supports--> v15\n- v03 --refutes--> v0e\n- v06 --supports--> v10\n- v0f --refutes--> v04\n- v01 --refutes--> v0f\n- v06 --refutes--> v03\n- v0b --refutes--> v13\n- v0f --refutes--> v07\n- v11 --refutes--> v00\n- v06 --supports--> v0d\n- v0e --supports--> v0c\n- v01 --supports--> v02\n- v00 --refutes--> v09\n- v08 --refutes--> v14\n- v04 --supports--> v0b\n- v00 --refutes--> v08\n- v12 --refutes--> v06\n- v0e --supports--> v11\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ef25b050f914f67638303d10511adf648d3fb959a185e17a4ad9a9136adfd935","schema":"livegraph.boolean-propagation.v1","spec_hash":"50043444ae22c161285e976db8efd06600eb5f0515f6cc45fff51501682b0d5b","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-e962ccbef9541fd8f4a5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v02\n- v08 --refutes--> v03\n- v13 --supports--> v10\n- v10 --supports--> v06\n- v13 --supports--> v04\n- v13 --refutes--> v09\n- v06 --supports--> v05\n- v0b --refutes--> v13\n- v0f --supports--> v08\n- v0a --refutes--> v11\n- v01 --supports--> v0c\n- v14 --supports--> v0b\n- v08 --supports--> v14\n- v02 --supports--> v0a\n- v05 --supports--> v12\n- v01 --refutes--> v07\n- v05 --supports--> v0d\n- v11 --refutes--> v01\n- v0f --refutes--> v15\n- v08 --supports--> v0e\n- v00 --refutes--> v0f\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a667e48df11f016b352b93a6c1d9c41b461bb13f18d30ef0c35baef472d8e97b","schema":"livegraph.boolean-propagation.v1","spec_hash":"e962ccbef9541fd8f4a5dca0a6e7b35e7e040bcf331b04c1b6e31d0a5ee8feca","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-13f78a4c58d00e4c5fcc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v08\n- v14 --refutes--> v0a\n- v11 --refutes--> v09\n- v13 --refutes--> v05\n- v15 --supports--> v0e\n- v0f --supports--> v13\n- v07 --refutes--> v12\n- v07 --supports--> v02\n- v08 --refutes--> v0f\n- v02 --refutes--> v01\n- v06 --refutes--> v15\n- v15 --refutes--> v0b\n- v09 --refutes--> v06\n- v0a --supports--> v07\n- v01 --refutes--> v11\n- v13 --supports--> v0d\n- v13 --supports--> v04\n- v04 --supports--> v10\n- v15 --supports--> v00\n- v02 --supports--> v0c\n- v04 --refutes--> v03\n\n[VERIFIED ROOT]\nv14 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"423864741654ba7a5a86e941ae4310e53385e83a4fd690e0c8e8676fecd8605a","schema":"livegraph.boolean-propagation.v1","spec_hash":"13f78a4c58d00e4c5fccdeff32df62b4aab7d659f1c563f6a5ef5ad8ec1a8b85","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v14\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-7a024b547d40d244e179","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v04\n- v05 --supports--> v11\n- v13 --supports--> v00\n- v05 --refutes--> v0b\n- v12 --supports--> v0d\n- v0a --refutes--> v0c\n- v00 --refutes--> v06\n- v0c --refutes--> v07\n- v02 --supports--> v13\n- v04 --supports--> v0a\n- v0a --refutes--> v01\n- v09 --supports--> v10\n- v08 --supports--> v12\n- v06 --refutes--> v08\n- v14 --refutes--> v02\n- v12 --supports--> v0e\n- v09 --refutes--> v05\n- v04 --supports--> v03\n- v0c --supports--> v09\n- v0c --supports--> v15\n- v0e --refutes--> v0f\n\n[VERIFIED ROOT]\nv14 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cec5959fba7c1f77c67e4b3b026937bca85f84e6ef2042b876465146d04f6ac4","schema":"livegraph.boolean-propagation.v1","spec_hash":"7a024b547d40d244e1793ccc2bb642c457a70254e376493475d06c008a096393","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v14\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-429585032eb4945dc04e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v0b\n- v15 --refutes--> v14\n- v0a --supports--> v03\n- v09 --refutes--> v01\n- v07 --refutes--> v06\n- v11 --supports--> v13\n- v04 --refutes--> v07\n- v0f --refutes--> v08\n- v0e --refutes--> v15\n- v06 --refutes--> v0a\n- v11 --supports--> v12\n- v10 --supports--> v0d\n- v02 --supports--> v11\n- v0b --refutes--> v0e\n- v0d --supports--> v02\n- v08 --supports--> v00\n- v15 --supports--> v10\n- v08 --supports--> v04\n- v0d --refutes--> v0c\n- v04 --refutes--> v05\n- v09 --refutes--> v0f\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ab12a46772dff9832c107ab2c3cfe2115d84636d47bbc3ce37e3f94087d13b2e","schema":"livegraph.boolean-propagation.v1","spec_hash":"429585032eb4945dc04e2bde1518aa71065b08e6e0d9c7444944bb76e6a1c0dd","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-8627a773061e859d2572","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v04\n- v14 --refutes--> v13\n- v08 --supports--> v0f\n- v0e --supports--> v01\n- v14 --supports--> v05\n- v06 --supports--> v0b\n- v02 --supports--> v11\n- v04 --supports--> v15\n- v14 --refutes--> v07\n- v0f --refutes--> v14\n- v03 --supports--> v0a\n- v01 --supports--> v0c\n- v12 --supports--> v09\n- v0d --refutes--> v12\n- v06 --supports--> v0d\n- v09 --supports--> v08\n- v04 --refutes--> v02\n- v0a --refutes--> v06\n- v02 --supports--> v03\n- v02 --supports--> v00\n- v0e --refutes--> v10\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"197f089793c9f4b686cf04d22245a66ba19b6f61707be2b9b4e78fc700fd9f93","schema":"livegraph.boolean-propagation.v1","spec_hash":"8627a773061e859d2572a19341699388a1ec07f2b7c80fb79e2fcabff3ae43ba","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-2d456ed2da96f7a1944d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --refutes--> v0c\n- v15 --refutes--> v00\n- v11 --refutes--> v15\n- v15 --refutes--> v04\n- v0e --refutes--> v07\n- v10 --refutes--> v0b\n- v0f --supports--> v03\n- v12 --refutes--> v0a\n- v0d --supports--> v09\n- v0b --supports--> v12\n- v0a --supports--> v05\n- v06 --refutes--> v10\n- v0a --refutes--> v01\n- v0d --refutes--> v0e\n- v01 --refutes--> v11\n- v03 --supports--> v0d\n- v0e --refutes--> v08\n- v0a --refutes--> v02\n- v13 --supports--> v06\n- v12 --refutes--> v14\n- v08 --refutes--> v13\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e73d9ff19dd228d4ccb62764e92995d0a743db0eee74b920b4a48218178b8ee8","schema":"livegraph.boolean-propagation.v1","spec_hash":"2d456ed2da96f7a1944d20981950040509e9527ab984cffebf4f60e0b81a2fed","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-415f1215021db4173120","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v0d\n- v10 --refutes--> v03\n- v09 --supports--> v02\n- v0d --supports--> v00\n- v0c --supports--> v15\n- v01 --supports--> v09\n- v15 --supports--> v01\n- v0d --supports--> v08\n- v11 --refutes--> v0e\n- v0e --supports--> v04\n- v03 --refutes--> v0c\n- v0c --supports--> v0a\n- v12 --supports--> v14\n- v04 --supports--> v0f\n- v09 --supports--> v11\n- v03 --supports--> v06\n- v12 --refutes--> v07\n- v00 --refutes--> v05\n- v10 --refutes--> v13\n- v00 --refutes--> v12\n- v0b --supports--> v10\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6ca64638a314d4579f6859ae0f2977a5e7f0d5f937002dd567eb917311ca24a7","schema":"livegraph.boolean-propagation.v1","spec_hash":"415f1215021db4173120a2e9e9b4723a16df452d307728f7f2b966cb73f6790d","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-3e748a1f54d83cb9b369","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v05\n- v00 --supports--> v08\n- v0d --supports--> v09\n- v14 --refutes--> v02\n- v01 --supports--> v07\n- v09 --refutes--> v0c\n- v06 --supports--> v03\n- v0f --supports--> v0a\n- v0f --supports--> v10\n- v08 --refutes--> v12\n- v07 --refutes--> v06\n- v13 --refutes--> v0d\n- v0f --supports--> v01\n- v15 --refutes--> v13\n- v08 --refutes--> v15\n- v04 --supports--> v00\n- v13 --supports--> v0b\n- v02 --supports--> v0f\n- v13 --supports--> v0e\n- v0c --supports--> v11\n- v0c --supports--> v14\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a59705f8ad814a66b9fde5a4a16434b3edaa2a36e8291c966a9ae6223faad265","schema":"livegraph.boolean-propagation.v1","spec_hash":"3e748a1f54d83cb9b3691d7e7decfe767e13abdc796668df2532a44f4563209a","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-cfc33ac8b5fd506099f0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v15\n- v08 --refutes--> v13\n- v01 --refutes--> v0b\n- v0b --refutes--> v06\n- v04 --refutes--> v09\n- v09 --supports--> v0d\n- v0a --supports--> v03\n- v03 --supports--> v07\n- v08 --refutes--> v05\n- v10 --supports--> v02\n- v08 --refutes--> v14\n- v00 --refutes--> v08\n- v0d --refutes--> v01\n- v12 --refutes--> v10\n- v0a --refutes--> v11\n- v0b --refutes--> v0c\n- v00 --refutes--> v0e\n- v02 --refutes--> v0a\n- v00 --refutes--> v0f\n- v14 --supports--> v12\n- v0c --refutes--> v00\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e12854dd55f6ece7c1c55f165773a8c9f442b69a96d510b8770ff2e9a1c45cc9","schema":"livegraph.boolean-propagation.v1","spec_hash":"cfc33ac8b5fd506099f0a7f144662680b4e91389a3b8571574ac67a49bd621f0","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-4b9f04e9b5b979300eeb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v0c\n- v0f --supports--> v02\n- v0f --supports--> v06\n- v15 --supports--> v10\n- v0b --refutes--> v15\n- v09 --supports--> v05\n- v09 --refutes--> v00\n- v05 --supports--> v03\n- v0d --supports--> v0f\n- v0a --supports--> v11\n- v03 --supports--> v0a\n- v0d --supports--> v12\n- v14 --refutes--> v01\n- v10 --refutes--> v13\n- v08 --supports--> v14\n- v11 --supports--> v0b\n- v01 --refutes--> v09\n- v0b --refutes--> v07\n- v0f --refutes--> v0e\n- v13 --refutes--> v0d\n- v01 --refutes--> v04\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"99cf53bf0b75576e0596235f9bcd45bc8f3bcf0af643c556879009c1b4d9c1dc","schema":"livegraph.boolean-propagation.v1","spec_hash":"4b9f04e9b5b979300eeb89942af4817451f4cd37f738ca8e58cd6103fa95ce35","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-5ae1e5424dcea0a20ee8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v04\n- v00 --supports--> v07\n- v06 --supports--> v08\n- v12 --refutes--> v0b\n- v02 --refutes--> v0d\n- v01 --supports--> v0a\n- v14 --supports--> v15\n- v11 --refutes--> v13\n- v05 --refutes--> v11\n- v0f --refutes--> v01\n- v07 --supports--> v10\n- v0d --refutes--> v12\n- v14 --refutes--> v03\n- v10 --refutes--> v02\n- v15 --refutes--> v05\n- v0f --supports--> v0c\n- v13 --supports--> v0e\n- v13 --supports--> v0f\n- v01 --supports--> v00\n- v12 --refutes--> v06\n- v12 --refutes--> v09\n\n[VERIFIED ROOT]\nv14 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3822b94fa60946e28dae55f1ddc4c888902d4701f4b5a580e2c726865c64c676","schema":"livegraph.boolean-propagation.v1","spec_hash":"5ae1e5424dcea0a20ee8dc9571bb930ca5639ff353201cf7f1c4d9af83e5791f","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v14\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-4b941a4571ea5961e8d9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --supports--> v0c\n- v03 --supports--> v12\n- v0e --refutes--> v15\n- v14 --refutes--> v13\n- v04 --supports--> v0d\n- v0c --supports--> v00\n- v11 --supports--> v01\n- v08 --supports--> v05\n- v0a --refutes--> v14\n- v0e --refutes--> v02\n- v0c --supports--> v0b\n- v0b --refutes--> v07\n- v07 --refutes--> v03\n- v05 --refutes--> v06\n- v0e --supports--> v11\n- v0d --supports--> v0e\n- v0d --refutes--> v0f\n- v12 --supports--> v08\n- v07 --refutes--> v10\n- v0b --supports--> v09\n- v06 --supports--> v0a\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6d87d31decdbf854b53cdae22e16cb283b6d953c5ec2b859c54d56ce8600ea4c","schema":"livegraph.boolean-propagation.v1","spec_hash":"4b941a4571ea5961e8d967cfc3f0bfd07ad0351acf0c037968e25177a1b12f31","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-a1e0551c45ab755dc621","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v03\n- v11 --refutes--> v06\n- v12 --refutes--> v02\n- v00 --refutes--> v0a\n- v0c --refutes--> v10\n- v15 --supports--> v11\n- v0c --supports--> v07\n- v11 --supports--> v0f\n- v00 --supports--> v0b\n- v0e --refutes--> v01\n- v06 --supports--> v00\n- v05 --supports--> v14\n- v01 --refutes--> v0d\n- v07 --refutes--> v0e\n- v00 --supports--> v04\n- v05 --supports--> v13\n- v0a --supports--> v12\n- v02 --supports--> v09\n- v14 --supports--> v15\n- v00 --supports--> v08\n- v0d --refutes--> v05\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cafd9f5759e9d1fbb34322382b7f265c4e5ab686027374195d4b26b608a27ddf","schema":"livegraph.boolean-propagation.v1","spec_hash":"a1e0551c45ab755dc62110c63b0e479a7fe740c892d0879c9ebbe0d4f17cf2a1","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-3236ce7b61318a54fd68","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --refutes--> v00\n- v12 --supports--> v04\n- v09 --refutes--> v01\n- v0e --refutes--> v09\n- v10 --supports--> v12\n- v11 --refutes--> v13\n- v12 --supports--> v14\n- v0b --supports--> v07\n- v0c --refutes--> v02\n- v02 --supports--> v03\n- v10 --refutes--> v06\n- v12 --supports--> v15\n- v0f --refutes--> v0b\n- v15 --supports--> v0c\n- v0d --refutes--> v08\n- v03 --refutes--> v11\n- v0f --supports--> v0a\n- v0d --supports--> v10\n- v13 --refutes--> v05\n- v09 --refutes--> v0d\n- v00 --refutes--> v0f\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"99040888fdd2bcefdd674aeb8ec1c0d20203853b9398b786b061399e19205ca4","schema":"livegraph.boolean-propagation.v1","spec_hash":"3236ce7b61318a54fd6853858d123061a50d8d35ea5f13978644a9ec914b6486","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-136515e11c292a42368a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v07\n- v13 --supports--> v06\n- v0a --refutes--> v11\n- v13 --refutes--> v14\n- v00 --refutes--> v08\n- v04 --refutes--> v12\n- v08 --refutes--> v0e\n- v07 --supports--> v03\n- v07 --refutes--> v10\n- v07 --refutes--> v13\n- v01 --supports--> v0c\n- v0e --refutes--> v0a\n- v0b --supports--> v05\n- v0a --refutes--> v0f\n- v12 --refutes--> v0b\n- v11 --refutes--> v0d\n- v11 --supports--> v02\n- v15 --supports--> v04\n- v11 --supports--> v09\n- v0c --supports--> v15\n- v05 --refutes--> v00\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv14\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e650109fe619606cc82ffcb4c9300ed852ca673d76e0ab16ff9b530cb2b69749","schema":"livegraph.boolean-propagation.v1","spec_hash":"136515e11c292a42368afc4655f5671efc74cd5f6b9c2355d90cc9dd204c8f7e","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-26184057f8c8c2318269","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v13\n- v03 --refutes--> v04\n- v0d --refutes--> v14\n- v0a --refutes--> v0c\n- v0b --refutes--> v05\n- v0f --refutes--> v01\n- v03 --refutes--> v0d\n- v0b --refutes--> v09\n- v06 --supports--> v03\n- v02 --supports--> v08\n- v14 --supports--> v10\n- v0e --supports--> v0b\n- v0c --refutes--> v0e\n- v14 --supports--> v07\n- v0f --supports--> v11\n- v10 --supports--> v0a\n- v08 --refutes--> v06\n- v01 --refutes--> v00\n- v0b --supports--> v0f\n- v08 --refutes--> v12\n- v00 --refutes--> v15\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv15\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c7e12eaaa2ce99b709028639e83da1d0a268cd4f440d8e41a744bddb4b8510fd","schema":"livegraph.boolean-propagation.v1","spec_hash":"26184057f8c8c2318269e3c565cf02bfa9c4e74102d978935d2dbc30b252507d","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-198b9db4e80c7d5ebcae","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v0d\n- v0c --supports--> v05\n- v06 --refutes--> v02\n- v09 --refutes--> v07\n- v0e --refutes--> v14\n- v07 --supports--> v13\n- v11 --supports--> v01\n- v11 --supports--> v00\n- v0c --supports--> v10\n- v0f --refutes--> v11\n- v13 --supports--> v0e\n- v06 --supports--> v08\n- v00 --supports--> v12\n- v06 --supports--> v04\n- v0b --supports--> v06\n- v14 --refutes--> v0f\n- v02 --refutes--> v0a\n- v10 --refutes--> v09\n- v0a --refutes--> v0c\n- v09 --supports--> v03\n- v0a --refutes--> v15\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cfe721bc72a0535e24704458bd14d1a2224dcd77e4715460cf57d9ed1fcef9c0","schema":"livegraph.boolean-propagation.v1","spec_hash":"198b9db4e80c7d5ebcaef247fca54d370639de7f8911acdbdfef1b771de007f4","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-33ec4c14f2578c6abba7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --supports--> v11\n- v15 --refutes--> v05\n- v03 --refutes--> v15\n- v04 --refutes--> v06\n- v01 --refutes--> v0c\n- v0e --refutes--> v12\n- v04 --refutes--> v0f\n- v03 --supports--> v08\n- v07 --refutes--> v10\n- v0d --refutes--> v0e\n- v10 --supports--> v14\n- v10 --refutes--> v09\n- v07 --supports--> v13\n- v0e --supports--> v00\n- v0c --supports--> v04\n- v06 --supports--> v07\n- v0a --supports--> v0d\n- v0b --refutes--> v03\n- v10 --supports--> v0a\n- v02 --refutes--> v0b\n- v11 --supports--> v01\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e7f06bac010180cc11f1b622b836cbc9c42ec173ea16f66c75d82933dded1a14","schema":"livegraph.boolean-propagation.v1","spec_hash":"33ec4c14f2578c6abba756b205fbf1d7126a557ff61183e479c3f34517e876b0","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-92a4dd51d639e55185ae","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v05\n- v08 --refutes--> v0a\n- v09 --refutes--> v06\n- v12 --refutes--> v09\n- v0d --refutes--> v13\n- v10 --refutes--> v08\n- v03 --refutes--> v00\n- v01 --supports--> v0f\n- v15 --refutes--> v14\n- v06 --refutes--> v10\n- v10 --supports--> v0b\n- v0a --refutes--> v01\n- v11 --supports--> v15\n- v11 --refutes--> v07\n- v03 --supports--> v04\n- v13 --supports--> v11\n- v14 --refutes--> v02\n- v14 --refutes--> v03\n- v03 --supports--> v0c\n- v01 --supports--> v0e\n- v00 --refutes--> v12\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b48c95a01c965c94251624f1d0e4c8f0212e8f744de8ac064d35ba9b1bc89da1","schema":"livegraph.boolean-propagation.v1","spec_hash":"92a4dd51d639e55185ae581e63f115a44183fc7bfa105ca56d5ea527dbf88398","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-db2a4dbe3614d5a0fd3e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v03\n- v14 --supports--> v09\n- v0d --refutes--> v00\n- v01 --supports--> v08\n- v08 --supports--> v0e\n- v00 --refutes--> v04\n- v13 --refutes--> v01\n- v05 --refutes--> v14\n- v09 --supports--> v0d\n- v08 --refutes--> v12\n- v0e --refutes--> v0c\n- v0f --refutes--> v05\n- v08 --supports--> v0b\n- v07 --refutes--> v15\n- v01 --supports--> v06\n- v15 --supports--> v0a\n- v10 --refutes--> v0f\n- v00 --supports--> v07\n- v0a --refutes--> v13\n- v15 --supports--> v02\n- v09 --refutes--> v11\n\n[VERIFIED ROOT]\nv10 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"799431dd5641e52f4718814217c559de05af3301cadd8979138ba721b5c90bf5","schema":"livegraph.boolean-propagation.v1","spec_hash":"db2a4dbe3614d5a0fd3e56c228b40d4c45c94757272cb3c78314cc4060216819","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-918600983f92f94301c6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --supports--> v09\n- v06 --supports--> v0a\n- v03 --refutes--> v0d\n- v10 --supports--> v15\n- v0f --refutes--> v13\n- v15 --refutes--> v02\n- v0d --supports--> v14\n- v11 --refutes--> v12\n- v12 --refutes--> v06\n- v14 --supports--> v0f\n- v01 --refutes--> v03\n- v02 --refutes--> v05\n- v0e --refutes--> v08\n- v0d --refutes--> v0b\n- v0a --refutes--> v04\n- v00 --supports--> v01\n- v0a --supports--> v07\n- v0a --supports--> v0e\n- v02 --refutes--> v00\n- v0e --refutes--> v10\n- v12 --refutes--> v0c\n\n[VERIFIED ROOT]\nv11 = FALSE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9b1bb32554530b027bc9319972d6281f21a7020b9637fd20e3ae42c989e6905a","schema":"livegraph.boolean-propagation.v1","spec_hash":"918600983f92f94301c6c3beb5ca099711bd0f11e429e2ffae4104aa66deb637","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-1cd5bca82f4ec679e0a1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v10\n- v12 --refutes--> v0f\n- v04 --supports--> v0a\n- v07 --supports--> v12\n- v0a --supports--> v03\n- v09 --refutes--> v06\n- v10 --refutes--> v04\n- v0a --supports--> v00\n- v11 --supports--> v05\n- v06 --supports--> v0c\n- v05 --refutes--> v09\n- v0d --refutes--> v11\n- v05 --supports--> v01\n- v05 --supports--> v02\n- v14 --supports--> v08\n- v03 --refutes--> v0d\n- v15 --supports--> v14\n- v05 --refutes--> v0b\n- v11 --supports--> v0e\n- v12 --refutes--> v15\n- v08 --refutes--> v13\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a50fc573da63084cf762c00bf46f724c377ec588293823b1c397bb5871a60fb7","schema":"livegraph.boolean-propagation.v1","spec_hash":"1cd5bca82f4ec679e0a1a4e861395176ab88da97121100fa60b91e7c1d478893","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-771af5834a32129b5b27","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --refutes--> v0d\n- v06 --supports--> v01\n- v07 --refutes--> v14\n- v09 --supports--> v13\n- v01 --refutes--> v02\n- v07 --refutes--> v12\n- v02 --refutes--> v05\n- v01 --supports--> v00\n- v15 --refutes--> v0f\n- v0e --supports--> v09\n- v02 --supports--> v0a\n- v0a --supports--> v04\n- v0a --supports--> v0b\n- v07 --supports--> v0e\n- v0f --refutes--> v03\n- v11 --supports--> v15\n- v04 --refutes--> v08\n- v0c --supports--> v07\n- v10 --refutes--> v0c\n- v13 --supports--> v06\n- v0f --refutes--> v10\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2e9c3175ba29c48efc129f6245d9bdbd3f8ec7fd54b2ceb33ae670e29d04f4b1","schema":"livegraph.boolean-propagation.v1","spec_hash":"771af5834a32129b5b27d26e78f250445eed40edc955cc94821f0152a9c4d34a","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-8c06674c32a8021d8674","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v09\n- v0a --supports--> v12\n- v04 --supports--> v05\n- v03 --supports--> v04\n- v10 --refutes--> v0a\n- v09 --refutes--> v02\n- v0e --refutes--> v01\n- v0c --refutes--> v15\n- v01 --supports--> v11\n- v06 --supports--> v03\n- v01 --refutes--> v08\n- v09 --refutes--> v13\n- v01 --refutes--> v0c\n- v00 --refutes--> v0f\n- v0d --refutes--> v10\n- v0c --supports--> v0b\n- v00 --supports--> v06\n- v13 --supports--> v07\n- v0d --supports--> v14\n- v0a --refutes--> v00\n- v05 --supports--> v0e\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3674f102c4718d16264926e046c14b2780b97a1bbfe9c886ae4c4b0fa5a13ae1","schema":"livegraph.boolean-propagation.v1","spec_hash":"8c06674c32a8021d8674479accb4a32510ebd6b75d87b6ca112d1afa93bfd9ed","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":14,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-18f5e19fa867a689a778","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v14\n- v10 --supports--> v02\n- v06 --refutes--> v11\n- v04 --refutes--> v0a\n- v04 --refutes--> v07\n- v15 --supports--> v05\n- v0d --refutes--> v12\n- v11 --supports--> v0e\n- v02 --refutes--> v15\n- v15 --refutes--> v03\n- v15 --refutes--> v00\n- v01 --supports--> v10\n- v08 --supports--> v04\n- v04 --supports--> v13\n- v12 --refutes--> v01\n- v0f --supports--> v0c\n- v09 --refutes--> v0f\n- v07 --supports--> v06\n- v08 --supports--> v0b\n- v05 --supports--> v09\n- v11 --supports--> v0d\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"63c2c0370036c50038b7e469a9fa50d242f238422dbe7f5efc19c4a3258b0280","schema":"livegraph.boolean-propagation.v1","spec_hash":"18f5e19fa867a689a778a95d2035464e62f32103b7203332128a8641ffee6a84","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-1333cae33d84341d443d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v12\n- v08 --supports--> v05\n- v12 --supports--> v0c\n- v04 --refutes--> v10\n- v04 --refutes--> v03\n- v01 --refutes--> v11\n- v13 --refutes--> v01\n- v05 --refutes--> v09\n- v0a --supports--> v08\n- v0d --refutes--> v0f\n- v0d --refutes--> v13\n- v09 --refutes--> v0e\n- v01 --supports--> v04\n- v07 --supports--> v02\n- v00 --supports--> v0b\n- v07 --refutes--> v0d\n- v08 --refutes--> v06\n- v0c --refutes--> v00\n- v07 --supports--> v15\n- v0b --supports--> v0a\n- v04 --supports--> v14\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6d1ac0e93a62fc3877c588e16f2b6d82587fa8505822a44845d1ada33e955382","schema":"livegraph.boolean-propagation.v1","spec_hash":"1333cae33d84341d443dc9e28bd0aec998828f1df1ba115b8ba37a214f01a570","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":14,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-9e7741430eff04f8d6bb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --supports--> v09\n- v05 --supports--> v0c\n- v06 --supports--> v00\n- v08 --supports--> v0b\n- v0b --refutes--> v05\n- v07 --supports--> v11\n- v05 --refutes--> v07\n- v09 --refutes--> v0f\n- v01 --supports--> v14\n- v06 --refutes--> v01\n- v11 --refutes--> v04\n- v0d --refutes--> v06\n- v12 --supports--> v0a\n- v04 --supports--> v02\n- v09 --supports--> v08\n- v06 --supports--> v15\n- v12 --refutes--> v03\n- v02 --supports--> v13\n- v05 --supports--> v0e\n- v09 --supports--> v10\n- v13 --refutes--> v12\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"da6744c555347c4096c24b717302400d116dde087a9dcf44e908984b22c2bd2e","schema":"livegraph.boolean-propagation.v1","spec_hash":"9e7741430eff04f8d6bbd80653028ce6b7a65133237b02e2ce004e25231ae6d7","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-f7f7856c6648b904af89","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v09\n- v0a --refutes--> v05\n- v0e --supports--> v0d\n- v09 --supports--> v06\n- v13 --refutes--> v0f\n- v06 --refutes--> v00\n- v0d --supports--> v10\n- v13 --refutes--> v03\n- v0c --refutes--> v13\n- v0b --supports--> v07\n- v00 --supports--> v12\n- v03 --supports--> v14\n- v03 --refutes--> v01\n- v10 --refutes--> v04\n- v14 --refutes--> v0b\n- v05 --supports--> v11\n- v03 --supports--> v08\n- v00 --supports--> v0e\n- v13 --refutes--> v15\n- v11 --supports--> v0c\n- v00 --refutes--> v02\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"51ceec26860085e274254458c83fe9f8bed118bf1db67a1c316b173766a645ea","schema":"livegraph.boolean-propagation.v1","spec_hash":"f7f7856c6648b904af89940072a2167bd45e81e6f9c4beaacd48595672c7893b","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-f696132886c276e2f824","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v0f\n- v08 --refutes--> v11\n- v02 --refutes--> v14\n- v06 --supports--> v02\n- v13 --refutes--> v00\n- v10 --refutes--> v0d\n- v12 --supports--> v08\n- v05 --supports--> v0a\n- v02 --refutes--> v10\n- v0e --refutes--> v03\n- v08 --refutes--> v15\n- v02 --refutes--> v01\n- v04 --refutes--> v06\n- v0d --supports--> v05\n- v07 --supports--> v0e\n- v03 --refutes--> v12\n- v0c --supports--> v07\n- v07 --supports--> v0b\n- v13 --supports--> v0c\n- v05 --refutes--> v13\n- v09 --supports--> v04\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv15\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2c8aa56161a8e4c8e2a0e35e4ed0fe0e72909c078deddddf738a81e311d47350","schema":"livegraph.boolean-propagation.v1","spec_hash":"f696132886c276e2f824cb3080c9727677260a76d3d3cfab006d9839142027e0","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":14,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-429cae07ca5bdc82da13","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --supports--> v00\n- v04 --supports--> v11\n- v13 --refutes--> v09\n- v0e --refutes--> v13\n- v14 --refutes--> v0d\n- v0e --refutes--> v0a\n- v10 --supports--> v0b\n- v03 --supports--> v04\n- v04 --supports--> v0f\n- v08 --refutes--> v01\n- v02 --refutes--> v05\n- v0c --supports--> v14\n- v12 --refutes--> v03\n- v01 --refutes--> v06\n- v0b --supports--> v0e\n- v04 --supports--> v0c\n- v06 --supports--> v15\n- v0c --refutes--> v07\n- v06 --refutes--> v10\n- v14 --refutes--> v02\n- v09 --supports--> v12\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b095a038d2b477c66f279fa6d136766d48cf7d4a5788b9cd7782743ba2deb02f","schema":"livegraph.boolean-propagation.v1","spec_hash":"429cae07ca5bdc82da13d853189e687ba9893175ff971b7a31ad0a85a0be0f78","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-c0e53df80972c447458b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v0b\n- v14 --refutes--> v03\n- v0a --refutes--> v00\n- v09 --refutes--> v0d\n- v06 --refutes--> v0c\n- v0e --supports--> v0f\n- v0c --supports--> v10\n- v05 --refutes--> v02\n- v0a --supports--> v05\n- v0c --refutes--> v15\n- v0c --refutes--> v0a\n- v0c --refutes--> v04\n- v08 --refutes--> v07\n- v02 --refutes--> v14\n- v05 --refutes--> v13\n- v07 --supports--> v01\n- v01 --refutes--> v09\n- v11 --supports--> v06\n- v0d --refutes--> v11\n- v0f --supports--> v08\n- v14 --refutes--> v12\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"485899fcee653dee09560dd0e803f12a4f608deff29b30c402dca8d4ef734022","schema":"livegraph.boolean-propagation.v1","spec_hash":"c0e53df80972c447458b22e26b14009b6454d81067d4c5f62738cd6e64d0c660","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-f8e5f881ba101938a073","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --supports--> v0a\n- v07 --refutes--> v15\n- v11 --supports--> v03\n- v11 --refutes--> v13\n- v09 --refutes--> v08\n- v04 --supports--> v00\n- v14 --refutes--> v07\n- v05 --supports--> v0d\n- v10 --refutes--> v12\n- v07 --refutes--> v0b\n- v15 --refutes--> v04\n- v03 --refutes--> v02\n- v15 --supports--> v0e\n- v06 --refutes--> v10\n- v0f --refutes--> v09\n- v02 --supports--> v05\n- v09 --supports--> v01\n- v06 --supports--> v0c\n- v12 --refutes--> v0f\n- v04 --refutes--> v06\n- v08 --refutes--> v11\n\n[VERIFIED ROOT]\nv14 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"323078ec0eb98a69b76f53f2c7f1ef9c9775def3e5b493b797fbfbe80331d657","schema":"livegraph.boolean-propagation.v1","spec_hash":"f8e5f881ba101938a07318c2d01fb07be308a44b3660a66152fa4cb0a2608593","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v14\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-38eaf8bcbd6d0aac8a4f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v0f\n- v0c --refutes--> v09\n- v14 --supports--> v0b\n- v11 --refutes--> v01\n- v05 --supports--> v0e\n- v05 --refutes--> v15\n- v06 --supports--> v13\n- v12 --refutes--> v07\n- v06 --supports--> v00\n- v07 --refutes--> v04\n- v06 --supports--> v10\n- v15 --refutes--> v14\n- v01 --supports--> v06\n- v0b --refutes--> v0c\n- v06 --supports--> v0d\n- v0c --refutes--> v02\n- v0a --refutes--> v12\n- v07 --refutes--> v03\n- v03 --supports--> v08\n- v08 --refutes--> v11\n- v00 --supports--> v05\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"059657a8082e2f0f8cf7baa16a7872bfc57376c5804377705e3c0aa41a559f11","schema":"livegraph.boolean-propagation.v1","spec_hash":"38eaf8bcbd6d0aac8a4fe9aa97d8b30d1596608adbe44a2b54b1dafb72eec5cf","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-d72d0c1a441fd2a73c84","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v09\n- v0d --supports--> v12\n- v04 --supports--> v03\n- v0a --supports--> v14\n- v04 --supports--> v15\n- v10 --refutes--> v0f\n- v0b --supports--> v07\n- v03 --refutes--> v02\n- v0b --supports--> v0a\n- v09 --supports--> v0e\n- v05 --supports--> v10\n- v0a --supports--> v0d\n- v00 --refutes--> v05\n- v12 --refutes--> v08\n- v09 --refutes--> v01\n- v0f --refutes--> v0b\n- v04 --supports--> v06\n- v03 --refutes--> v13\n- v08 --supports--> v11\n- v00 --refutes--> v0c\n- v01 --refutes--> v00\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4960c6d87ec6bed74135a477e7c45a4f957596ea72be84a3bae0a835da647ed7","schema":"livegraph.boolean-propagation.v1","spec_hash":"d72d0c1a441fd2a73c846f732ca50079fc965994e33ef17505f0d5b5c98fa0cc","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":14,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-724e662ea1b3bbabbd64","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v0e\n- v03 --refutes--> v07\n- v12 --supports--> v02\n- v0c --refutes--> v00\n- v0d --refutes--> v03\n- v13 --supports--> v09\n- v08 --supports--> v13\n- v12 --supports--> v0f\n- v05 --supports--> v14\n- v0f --supports--> v0b\n- v0c --refutes--> v08\n- v0e --refutes--> v0a\n- v04 --refutes--> v10\n- v11 --supports--> v0d\n- v09 --supports--> v05\n- v03 --supports--> v06\n- v0a --refutes--> v11\n- v05 --supports--> v12\n- v0e --refutes--> v01\n- v07 --supports--> v04\n- v12 --supports--> v15\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cdfda24fabd7ad48ca150ef4f9ee1da638b7d3e8f599efb27c99629a833f617f","schema":"livegraph.boolean-propagation.v1","spec_hash":"724e662ea1b3bbabbd64c9a635bf9e3e222ac6163b31ac5c24b90905154cde14","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":14,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-7222d096b273eb8727da","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v08\n- v0e --refutes--> v0b\n- v03 --supports--> v13\n- v10 --refutes--> v0a\n- v12 --supports--> v00\n- v06 --refutes--> v01\n- v12 --supports--> v0e\n- v0d --supports--> v03\n- v02 --refutes--> v0c\n- v11 --refutes--> v12\n- v08 --supports--> v07\n- v11 --supports--> v04\n- v0a --refutes--> v05\n- v0f --refutes--> v15\n- v05 --supports--> v02\n- v0e --refutes--> v0d\n- v0f --supports--> v14\n- v15 --refutes--> v06\n- v02 --supports--> v11\n- v0a --refutes--> v09\n- v07 --refutes--> v0f\n\n[VERIFIED ROOT]\nv10 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f38565860479a49cf090b010ae6dc8a6acd23849e25f0567606d9a1941dc42a5","schema":"livegraph.boolean-propagation.v1","spec_hash":"7222d096b273eb8727da37abb7d7d4da96a905cc6a4061186e46f55818b56f09","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-cae06c7ff5712c414ff0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --refutes--> v06\n- v05 --supports--> v0c\n- v09 --refutes--> v02\n- v10 --supports--> v11\n- v12 --refutes--> v01\n- v0c --refutes--> v13\n- v02 --refutes--> v03\n- v0c --refutes--> v15\n- v0b --supports--> v05\n- v08 --supports--> v0e\n- v00 --supports--> v12\n- v11 --supports--> v07\n- v0d --supports--> v0f\n- v13 --supports--> v04\n- v12 --refutes--> v14\n- v13 --supports--> v08\n- v0a --refutes--> v0b\n- v08 --refutes--> v09\n- v07 --supports--> v0d\n- v02 --refutes--> v10\n- v0d --supports--> v00\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv14\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"728c83de0f98255c59a721dfa109bf2327b6e16ae26198a172b160819cf541f3","schema":"livegraph.boolean-propagation.v1","spec_hash":"cae06c7ff5712c414ff088a6d218f1eaeb809a0b3ae1d02a44768767f83f196a","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":14,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-d5c928ea4698b290b62a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --refutes--> v00\n- v06 --supports--> v04\n- v01 --refutes--> v14\n- v00 --supports--> v0f\n- v0b --supports--> v12\n- v03 --supports--> v02\n- v06 --refutes--> v07\n- v07 --refutes--> v01\n- v09 --refutes--> v15\n- v09 --refutes--> v06\n- v0a --supports--> v0e\n- v0b --refutes--> v11\n- v14 --supports--> v0d\n- v0f --refutes--> v03\n- v02 --refutes--> v0b\n- v00 --supports--> v05\n- v11 --supports--> v13\n- v06 --refutes--> v08\n- v0e --refutes--> v09\n- v14 --refutes--> v0c\n- v13 --refutes--> v10\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3a096aee425c140e791ba41beed1c70d5a26e197c5ab24c2c92ae8c2b755455b","schema":"livegraph.boolean-propagation.v1","spec_hash":"d5c928ea4698b290b62ac9bd78334bcfe51d5cf254fb4b42d925f0fbc0b31c72","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-34cb3dd6e2d100306e8c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v00\n- v10 --supports--> v07\n- v08 --supports--> v02\n- v0b --supports--> v0f\n- v0d --refutes--> v14\n- v0e --supports--> v09\n- v14 --refutes--> v01\n- v03 --refutes--> v0d\n- v0b --refutes--> v04\n- v0c --refutes--> v11\n- v01 --supports--> v13\n- v0e --refutes--> v0a\n- v0c --supports--> v05\n- v04 --refutes--> v03\n- v07 --refutes--> v0b\n- v02 --refutes--> v15\n- v01 --supports--> v12\n- v02 --supports--> v0e\n- v14 --refutes--> v06\n- v0e --refutes--> v0c\n- v13 --supports--> v08\n\n[VERIFIED ROOT]\nv10 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d0361ece896e473f73c2654220728c6448cd71c33a86c93b715de5ae7efaf3bc","schema":"livegraph.boolean-propagation.v1","spec_hash":"34cb3dd6e2d100306e8c12efb88104612791d70582399721bf8861230c0f950b","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-42ce8b85fb6aa7fa7997","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v10\n- v0d --refutes--> v0b\n- v15 --refutes--> v0e\n- v10 --refutes--> v0f\n- v0e --supports--> v0d\n- v02 --refutes--> v03\n- v0f --refutes--> v02\n- v15 --refutes--> v01\n- v0b --supports--> v05\n- v0c --supports--> v13\n- v03 --supports--> v08\n- v04 --refutes--> v11\n- v14 --supports--> v00\n- v0c --refutes--> v14\n- v14 --refutes--> v0a\n- v06 --supports--> v0c\n- v00 --supports--> v12\n- v03 --supports--> v07\n- v03 --refutes--> v15\n- v12 --refutes--> v04\n- v0e --supports--> v09\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"915ec77195f31d11a8837ddda3910b935195d7844c98bf7f1718acb91cd96857","schema":"livegraph.boolean-propagation.v1","spec_hash":"42ce8b85fb6aa7fa7997b8d3d9e207993b59090585b5a9dd8d1a1502e0402443","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-37ccc78c0470b620a691","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --refutes--> v10\n- v11 --supports--> v00\n- v12 --supports--> v13\n- v0b --supports--> v0c\n- v01 --refutes--> v0f\n- v0b --supports--> v0a\n- v12 --refutes--> v0b\n- v06 --refutes--> v09\n- v03 --supports--> v14\n- v15 --supports--> v04\n- v14 --refutes--> v02\n- v08 --refutes--> v03\n- v07 --supports--> v05\n- v09 --refutes--> v01\n- v05 --refutes--> v15\n- v0c --supports--> v11\n- v07 --supports--> v0d\n- v14 --supports--> v07\n- v11 --refutes--> v08\n- v10 --supports--> v0e\n- v10 --refutes--> v06\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"67c875330b2c71e07d3848cdaaa37102374548559df8b6fc8be9ba29e30e008b","schema":"livegraph.boolean-propagation.v1","spec_hash":"37ccc78c0470b620a691f33b8af00ed8928c8278fdc3d8b2ca3b09bb596290eb","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-492e9f4c32540ffbc957","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --supports--> v08\n- v08 --refutes--> v0a\n- v15 --refutes--> v03\n- v0d --refutes--> v0b\n- v11 --refutes--> v0f\n- v10 --refutes--> v13\n- v0d --supports--> v00\n- v02 --refutes--> v05\n- v0c --refutes--> v10\n- v13 --supports--> v14\n- v0c --supports--> v06\n- v07 --supports--> v15\n- v04 --refutes--> v01\n- v0a --supports--> v09\n- v0a --refutes--> v0e\n- v03 --supports--> v0d\n- v0e --refutes--> v07\n- v13 --refutes--> v02\n- v05 --supports--> v12\n- v04 --supports--> v11\n- v0d --supports--> v0c\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c66a31d0763339e11aad518f43a0e3bbb7f6d415bdae0578670c6d745eeb7637","schema":"livegraph.boolean-propagation.v1","spec_hash":"492e9f4c32540ffbc9578b56780e334f9bf2b79bf02b209a1f6c011fdeecf8f8","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-d8b7501bb8b191b48c5a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v0e\n- v07 --refutes--> v0a\n- v05 --supports--> v0c\n- v14 --supports--> v05\n- v0a --refutes--> v01\n- v0d --refutes--> v0b\n- v0f --supports--> v11\n- v08 --refutes--> v00\n- v05 --supports--> v04\n- v0e --supports--> v0f\n- v0f --supports--> v02\n- v05 --refutes--> v15\n- v02 --refutes--> v0d\n- v08 --refutes--> v10\n- v0b --supports--> v07\n- v01 --supports--> v03\n- v08 --supports--> v14\n- v04 --refutes--> v09\n- v12 --refutes--> v08\n- v01 --refutes--> v06\n- v08 --supports--> v13\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ac3846a98efe975f6e28cdab0c557670be54ac423b6a2d2886288e379e7cb17e","schema":"livegraph.boolean-propagation.v1","spec_hash":"d8b7501bb8b191b48c5ad07032ff10d0ea8b2bf454bf03159cd38be7f0ae476e","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-90ebbeff256687ae45a4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v12\n- v02 --supports--> v0e\n- v09 --supports--> v0f\n- v03 --supports--> v15\n- v12 --refutes--> v07\n- v06 --refutes--> v13\n- v03 --supports--> v04\n- v0a --refutes--> v09\n- v0c --supports--> v05\n- v0f --supports--> v11\n- v0f --supports--> v02\n- v0c --refutes--> v08\n- v00 --supports--> v06\n- v0e --refutes--> v0b\n- v06 --refutes--> v10\n- v07 --supports--> v01\n- v0f --refutes--> v14\n- v13 --refutes--> v0a\n- v08 --supports--> v03\n- v04 --supports--> v00\n- v06 --supports--> v0d\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1c5b99fb19900f65b8ebed94049608e3a833843d3236e8478ef0950d254097cf","schema":"livegraph.boolean-propagation.v1","spec_hash":"90ebbeff256687ae45a45a98c765354058fafb5a3ebd95d459ef459d41e5e865","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":14,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-50b82f208acb8c96aa52","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v02\n- v0f --supports--> v07\n- v14 --refutes--> v0c\n- v0e --supports--> v14\n- v11 --supports--> v0f\n- v05 --refutes--> v03\n- v03 --refutes--> v15\n- v08 --supports--> v0e\n- v12 --refutes--> v0b\n- v08 --refutes--> v04\n- v14 --refutes--> v01\n- v00 --supports--> v09\n- v14 --refutes--> v13\n- v08 --supports--> v10\n- v09 --supports--> v08\n- v02 --refutes--> v0a\n- v03 --refutes--> v00\n- v0c --supports--> v06\n- v07 --supports--> v12\n- v0b --supports--> v05\n- v05 --refutes--> v0d\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"964220d5935a4b5f3a41eac0dd9fb45facab2fca80061591c200c6bc91d0e9d6","schema":"livegraph.boolean-propagation.v1","spec_hash":"50b82f208acb8c96aa5260724c61acae83b0d49a7673ce5cca91d7017162ea1e","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-2b558c0e0742480b5225","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v07\n- v01 --supports--> v05\n- v11 --supports--> v10\n- v0d --supports--> v02\n- v0f --supports--> v09\n- v01 --supports--> v08\n- v12 --refutes--> v0c\n- v14 --refutes--> v13\n- v15 --refutes--> v12\n- v08 --refutes--> v0d\n- v07 --supports--> v11\n- v06 --supports--> v00\n- v15 --refutes--> v03\n- v0d --refutes--> v0a\n- v06 --supports--> v0b\n- v13 --supports--> v15\n- v0d --supports--> v0e\n- v00 --supports--> v14\n- v15 --refutes--> v04\n- v10 --supports--> v01\n- v0c --supports--> v0f\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f4840d33a9b766a4adf9af7d757333ff4eac0d7cc24704309525eb740610a7ab","schema":"livegraph.boolean-propagation.v1","spec_hash":"2b558c0e0742480b52254421dd3a6b5270df1b6d275918f65362437120ba8d2b","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-e519bdb8a6edaeb15a87","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v08\n- v03 --refutes--> v0b\n- v0b --supports--> v0a\n- v00 --supports--> v11\n- v14 --refutes--> v12\n- v05 --refutes--> v0d\n- v00 --supports--> v05\n- v13 --refutes--> v02\n- v14 --refutes--> v07\n- v14 --supports--> v06\n- v0d --refutes--> v0e\n- v09 --refutes--> v01\n- v04 --supports--> v0f\n- v12 --refutes--> v03\n- v0f --supports--> v15\n- v03 --supports--> v0c\n- v0e --supports--> v13\n- v0d --refutes--> v10\n- v15 --supports--> v14\n- v01 --supports--> v00\n- v13 --refutes--> v04\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7e986a3718f8479d43ec76bc8572f0ca00023b9ba21c1c387336fc3509a72ba0","schema":"livegraph.boolean-propagation.v1","spec_hash":"e519bdb8a6edaeb15a879a8b63cc5a0420b52ea0ebf02f80b3313af6a796b610","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-e8e8eaf577890f2eb8d0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --supports--> v0c\n- v13 --supports--> v01\n- v08 --supports--> v14\n- v09 --refutes--> v12\n- v10 --supports--> v02\n- v0d --supports--> v06\n- v05 --refutes--> v0e\n- v0f --supports--> v03\n- v13 --supports--> v0d\n- v09 --supports--> v15\n- v0e --refutes--> v13\n- v06 --refutes--> v08\n- v0f --refutes--> v09\n- v15 --supports--> v0b\n- v0c --supports--> v11\n- v14 --refutes--> v0f\n- v02 --refutes--> v07\n- v0d --supports--> v0a\n- v11 --refutes--> v10\n- v06 --refutes--> v04\n- v11 --refutes--> v00\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"562bf8984dd313f1a185ff8f861c289b69cd14e01740bbbf3e94a3125c14a9ea","schema":"livegraph.boolean-propagation.v1","spec_hash":"e8e8eaf577890f2eb8d0db24607145e236535f4172f18f8fd8c58e5f84c96243","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-86fb72e1a92c357f7426","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --supports--> v14\n- v09 --refutes--> v0f\n- v0b --refutes--> v02\n- v14 --refutes--> v06\n- v03 --refutes--> v0c\n- v14 --supports--> v10\n- v12 --refutes--> v08\n- v14 --refutes--> v07\n- v06 --refutes--> v15\n- v12 --refutes--> v0d\n- v0d --refutes--> v11\n- v02 --refutes--> v05\n- v00 --supports--> v09\n- v12 --refutes--> v01\n- v09 --refutes--> v0e\n- v05 --refutes--> v12\n- v0e --supports--> v03\n- v04 --supports--> v13\n- v0c --refutes--> v04\n- v06 --refutes--> v00\n- v03 --supports--> v0a\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6fb274b1b97819af05f816d80f03fcb1fb047e25cf0827daab366ce5b79821ea","schema":"livegraph.boolean-propagation.v1","spec_hash":"86fb72e1a92c357f7426452a3b9ade06a3295e6ddc6004b682ad085ebb3887a9","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-d1f023e7135c8a0fade4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --supports--> v05\n- v05 --supports--> v12\n- v09 --refutes--> v00\n- v00 --supports--> v13\n- v0e --supports--> v04\n- v02 --refutes--> v10\n- v0d --supports--> v03\n- v01 --refutes--> v02\n- v0d --refutes--> v0f\n- v05 --supports--> v0b\n- v02 --refutes--> v0c\n- v0f --supports--> v08\n- v0c --refutes--> v15\n- v00 --refutes--> v06\n- v12 --supports--> v0e\n- v06 --supports--> v01\n- v0d --supports--> v14\n- v01 --refutes--> v07\n- v15 --refutes--> v0d\n- v08 --refutes--> v11\n- v05 --supports--> v0a\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c362450d2362d18127cbfe1cce08832df585d64deb144d51f5fa4df912e3dd73","schema":"livegraph.boolean-propagation.v1","spec_hash":"d1f023e7135c8a0fade4347250683e09a2b8950c00b78e815898c3d3cc59df5a","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-a4a63c305b4af9602592","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v08\n- v04 --refutes--> v0f\n- v04 --refutes--> v14\n- v01 --supports--> v03\n- v06 --supports--> v09\n- v10 --supports--> v0a\n- v05 --supports--> v01\n- v03 --refutes--> v07\n- v14 --supports--> v05\n- v09 --supports--> v0e\n- v07 --supports--> v02\n- v01 --refutes--> v0c\n- v02 --refutes--> v11\n- v0e --refutes--> v10\n- v15 --refutes--> v04\n- v10 --supports--> v00\n- v11 --refutes--> v06\n- v00 --refutes--> v0d\n- v14 --supports--> v0b\n- v0e --refutes--> v13\n- v02 --supports--> v12\n\n[VERIFIED ROOT]\nv15 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"83a269f8da862dbab19f4d54a1e430ce08a57e0f1c2e0505d92840cfc55ad378","schema":"livegraph.boolean-propagation.v1","spec_hash":"a4a63c305b4af9602592b820972c33d8151078e0d8811bbc7a1325bb843d4542","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v15\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-970a4f7575c2689ab1e1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v13\n- v0d --refutes--> v09\n- v05 --supports--> v0f\n- v04 --supports--> v0c\n- v00 --supports--> v03\n- v03 --refutes--> v02\n- v0d --supports--> v06\n- v0e --supports--> v0a\n- v14 --supports--> v11\n- v0f --refutes--> v15\n- v0c --refutes--> v00\n- v0a --supports--> v10\n- v09 --refutes--> v0b\n- v03 --refutes--> v08\n- v0b --refutes--> v01\n- v0f --supports--> v0d\n- v05 --supports--> v12\n- v10 --supports--> v04\n- v08 --refutes--> v14\n- v14 --refutes--> v07\n- v0b --supports--> v0e\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"18a71680d53e0aed64e7c2e73266c97d8899e86443987caefc4d1214a604c64d","schema":"livegraph.boolean-propagation.v1","spec_hash":"970a4f7575c2689ab1e10faffbec94d1e1c8e7604907ae744b1cab6a68f749df","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-9c74d938626b58098930","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v10\n- v13 --supports--> v15\n- v00 --refutes--> v09\n- v09 --refutes--> v0b\n- v0a --refutes--> v00\n- v13 --supports--> v08\n- v0f --refutes--> v0d\n- v12 --refutes--> v13\n- v0d --refutes--> v0a\n- v02 --supports--> v0c\n- v0f --refutes--> v06\n- v0e --refutes--> v04\n- v04 --supports--> v01\n- v02 --refutes--> v0e\n- v10 --refutes--> v07\n- v13 --supports--> v11\n- v0b --supports--> v03\n- v07 --supports--> v0f\n- v09 --refutes--> v14\n- v0b --refutes--> v02\n- v0e --supports--> v05\n\n[VERIFIED ROOT]\nv12 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"31cea627528dc695a31ef488ed22716e263fff55ed6598a525ecb866123aa5d2","schema":"livegraph.boolean-propagation.v1","spec_hash":"9c74d938626b58098930a260ccfaffddb1223a55774d950951f1822d4a854607","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":14,\"root\":\"v12\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-5f9ee1d7b571637c2456","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --refutes--> v0d\n- v0b --supports--> v07\n- v07 --supports--> v15\n- v01 --supports--> v06\n- v12 --refutes--> v01\n- v10 --refutes--> v13\n- v15 --supports--> v0f\n- v12 --refutes--> v0a\n- v05 --refutes--> v0e\n- v06 --refutes--> v11\n- v14 --refutes--> v0c\n- v15 --refutes--> v12\n- v14 --supports--> v08\n- v09 --supports--> v14\n- v06 --refutes--> v02\n- v0e --supports--> v09\n- v08 --refutes--> v04\n- v04 --refutes--> v10\n- v01 --refutes--> v00\n- v12 --supports--> v03\n- v13 --supports--> v0b\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9e6cd1784ecee1d9f6da41278bf2590e4f6825784f45f7c45cae23bc200be83f","schema":"livegraph.boolean-propagation.v1","spec_hash":"5f9ee1d7b571637c2456b60a28328a9060c84de2d8a00c899a5de0046f02b32d","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-84dad8d6e4d172aa5f4e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v04\n- v0c --refutes--> v11\n- v0a --supports--> v15\n- v02 --refutes--> v12\n- v05 --refutes--> v0b\n- v01 --refutes--> v00\n- v0a --refutes--> v0e\n- v0f --supports--> v06\n- v04 --supports--> v09\n- v0d --refutes--> v0c\n- v0c --refutes--> v07\n- v04 --supports--> v13\n- v0e --refutes--> v01\n- v00 --supports--> v05\n- v14 --supports--> v03\n- v07 --refutes--> v0f\n- v13 --refutes--> v02\n- v12 --supports--> v08\n- v05 --refutes--> v10\n- v12 --refutes--> v14\n- v14 --supports--> v0a\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c7f82f7ddf6229a0846d9defd1a60decbbbb85fa8aee41a20425315625dcaf3e","schema":"livegraph.boolean-propagation.v1","spec_hash":"84dad8d6e4d172aa5f4e4760474f95211ff27e8faabf470a9413e28811597d7e","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-5c6c2e2a4b8a234e819e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v00\n- v08 --refutes--> v01\n- v15 --supports--> v12\n- v07 --refutes--> v11\n- v15 --refutes--> v02\n- v07 --supports--> v14\n- v10 --refutes--> v0e\n- v12 --refutes--> v0c\n- v01 --refutes--> v13\n- v0b --supports--> v08\n- v13 --supports--> v03\n- v11 --supports--> v10\n- v15 --supports--> v0a\n- v09 --supports--> v15\n- v00 --refutes--> v09\n- v13 --refutes--> v06\n- v07 --refutes--> v04\n- v10 --supports--> v0d\n- v08 --refutes--> v05\n- v0f --supports--> v07\n- v06 --supports--> v0f\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f79ad7cf123882645874180b8436a5689e24790c2177ff32d2c058d15dce4097","schema":"livegraph.boolean-propagation.v1","spec_hash":"5c6c2e2a4b8a234e819e1a64fe88e32a30977b3e1b8ec3ba838326e5ddb10fe7","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-a642b969b36d151ef3ba","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v15\n- v13 --supports--> v03\n- v04 --supports--> v05\n- v02 --refutes--> v12\n- v07 --refutes--> v09\n- v03 --refutes--> v04\n- v06 --supports--> v14\n- v06 --refutes--> v10\n- v12 --refutes--> v0d\n- v07 --refutes--> v13\n- v10 --refutes--> v02\n- v01 --supports--> v08\n- v15 --supports--> v01\n- v05 --supports--> v0a\n- v13 --supports--> v0e\n- v12 --supports--> v0f\n- v0a --refutes--> v00\n- v0f --refutes--> v11\n- v01 --supports--> v06\n- v03 --refutes--> v0b\n- v10 --refutes--> v0c\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f6cb52edef19bea5e7fa75c2e4787816fb29342bbad1337fc3c8e6d195573d6f","schema":"livegraph.boolean-propagation.v1","spec_hash":"a642b969b36d151ef3ba3de9516b23867cbcd203a7ef6a357275eab98532cd36","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-590c6dff4af2033674da","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v02\n- v02 --refutes--> v0b\n- v13 --supports--> v00\n- v05 --refutes--> v03\n- v08 --refutes--> v14\n- v0a --refutes--> v06\n- v0e --refutes--> v13\n- v00 --supports--> v0a\n- v02 --supports--> v04\n- v0b --refutes--> v05\n- v10 --refutes--> v15\n- v06 --supports--> v11\n- v0d --supports--> v01\n- v05 --supports--> v07\n- v0a --supports--> v0f\n- v14 --supports--> v10\n- v05 --supports--> v09\n- v00 --supports--> v12\n- v11 --refutes--> v0d\n- v13 --refutes--> v0c\n- v09 --supports--> v08\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv15\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3a4ae926f13acd2af6ee514ae0edff21370368bedbdad52bed1b8b98d969fcee","schema":"livegraph.boolean-propagation.v1","spec_hash":"590c6dff4af2033674da8d870c4ca6c66cd4634b10e0d5e0a4ee730b0254e14e","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-a77f4c117acbaa50ab45","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --refutes--> v0d\n- v11 --refutes--> v01\n- v0c --refutes--> v0f\n- v0a --supports--> v09\n- v0a --supports--> v12\n- v0f --refutes--> v11\n- v04 --supports--> v14\n- v14 --refutes--> v08\n- v0b --refutes--> v10\n- v0a --refutes--> v02\n- v15 --refutes--> v13\n- v14 --supports--> v0c\n- v0d --refutes--> v00\n- v06 --refutes--> v05\n- v13 --refutes--> v03\n- v00 --supports--> v0b\n- v11 --supports--> v0e\n- v11 --refutes--> v07\n- v0b --supports--> v0a\n- v12 --refutes--> v04\n- v05 --supports--> v15\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c385585b6cfda1a8495b8cc1ec190666817b10fe2673093f9b29bd231fe921d2","schema":"livegraph.boolean-propagation.v1","spec_hash":"a77f4c117acbaa50ab45ed7057c9140f6964d17976b1d3197099cf9c4005347b","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-d5c228b73056ff36c664","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v09\n- v13 --refutes--> v01\n- v11 --refutes--> v0e\n- v02 --refutes--> v0a\n- v01 --refutes--> v07\n- v0e --supports--> v13\n- v0c --refutes--> v0f\n- v07 --refutes--> v03\n- v07 --supports--> v06\n- v0d --refutes--> v04\n- v0d --refutes--> v14\n- v06 --supports--> v12\n- v07 --supports--> v15\n- v08 --supports--> v0d\n- v0a --supports--> v05\n- v0c --supports--> v02\n- v0d --refutes--> v0b\n- v05 --supports--> v00\n- v05 --supports--> v08\n- v12 --supports--> v0c\n- v09 --refutes--> v10\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"51d3a2461cba48290e141f2225d11dade8aff9955ae36b67ac551014b05f69a5","schema":"livegraph.boolean-propagation.v1","spec_hash":"d5c228b73056ff36c664d03c1273d114d8ee7e782459f720e4d2b185000f1120","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":14,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-76cce6ab51d9b3d46caf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v02\n- v05 --supports--> v06\n- v04 --refutes--> v09\n- v05 --refutes--> v0c\n- v03 --supports--> v14\n- v02 --refutes--> v15\n- v12 --supports--> v01\n- v00 --supports--> v03\n- v13 --supports--> v07\n- v0a --supports--> v04\n- v10 --refutes--> v08\n- v11 --supports--> v0d\n- v13 --refutes--> v0f\n- v00 --supports--> v0e\n- v0d --refutes--> v0b\n- v02 --refutes--> v0a\n- v13 --refutes--> v11\n- v09 --refutes--> v05\n- v10 --supports--> v12\n- v01 --refutes--> v00\n- v03 --refutes--> v13\n\n[VERIFIED ROOT]\nv10 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"51a1f93433bea1d89b20a7553b32a1613c724cc7bedd78ec5036b23a813f5324","schema":"livegraph.boolean-propagation.v1","spec_hash":"76cce6ab51d9b3d46cafda6fd17540aac4afbed798d0839baae91049f545a24d","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":14,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-3c1c4c4b1a94b1a2eb11","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v08\n- v0a --refutes--> v13\n- v02 --refutes--> v12\n- v12 --refutes--> v10\n- v13 --refutes--> v02\n- v15 --supports--> v0b\n- v14 --refutes--> v11\n- v08 --refutes--> v06\n- v0b --refutes--> v07\n- v00 --supports--> v0e\n- v08 --supports--> v01\n- v08 --refutes--> v0c\n- v14 --refutes--> v15\n- v08 --supports--> v04\n- v0f --supports--> v0a\n- v0c --refutes--> v00\n- v0c --supports--> v09\n- v05 --supports--> v03\n- v05 --supports--> v14\n- v10 --supports--> v05\n- v0b --supports--> v0d\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"244f59166666ad039bfdfafd002c4eced9f516e33465bba0989855afaf94be92","schema":"livegraph.boolean-propagation.v1","spec_hash":"3c1c4c4b1a94b1a2eb11a3dff6de38c0b5df07880312da3f80482c557c555ae6","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-916de812ab30f24406ad","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v07\n- v0e --supports--> v10\n- v0b --refutes--> v11\n- v15 --refutes--> v0d\n- v07 --refutes--> v15\n- v12 --supports--> v0c\n- v0c --supports--> v14\n- v00 --refutes--> v08\n- v01 --supports--> v0b\n- v06 --supports--> v09\n- v04 --refutes--> v06\n- v14 --refutes--> v04\n- v14 --refutes--> v13\n- v07 --refutes--> v03\n- v09 --supports--> v01\n- v07 --supports--> v05\n- v00 --refutes--> v0f\n- v0d --supports--> v12\n- v15 --refutes--> v02\n- v11 --supports--> v00\n- v09 --refutes--> v0a\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"61234891ee7ccace9333bba0ba0d49330116c30a4f4478d230c820b847bf6f44","schema":"livegraph.boolean-propagation.v1","spec_hash":"916de812ab30f24406adb3138dcd3958bbc61845410bfea7f015b63cab8aa8a8","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-a75488437d998bf08962","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v04\n- v15 --refutes--> v0a\n- v0f --supports--> v05\n- v05 --supports--> v13\n- v0d --supports--> v0c\n- v08 --supports--> v0f\n- v11 --supports--> v0e\n- v13 --refutes--> v01\n- v11 --refutes--> v12\n- v0d --supports--> v06\n- v12 --supports--> v0b\n- v0a --refutes--> v00\n- v04 --refutes--> v02\n- v0c --refutes--> v03\n- v04 --supports--> v0d\n- v0b --supports--> v08\n- v03 --refutes--> v09\n- v09 --supports--> v14\n- v08 --refutes--> v07\n- v0d --supports--> v10\n- v13 --supports--> v15\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv14\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4004489c91a6473d279ac63496a458d38dd9a3b756600d236856ff24f897cf36","schema":"livegraph.boolean-propagation.v1","spec_hash":"a75488437d998bf089625e4477509083b466100acc1e4ecb4845022cef067a34","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-2e98ae78ea791ed45df4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v01\n- v12 --supports--> v09\n- v0a --refutes--> v10\n- v02 --refutes--> v00\n- v15 --refutes--> v0d\n- v04 --supports--> v14\n- v02 --supports--> v0b\n- v0d --refutes--> v02\n- v14 --supports--> v13\n- v02 --refutes--> v0e\n- v0c --refutes--> v06\n- v09 --refutes--> v11\n- v01 --supports--> v0a\n- v06 --supports--> v05\n- v11 --refutes--> v03\n- v01 --refutes--> v0f\n- v09 --refutes--> v08\n- v03 --supports--> v0c\n- v0b --supports--> v12\n- v11 --refutes--> v07\n- v14 --refutes--> v15\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fbab5968fe16c8b63e6bd32d0ded2b6d28fe1fc31aea68fe527d6ea6e0f0d2e5","schema":"livegraph.boolean-propagation.v1","spec_hash":"2e98ae78ea791ed45df498ba5982021d5a8e657dfd322d85daaa93217c33ef50","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-e0eb42fb81f0708e31e8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v0b\n- v15 --supports--> v14\n- v01 --refutes--> v06\n- v03 --refutes--> v04\n- v0b --supports--> v0f\n- v03 --refutes--> v01\n- v05 --supports--> v15\n- v08 --refutes--> v12\n- v0c --supports--> v0e\n- v03 --refutes--> v10\n- v11 --refutes--> v08\n- v03 --refutes--> v07\n- v0f --refutes--> v05\n- v0d --refutes--> v00\n- v11 --supports--> v09\n- v14 --refutes--> v11\n- v00 --refutes--> v0c\n- v11 --supports--> v13\n- v12 --refutes--> v0d\n- v0b --supports--> v0a\n- v0c --supports--> v02\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fa8138cb03321f9cc2f1e325e2041ef24302696557b53937f3abe6b0a84a7644","schema":"livegraph.boolean-propagation.v1","spec_hash":"e0eb42fb81f0708e31e811c8962e6310b6441677638d6b81347abbcff899410d","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-920e45af05927e791d9f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v0a\n- v0b --refutes--> v00\n- v0d --supports--> v15\n- v01 --refutes--> v04\n- v15 --supports--> v13\n- v08 --supports--> v0c\n- v04 --refutes--> v09\n- v03 --supports--> v10\n- v12 --supports--> v11\n- v09 --supports--> v07\n- v12 --refutes--> v0b\n- v08 --refutes--> v01\n- v03 --refutes--> v02\n- v13 --supports--> v03\n- v0b --supports--> v0e\n- v02 --refutes--> v0f\n- v0f --refutes--> v12\n- v0e --supports--> v05\n- v0a --refutes--> v0d\n- v12 --supports--> v06\n- v0b --refutes--> v14\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c635a3512bbdb3f6e36212f81fffc819c31ac3af4ef078097265491849587326","schema":"livegraph.boolean-propagation.v1","spec_hash":"920e45af05927e791d9f2b77d97a4339125ca14690961f8e38b70c4da68d13a6","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-d4df73f40c4d80bd3dec","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --refutes--> v14\n- v04 --refutes--> v07\n- v13 --supports--> v05\n- v12 --refutes--> v0d\n- v08 --refutes--> v02\n- v0d --refutes--> v11\n- v0d --refutes--> v06\n- v0e --refutes--> v03\n- v01 --refutes--> v0b\n- v08 --refutes--> v00\n- v0a --supports--> v01\n- v02 --supports--> v13\n- v05 --supports--> v15\n- v01 --refutes--> v08\n- v09 --refutes--> v0c\n- v0d --supports--> v0e\n- v04 --refutes--> v12\n- v15 --supports--> v04\n- v03 --supports--> v10\n- v01 --refutes--> v0f\n- v0c --supports--> v0a\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ef68b4a79e83935161ff077e52950dba587e6f4426a09e78479bc313f7466f92","schema":"livegraph.boolean-propagation.v1","spec_hash":"d4df73f40c4d80bd3dece0a7646bb4328b2a9bab79027620db1bf8d6c694ec39","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-1714ff64f3b1ccf77259","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v10\n- v01 --refutes--> v03\n- v06 --supports--> v15\n- v0e --supports--> v0b\n- v04 --refutes--> v01\n- v02 --refutes--> v11\n- v03 --supports--> v0a\n- v02 --refutes--> v0c\n- v05 --refutes--> v13\n- v14 --supports--> v0f\n- v0f --refutes--> v00\n- v14 --supports--> v12\n- v0f --refutes--> v04\n- v15 --supports--> v0e\n- v10 --refutes--> v05\n- v11 --supports--> v06\n- v03 --refutes--> v07\n- v14 --refutes--> v08\n- v05 --refutes--> v14\n- v06 --supports--> v0d\n- v09 --supports--> v02\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"07be7ebefc65614d73801f9347b6a871cf390505516d46baf0c067b187c6630d","schema":"livegraph.boolean-propagation.v1","spec_hash":"1714ff64f3b1ccf77259f8a99fd49c1aab6f99c73b298e8f4e8db366183df88f","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-588ec011a4de809f4dae","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --supports--> v12\n- v06 --supports--> v09\n- v02 --refutes--> v0e\n- v00 --refutes--> v0c\n- v10 --supports--> v02\n- v0b --supports--> v07\n- v02 --supports--> v03\n- v0c --supports--> v13\n- v11 --refutes--> v0f\n- v04 --supports--> v0a\n- v0c --refutes--> v0d\n- v0a --refutes--> v10\n- v09 --supports--> v05\n- v04 --supports--> v08\n- v01 --refutes--> v14\n- v01 --supports--> v11\n- v0e --supports--> v01\n- v09 --refutes--> v0b\n- v07 --supports--> v15\n- v0c --refutes--> v04\n- v15 --refutes--> v00\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4c11087d0f6241df9bb7adb69e537db1f53297f2af07ff91e9d214d5414ecba6","schema":"livegraph.boolean-propagation.v1","spec_hash":"588ec011a4de809f4dae996fd3a4c1d45b725a4704c02a440a5568cab6c050d7","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-bfb03e35209e899687f7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v0e\n- v03 --supports--> v15\n- v03 --supports--> v09\n- v02 --supports--> v01\n- v0d --supports--> v08\n- v07 --supports--> v10\n- v0e --refutes--> v07\n- v10 --supports--> v00\n- v03 --supports--> v05\n- v13 --supports--> v11\n- v0c --refutes--> v13\n- v08 --refutes--> v03\n- v04 --supports--> v0f\n- v0f --refutes--> v06\n- v0f --supports--> v02\n- v0d --supports--> v0a\n- v13 --refutes--> v04\n- v00 --supports--> v0c\n- v08 --refutes--> v12\n- v03 --refutes--> v14\n- v14 --refutes--> v0b\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"213d9c36b39ca2161c1b50722859c088409a3bd794508b6cea1dfb4b6cf32181","schema":"livegraph.boolean-propagation.v1","spec_hash":"bfb03e35209e899687f7ab4ef7581331af567cb7998099ff932c55eadefdb672","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-083816d027f5c418d165","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v01\n- v01 --refutes--> v08\n- v0d --refutes--> v11\n- v08 --refutes--> v07\n- v08 --supports--> v15\n- v10 --refutes--> v04\n- v0a --supports--> v0b\n- v13 --refutes--> v0d\n- v0b --refutes--> v05\n- v08 --refutes--> v0f\n- v14 --refutes--> v00\n- v03 --refutes--> v13\n- v13 --supports--> v12\n- v00 --refutes--> v02\n- v0c --supports--> v10\n- v02 --refutes--> v03\n- v08 --refutes--> v0c\n- v14 --supports--> v0e\n- v0b --refutes--> v06\n- v09 --supports--> v0a\n- v04 --supports--> v09\n\n[VERIFIED ROOT]\nv14 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6dde0decf57a782e262b5cf066f15a6d370474bd68e6ddde9a648018a25f6a77","schema":"livegraph.boolean-propagation.v1","spec_hash":"083816d027f5c418d165d28f8c309de24b8237df75a33024bf346412175e43bc","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v14\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-c3f511149af75acf03d4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v0b\n- v0e --refutes--> v08\n- v0f --supports--> v02\n- v10 --refutes--> v11\n- v08 --refutes--> v0d\n- v05 --supports--> v01\n- v09 --refutes--> v05\n- v0b --refutes--> v0a\n- v04 --refutes--> v06\n- v04 --supports--> v14\n- v14 --refutes--> v07\n- v15 --supports--> v00\n- v13 --refutes--> v09\n- v08 --refutes--> v04\n- v15 --supports--> v0e\n- v02 --supports--> v13\n- v0e --supports--> v12\n- v09 --refutes--> v03\n- v10 --refutes--> v0c\n- v0c --supports--> v15\n- v07 --refutes--> v0f\n\n[VERIFIED ROOT]\nv10 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a9e72c311de125f8f4ab21cb018f3a02db11f24a0b1e53a9ce12762fac5f45d6","schema":"livegraph.boolean-propagation.v1","spec_hash":"c3f511149af75acf03d4a5f259579a0dce3494621ccdc440948aa5c3a7886bc5","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-d1192e36ef22b7eaf7da","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v05\n- v08 --refutes--> v11\n- v05 --refutes--> v07\n- v00 --refutes--> v0b\n- v10 --supports--> v01\n- v0e --refutes--> v04\n- v05 --supports--> v0f\n- v04 --refutes--> v0a\n- v15 --refutes--> v0c\n- v11 --refutes--> v15\n- v15 --refutes--> v02\n- v0a --supports--> v10\n- v13 --refutes--> v00\n- v00 --refutes--> v0e\n- v05 --refutes--> v0d\n- v0d --refutes--> v13\n- v11 --refutes--> v12\n- v06 --supports--> v09\n- v11 --supports--> v14\n- v02 --refutes--> v03\n- v10 --refutes--> v06\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"04691d4e9deab22c2663e550512bfb8404adc05a6f2727c419cfb7a083f579b7","schema":"livegraph.boolean-propagation.v1","spec_hash":"d1192e36ef22b7eaf7dab64dfc1a16f1ce4e17344a4a0b464da0380895ddd0cd","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-10942750fe04f3261f9e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v12\n- v12 --refutes--> v10\n- v09 --refutes--> v07\n- v01 --supports--> v0a\n- v11 --supports--> v06\n- v14 --supports--> v0b\n- v0f --supports--> v0e\n- v03 --refutes--> v14\n- v0c --refutes--> v13\n- v0d --supports--> v0c\n- v02 --supports--> v00\n- v07 --refutes--> v08\n- v0e --supports--> v01\n- v0a --refutes--> v03\n- v11 --refutes--> v0f\n- v0f --supports--> v04\n- v14 --supports--> v02\n- v09 --supports--> v05\n- v0c --refutes--> v11\n- v09 --supports--> v15\n- v02 --supports--> v09\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"618cf9438f8cfa52ea9d13e2266d8341cdc871079e7e4b0961a561a758cb641c","schema":"livegraph.boolean-propagation.v1","spec_hash":"10942750fe04f3261f9e6be8cfc487fa17edf313162e4f6b369dd6934ca45ec6","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-cd41ac2b2c6bf61fa601","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --supports--> v0a\n- v0a --refutes--> v03\n- v01 --refutes--> v14\n- v0a --supports--> v10\n- v11 --supports--> v0f\n- v09 --refutes--> v12\n- v0e --refutes--> v0b\n- v09 --supports--> v0d\n- v07 --supports--> v09\n- v0c --refutes--> v04\n- v15 --supports--> v00\n- v11 --supports--> v13\n- v0b --refutes--> v07\n- v0f --supports--> v0c\n- v0c --supports--> v15\n- v12 --refutes--> v11\n- v15 --refutes--> v02\n- v07 --refutes--> v06\n- v00 --supports--> v05\n- v05 --refutes--> v01\n- v0b --supports--> v08\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8f21ba41050e53d9902705d69782975afdd0b6bbdbc8581467b2a91d21d6ca4d","schema":"livegraph.boolean-propagation.v1","spec_hash":"cd41ac2b2c6bf61fa601fbee96fed83ce1e667b9c052697177b92f97a03a02ed","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-d933ec59ddeb2cb3320e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v07\n- v02 --refutes--> v11\n- v07 --refutes--> v15\n- v05 --refutes--> v10\n- v05 --refutes--> v09\n- v0c --supports--> v01\n- v10 --supports--> v03\n- v03 --refutes--> v08\n- v0d --supports--> v00\n- v02 --supports--> v12\n- v08 --supports--> v04\n- v04 --refutes--> v0a\n- v04 --refutes--> v0e\n- v0e --supports--> v02\n- v13 --supports--> v0d\n- v0d --refutes--> v06\n- v00 --refutes--> v0c\n- v13 --supports--> v0f\n- v0d --refutes--> v0b\n- v15 --refutes--> v05\n- v10 --supports--> v14\n\n[VERIFIED ROOT]\nv13 = FALSE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"271ec62dc37f46d258c8c3f3095b877886219a46fc2eb36c41619ebd9a631bfc","schema":"livegraph.boolean-propagation.v1","spec_hash":"d933ec59ddeb2cb3320e41fbd5b0addde271d5d9c36f2bbd1705c03b6d888f5a","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v13\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-47555a061d7b7a96150a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v00\n- v0a --refutes--> v07\n- v15 --supports--> v0e\n- v10 --refutes--> v14\n- v14 --supports--> v0b\n- v01 --refutes--> v0a\n- v03 --supports--> v05\n- v0a --refutes--> v10\n- v12 --supports--> v0c\n- v0e --supports--> v13\n- v11 --refutes--> v04\n- v0c --refutes--> v09\n- v09 --refutes--> v06\n- v11 --refutes--> v0f\n- v15 --refutes--> v08\n- v05 --refutes--> v01\n- v14 --refutes--> v02\n- v11 --refutes--> v03\n- v09 --supports--> v11\n- v0d --supports--> v12\n- v13 --supports--> v0d\n\n[VERIFIED ROOT]\nv15 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c647614e45d1ffe011cb5eb3d92e5472d7e4108eccb4a0211540e69f651f52b4","schema":"livegraph.boolean-propagation.v1","spec_hash":"47555a061d7b7a96150afed56c614911af316e9783062cce3c62679690eb9818","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v15\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-09cbef2463e7928b851e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v03\n- v07 --refutes--> v04\n- v10 --refutes--> v02\n- v0b --refutes--> v01\n- v12 --refutes--> v15\n- v0e --refutes--> v05\n- v0c --supports--> v14\n- v0d --supports--> v13\n- v15 --refutes--> v0e\n- v0b --refutes--> v0d\n- v0d --supports--> v0a\n- v08 --supports--> v06\n- v02 --refutes--> v0b\n- v0e --refutes--> v00\n- v0f --supports--> v08\n- v0e --refutes--> v07\n- v0a --refutes--> v0c\n- v07 --refutes--> v0f\n- v0a --supports--> v11\n- v03 --supports--> v12\n- v08 --refutes--> v10\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv14\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"51b2daadc96983093d78edd3754e82cea1659a205614a221af5ac1dccc09e2a3","schema":"livegraph.boolean-propagation.v1","spec_hash":"09cbef2463e7928b851e585116b611ea9886711657414db1f74c7ce6d955b0c7","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-411912397a2ae2ddaee8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v05\n- v05 --supports--> v11\n- v02 --refutes--> v08\n- v03 --supports--> v12\n- v15 --refutes--> v00\n- v00 --supports--> v09\n- v0f --refutes--> v10\n- v0d --supports--> v07\n- v0f --refutes--> v0a\n- v05 --refutes--> v0c\n- v06 --supports--> v0e\n- v0a --supports--> v03\n- v01 --supports--> v02\n- v12 --supports--> v15\n- v02 --refutes--> v04\n- v00 --refutes--> v14\n- v02 --supports--> v13\n- v0e --supports--> v01\n- v12 --refutes--> v0b\n- v13 --refutes--> v0d\n- v00 --supports--> v06\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"04a97beb4c4dcdb1b325b67508247dc0987e405f56a237a487665f6656014999","schema":"livegraph.boolean-propagation.v1","spec_hash":"411912397a2ae2ddaee8dbf685173a13002dc58de6ea5d7e412a88bcb0af0623","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-f4272d5afa6b3a77ce06","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v08\n- v0f --refutes--> v00\n- v02 --supports--> v10\n- v14 --supports--> v0e\n- v15 --supports--> v12\n- v0f --supports--> v0c\n- v10 --supports--> v05\n- v0b --refutes--> v06\n- v02 --supports--> v03\n- v06 --supports--> v04\n- v11 --refutes--> v13\n- v0a --supports--> v0f\n- v0e --refutes--> v01\n- v0e --supports--> v15\n- v10 --refutes--> v0b\n- v08 --refutes--> v0d\n- v06 --refutes--> v07\n- v13 --refutes--> v0a\n- v0e --supports--> v09\n- v0c --supports--> v14\n- v06 --supports--> v11\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7d91fc5664a5bb36c2dfbc6daf0b6d87c360e2ba2785b122c50535d2c3cc2c59","schema":"livegraph.boolean-propagation.v1","spec_hash":"f4272d5afa6b3a77ce06ecf2082b3c37dccbe40c1da8dadb6e8037f11e0e2af8","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-95991c92aeea4902d06f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v00\n- v03 --supports--> v07\n- v00 --refutes--> v04\n- v15 --supports--> v14\n- v15 --supports--> v0d\n- v10 --refutes--> v0b\n- v10 --refutes--> v12\n- v10 --refutes--> v0e\n- v0c --refutes--> v02\n- v0a --refutes--> v10\n- v08 --supports--> v13\n- v06 --supports--> v08\n- v13 --supports--> v05\n- v05 --refutes--> v0c\n- v04 --refutes--> v0a\n- v10 --supports--> v0f\n- v0d --supports--> v06\n- v08 --refutes--> v01\n- v08 --refutes--> v11\n- v06 --refutes--> v09\n- v0b --refutes--> v15\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8dae80d2e7e4759fbb911ed499098593a8aae3086b619a03efbbef6f33ffb245","schema":"livegraph.boolean-propagation.v1","spec_hash":"95991c92aeea4902d06f5e3961861091fdca690dff9450dcb3093cec96e6e705","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-281ee969f5c7a6a0cd6b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v10\n- v09 --supports--> v06\n- v13 --supports--> v09\n- v0c --refutes--> v03\n- v15 --refutes--> v07\n- v04 --refutes--> v00\n- v10 --supports--> v13\n- v0b --supports--> v02\n- v07 --supports--> v01\n- v0e --supports--> v11\n- v06 --refutes--> v15\n- v10 --supports--> v0f\n- v05 --supports--> v04\n- v06 --supports--> v0d\n- v10 --refutes--> v12\n- v00 --supports--> v14\n- v03 --refutes--> v08\n- v0b --refutes--> v0c\n- v0e --supports--> v0a\n- v11 --refutes--> v05\n- v07 --refutes--> v0b\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"61e5c277c758bf002dbc225a5470e627bd9d9c52758e45641bd9cc60a661681d","schema":"livegraph.boolean-propagation.v1","spec_hash":"281ee969f5c7a6a0cd6bd55d8e7f6b83822caab497cbc633c9591d7ade5804d1","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-b62c07a59c0b24aaf368","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v14\n- v09 --supports--> v13\n- v0d --supports--> v09\n- v0d --refutes--> v04\n- v13 --supports--> v11\n- v14 --supports--> v0f\n- v0b --refutes--> v0d\n- v0d --refutes--> v00\n- v13 --supports--> v0e\n- v07 --supports--> v0b\n- v03 --supports--> v02\n- v13 --supports--> v12\n- v08 --refutes--> v10\n- v0a --supports--> v06\n- v14 --supports--> v15\n- v03 --refutes--> v01\n- v01 --refutes--> v0a\n- v0f --refutes--> v05\n- v06 --refutes--> v08\n- v13 --supports--> v0c\n- v05 --supports--> v03\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9a45f01051c311e24fd33b19c687e8249ed0d83ae5a435c11aa6eb851feb49f3","schema":"livegraph.boolean-propagation.v1","spec_hash":"b62c07a59c0b24aaf3686e1d24a9efe90f2a00cbb383eb8f01a1e5fd53dc9576","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-c707c211b6d38dbb0ac2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --supports--> v0b\n- v15 --supports--> v06\n- v14 --supports--> v12\n- v06 --refutes--> v0c\n- v02 --refutes--> v0d\n- v00 --refutes--> v03\n- v0f --supports--> v02\n- v12 --refutes--> v10\n- v06 --supports--> v14\n- v01 --refutes--> v05\n- v07 --supports--> v09\n- v05 --refutes--> v04\n- v07 --refutes--> v15\n- v07 --refutes--> v11\n- v08 --refutes--> v07\n- v0b --refutes--> v0a\n- v02 --refutes--> v01\n- v00 --supports--> v0e\n- v15 --refutes--> v13\n- v05 --supports--> v00\n- v03 --refutes--> v08\n\n[VERIFIED ROOT]\nv0f = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"700954cd15c93933039ceeebbcab8543f784aeb71739dfbf71244cb9637824c2","schema":"livegraph.boolean-propagation.v1","spec_hash":"c707c211b6d38dbb0ac276638f7f4f3655a97698680d0abce6ff75beba1e3d8b","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-4e564f1d1fec1bf48e8c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v10\n- v05 --refutes--> v0e\n- v13 --refutes--> v07\n- v12 --supports--> v01\n- v07 --refutes--> v14\n- v15 --supports--> v13\n- v09 --supports--> v0b\n- v14 --supports--> v0a\n- v14 --supports--> v12\n- v0d --supports--> v0c\n- v05 --refutes--> v09\n- v04 --supports--> v02\n- v15 --supports--> v0f\n- v0c --supports--> v03\n- v09 --refutes--> v08\n- v10 --supports--> v00\n- v12 --refutes--> v0d\n- v04 --supports--> v06\n- v06 --refutes--> v05\n- v05 --supports--> v11\n- v00 --refutes--> v04\n\n[VERIFIED ROOT]\nv15 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"610dafcfb1e365ca4902df5a634f41f4ffa7089cb29d1ecc385e06c52a107ff6","schema":"livegraph.boolean-propagation.v1","spec_hash":"4e564f1d1fec1bf48e8cc4d4d64c3b3ecc76d04ef627f08fbe275e83b22c8ba4","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v15\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-0344503dc5944892ed2b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --supports--> v0d\n- v0a --supports--> v08\n- v04 --supports--> v07\n- v09 --supports--> v02\n- v09 --supports--> v01\n- v14 --supports--> v00\n- v10 --refutes--> v13\n- v07 --refutes--> v0b\n- v0d --supports--> v03\n- v0e --refutes--> v0a\n- v09 --refutes--> v0e\n- v11 --refutes--> v12\n- v0a --supports--> v11\n- v0b --refutes--> v05\n- v0b --supports--> v06\n- v06 --supports--> v14\n- v0c --refutes--> v10\n- v0f --refutes--> v04\n- v10 --refutes--> v0f\n- v10 --refutes--> v15\n- v00 --supports--> v09\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e725556e51920bc90040a615bfba7508609d72c67744f156cbff51668b2c39d3","schema":"livegraph.boolean-propagation.v1","spec_hash":"0344503dc5944892ed2b70a669b47bc69b6a95f5ddb61f644bb07410c915072b","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-dd4705e3d40e2070ff35","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v0c\n- v01 --supports--> v11\n- v10 --refutes--> v12\n- v03 --supports--> v0b\n- v12 --supports--> v07\n- v11 --supports--> v10\n- v00 --refutes--> v06\n- v0f --refutes--> v03\n- v12 --refutes--> v0f\n- v06 --refutes--> v14\n- v02 --refutes--> v09\n- v05 --supports--> v04\n- v10 --supports--> v0a\n- v04 --refutes--> v01\n- v11 --refutes--> v15\n- v06 --supports--> v02\n- v0e --supports--> v0d\n- v09 --refutes--> v13\n- v03 --refutes--> v0e\n- v0d --refutes--> v00\n- v0d --supports--> v08\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c5ad69b1df969dbad253a3be8bd45be94d579c9e8afb25909926b5295552966e","schema":"livegraph.boolean-propagation.v1","spec_hash":"dd4705e3d40e2070ff35d8c6249d29614a8a3903bc05535cc3bff3120c729a85","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":14,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-f121a16e25a9bc9f0c1d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v0e\n- v00 --supports--> v12\n- v12 --supports--> v06\n- v0e --supports--> v0a\n- v10 --supports--> v13\n- v13 --supports--> v0d\n- v0a --refutes--> v09\n- v15 --refutes--> v02\n- v0d --refutes--> v14\n- v09 --supports--> v05\n- v0b --supports--> v15\n- v08 --supports--> v00\n- v09 --refutes--> v01\n- v00 --refutes--> v11\n- v0d --refutes--> v04\n- v0c --refutes--> v10\n- v0d --supports--> v0b\n- v15 --supports--> v03\n- v01 --supports--> v07\n- v15 --refutes--> v08\n- v01 --supports--> v0c\n\n[VERIFIED ROOT]\nv0f = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"654343eb1aaad8d4957b2723f22d87c23612dc153b43df3eb9d9cd48657c9427","schema":"livegraph.boolean-propagation.v1","spec_hash":"f121a16e25a9bc9f0c1d77c3ed79fe2421d4819696e7f9fbac68b6257b558b76","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-caeed8264032aee9cf5d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v11\n- v0e --refutes--> v10\n- v07 --refutes--> v15\n- v01 --refutes--> v08\n- v0a --refutes--> v0e\n- v00 --supports--> v12\n- v12 --refutes--> v0c\n- v11 --refutes--> v05\n- v02 --refutes--> v09\n- v02 --refutes--> v00\n- v00 --refutes--> v04\n- v0e --refutes--> v0d\n- v0b --refutes--> v07\n- v0e --supports--> v0f\n- v14 --supports--> v0a\n- v11 --supports--> v06\n- v08 --supports--> v03\n- v0e --supports--> v13\n- v06 --supports--> v02\n- v15 --refutes--> v01\n- v12 --supports--> v0b\n\n[VERIFIED ROOT]\nv14 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fc6bcc4be296c64b2980f7d7e012dcf24b460e963f3d1594efb743bc5c410fb3","schema":"livegraph.boolean-propagation.v1","spec_hash":"caeed8264032aee9cf5d03b7fb2a2f85c0c180689562303c70af3b11879bef28","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v14\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-e028d66e6569d29f9986","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v10\n- v05 --supports--> v08\n- v00 --refutes--> v0d\n- v11 --supports--> v02\n- v15 --supports--> v03\n- v10 --refutes--> v04\n- v07 --refutes--> v11\n- v00 --refutes--> v13\n- v10 --supports--> v0b\n- v0e --supports--> v12\n- v10 --refutes--> v0e\n- v15 --supports--> v05\n- v07 --supports--> v0a\n- v01 --supports--> v07\n- v0e --refutes--> v0f\n- v06 --supports--> v00\n- v13 --refutes--> v15\n- v09 --supports--> v06\n- v11 --refutes--> v0c\n- v0c --supports--> v09\n- v14 --supports--> v01\n\n[VERIFIED ROOT]\nv14 = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9dd9a44797b2ae58d584f88332b213541ab39008fe7570cb5cd21d282c9ccde7","schema":"livegraph.boolean-propagation.v1","spec_hash":"e028d66e6569d29f99865e3317536a2215fb10e20237ed70f5da25deb94b2ca9","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v14\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-7a7920e5eb9666715951","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v09\n- v11 --refutes--> v14\n- v0f --refutes--> v10\n- v0c --supports--> v0a\n- v10 --supports--> v02\n- v15 --refutes--> v05\n- v04 --supports--> v00\n- v0e --refutes--> v0f\n- v0a --supports--> v06\n- v0d --supports--> v11\n- v10 --refutes--> v04\n- v0f --refutes--> v08\n- v04 --supports--> v0d\n- v09 --supports--> v0b\n- v09 --refutes--> v13\n- v11 --supports--> v0c\n- v0b --supports--> v12\n- v01 --refutes--> v0e\n- v07 --supports--> v15\n- v0b --refutes--> v03\n- v03 --refutes--> v07\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"753d948fb48321ca4aab65c985d4063f108b880c541dd706c9ae2b560bedeb2b","schema":"livegraph.boolean-propagation.v1","spec_hash":"7a7920e5eb9666715951ad9f1a7c0394bb4093e331797a5bc9317027c8c4372c","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-6c5a0e26b84aa281deca","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v05\n- v02 --refutes--> v04\n- v0d --supports--> v09\n- v0a --refutes--> v02\n- v01 --supports--> v14\n- v10 --supports--> v08\n- v00 --refutes--> v10\n- v02 --supports--> v00\n- v15 --supports--> v0b\n- v10 --refutes--> v03\n- v0b --supports--> v0e\n- v0e --refutes--> v11\n- v0e --supports--> v13\n- v01 --refutes--> v0a\n- v11 --refutes--> v12\n- v12 --supports--> v0d\n- v06 --refutes--> v0f\n- v11 --refutes--> v07\n- v0f --supports--> v01\n- v02 --refutes--> v0c\n- v03 --refutes--> v15\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"24072c5da6ff45121298e391a068a4d40bad75d6fecd0f8e1ee7d397808a8887","schema":"livegraph.boolean-propagation.v1","spec_hash":"6c5a0e26b84aa281decaa68acebd7cbea76e8a4f53d0b9353bcf9a0cf9411ef5","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-c07949c43e4bbf1d94ad","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v09\n- v0a --supports--> v07\n- v0a --supports--> v10\n- v07 --refutes--> v15\n- v0b --supports--> v02\n- v05 --supports--> v0b\n- v0c --refutes--> v14\n- v0d --refutes--> v01\n- v00 --supports--> v0d\n- v15 --supports--> v03\n- v01 --supports--> v0c\n- v11 --supports--> v00\n- v03 --refutes--> v12\n- v13 --supports--> v05\n- v11 --supports--> v06\n- v12 --refutes--> v11\n- v01 --refutes--> v08\n- v0a --supports--> v04\n- v01 --refutes--> v0f\n- v09 --refutes--> v0a\n- v03 --supports--> v0e\n\n[VERIFIED ROOT]\nv13 = TRUE\n\n[TARGET]\nv14\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4751cbc91e9b84941cbf9b6a9ca879909392b28dcfb0005e6df756b6b45185b2","schema":"livegraph.boolean-propagation.v1","spec_hash":"c07949c43e4bbf1d94add5f3228a34a79412a65a295a7e5880a8bf8fa33ae2e0","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-b31ecf2ce836723d095b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --supports--> v07\n- v05 --supports--> v0a\n- v11 --supports--> v0c\n- v10 --supports--> v12\n- v15 --supports--> v10\n- v08 --refutes--> v15\n- v03 --supports--> v09\n- v05 --supports--> v02\n- v11 --refutes--> v03\n- v05 --refutes--> v0e\n- v14 --supports--> v0b\n- v07 --supports--> v01\n- v14 --supports--> v06\n- v0f --supports--> v0d\n- v0f --supports--> v13\n- v09 --refutes--> v14\n- v00 --refutes--> v11\n- v04 --refutes--> v00\n- v0d --refutes--> v05\n- v07 --supports--> v0f\n- v0b --refutes--> v08\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"446b029069618aa5c624c60c6cfef2d0c85a474d559da120fe740c0a898040e4","schema":"livegraph.boolean-propagation.v1","spec_hash":"b31ecf2ce836723d095b1a96066c04a4e328b75ddf99b51afcba764329246d85","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-f5b5d5b3d6f546f667d2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v02\n- v01 --refutes--> v08\n- v09 --refutes--> v15\n- v09 --refutes--> v03\n- v10 --supports--> v14\n- v0d --refutes--> v0c\n- v0b --supports--> v01\n- v13 --supports--> v12\n- v0a --refutes--> v00\n- v0a --refutes--> v0b\n- v14 --supports--> v0d\n- v0f --supports--> v13\n- v04 --refutes--> v0a\n- v12 --supports--> v07\n- v14 --supports--> v06\n- v01 --refutes--> v10\n- v0e --supports--> v04\n- v0a --supports--> v11\n- v12 --refutes--> v0e\n- v0c --supports--> v09\n- v05 --refutes--> v0f\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"70e24de07846bf47e8e8e52f5645fe4bbaf2da5b430e85089340d84bd412912d","schema":"livegraph.boolean-propagation.v1","spec_hash":"f5b5d5b3d6f546f667d20d055bddcc0ea7af606404cd6ab15e38fc25c8aeaf9f","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-a2b51aab326dd44873ab","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v0e\n- v00 --refutes--> v14\n- v0c --supports--> v12\n- v13 --refutes--> v06\n- v09 --refutes--> v10\n- v0a --supports--> v11\n- v0b --supports--> v0f\n- v06 --refutes--> v00\n- v15 --supports--> v13\n- v0e --supports--> v08\n- v15 --supports--> v0d\n- v08 --supports--> v04\n- v10 --refutes--> v0c\n- v04 --refutes--> v09\n- v08 --refutes--> v03\n- v0b --supports--> v07\n- v0a --refutes--> v05\n- v0c --refutes--> v02\n- v12 --refutes--> v0a\n- v01 --refutes--> v15\n- v0a --refutes--> v0b\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a98ad7efa4b239895a5c384d02c487120f0fd9e88e4254744a0184e931f97249","schema":"livegraph.boolean-propagation.v1","spec_hash":"a2b51aab326dd44873ab8814270b9ff999a18cfd67f98a08c2addd4e6a989dc9","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-69d04e3b8aac115dac4e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v06\n- v01 --refutes--> v13\n- v00 --refutes--> v0f\n- v00 --supports--> v11\n- v07 --supports--> v15\n- v08 --supports--> v07\n- v0f --refutes--> v04\n- v0c --supports--> v02\n- v05 --refutes--> v08\n- v05 --supports--> v0b\n- v13 --supports--> v00\n- v0e --supports--> v03\n- v07 --supports--> v0d\n- v12 --refutes--> v0e\n- v07 --supports--> v01\n- v02 --supports--> v0a\n- v00 --supports--> v10\n- v0a --supports--> v05\n- v00 --refutes--> v09\n- v14 --supports--> v0c\n- v04 --supports--> v12\n\n[VERIFIED ROOT]\nv14 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e51f9aad9a8ff0b685d40d99dc388a893a60c41acf910525f6f5c030cc0d526b","schema":"livegraph.boolean-propagation.v1","spec_hash":"69d04e3b8aac115dac4e7da785ca27863472b88fe84b0486cee8131d8861b025","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v14\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-4e9c2a5e39422a685f30","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v03\n- v11 --refutes--> v02\n- v0e --refutes--> v06\n- v02 --supports--> v0f\n- v04 --supports--> v0b\n- v0b --refutes--> v05\n- v14 --supports--> v15\n- v13 --supports--> v01\n- v04 --supports--> v00\n- v09 --refutes--> v11\n- v14 --refutes--> v10\n- v10 --refutes--> v04\n- v0e --refutes--> v0d\n- v05 --supports--> v09\n- v0a --supports--> v0e\n- v0c --supports--> v0a\n- v0d --refutes--> v12\n- v12 --refutes--> v14\n- v10 --supports--> v08\n- v13 --refutes--> v0c\n- v0d --refutes--> v07\n\n[VERIFIED ROOT]\nv13 = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"52432bdae3d96e20d4591dd12fec2ed832699f172c283d968be3710e21b6e595","schema":"livegraph.boolean-propagation.v1","spec_hash":"4e9c2a5e39422a685f30f8c54f9aff188780beba0a7ded245e9b79c2804edaf0","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-543ddb4ab2b5fb43277a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --supports--> v0d\n- v11 --refutes--> v02\n- v0d --supports--> v06\n- v07 --refutes--> v12\n- v0b --refutes--> v00\n- v15 --supports--> v11\n- v0d --refutes--> v0f\n- v0a --refutes--> v10\n- v0c --supports--> v09\n- v0b --refutes--> v04\n- v05 --refutes--> v0c\n- v0f --supports--> v05\n- v07 --supports--> v14\n- v08 --refutes--> v03\n- v0a --supports--> v01\n- v0d --supports--> v0e\n- v0c --refutes--> v13\n- v09 --supports--> v15\n- v11 --supports--> v0b\n- v03 --refutes--> v0a\n- v04 --refutes--> v08\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e6fbf831b8546b75541d438c4d68c7e003cf65d60112ee58cc9bc368e2a68fe3","schema":"livegraph.boolean-propagation.v1","spec_hash":"543ddb4ab2b5fb43277ae2531e29acaca1dde4ca614f0f6722fcbad7c6f25eb0","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-0b556d21a37bb190de6c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v04\n- v12 --refutes--> v0c\n- v10 --supports--> v13\n- v11 --refutes--> v03\n- v0a --refutes--> v01\n- v01 --supports--> v0f\n- v00 --supports--> v02\n- v14 --supports--> v0d\n- v06 --supports--> v05\n- v00 --supports--> v06\n- v15 --supports--> v11\n- v11 --refutes--> v00\n- v11 --refutes--> v0b\n- v08 --supports--> v10\n- v0d --refutes--> v0e\n- v14 --refutes--> v09\n- v10 --refutes--> v0a\n- v0c --refutes--> v14\n- v05 --supports--> v12\n- v01 --refutes--> v15\n- v07 --supports--> v08\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a683aec9a74221622cdf427f8b26856f445efe7dbec24ba7fa37a5507d0c2aad","schema":"livegraph.boolean-propagation.v1","spec_hash":"0b556d21a37bb190de6ce00ff750f7c44fb9ff75fca38ce173605d5fd030acc2","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-7034583ca6beb59fe8b8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v0a\n- v11 --refutes--> v09\n- v0d --supports--> v0f\n- v09 --supports--> v15\n- v08 --supports--> v04\n- v0d --supports--> v03\n- v02 --refutes--> v06\n- v0d --refutes--> v0e\n- v10 --supports--> v0b\n- v14 --refutes--> v08\n- v04 --refutes--> v01\n- v04 --supports--> v05\n- v06 --refutes--> v0c\n- v00 --supports--> v02\n- v0b --refutes--> v12\n- v09 --supports--> v00\n- v0b --refutes--> v13\n- v0d --refutes--> v11\n- v13 --supports--> v14\n- v0c --supports--> v07\n- v07 --supports--> v10\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"aa8fd495932643cfeab16d726c038df74ba73678f7aa704c010296930f5b39d4","schema":"livegraph.boolean-propagation.v1","spec_hash":"7034583ca6beb59fe8b81d74cc05657afbc384ebd384726a1d0e74aa3bc0d4d1","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-1f224c9dd69980b45bf4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --supports--> v14\n- v0d --refutes--> v06\n- v13 --supports--> v01\n- v0d --refutes--> v04\n- v0d --supports--> v0e\n- v06 --supports--> v0f\n- v10 --refutes--> v00\n- v02 --refutes--> v08\n- v03 --refutes--> v15\n- v0b --supports--> v10\n- v10 --supports--> v0c\n- v07 --supports--> v02\n- v0b --supports--> v0a\n- v11 --refutes--> v07\n- v13 --supports--> v09\n- v0f --refutes--> v12\n- v01 --supports--> v03\n- v06 --supports--> v05\n- v10 --supports--> v0d\n- v15 --refutes--> v11\n- v14 --supports--> v13\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"44ca0b2289102febde179495715fd6dad2128d81b24462c96ee47682ea7f8c03","schema":"livegraph.boolean-propagation.v1","spec_hash":"1f224c9dd69980b45bf4769bd63237cb72c6f18eb0877848db0ae3b674f5265f","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-a49548dbfd5173e5b161","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v10\n- v0b --refutes--> v11\n- v10 --refutes--> v0f\n- v13 --refutes--> v06\n- v0e --supports--> v13\n- v01 --supports--> v0d\n- v08 --supports--> v0a\n- v0e --refutes--> v15\n- v13 --supports--> v07\n- v0f --supports--> v05\n- v0a --refutes--> v04\n- v0f --refutes--> v01\n- v07 --supports--> v0b\n- v07 --supports--> v02\n- v03 --refutes--> v0c\n- v09 --supports--> v03\n- v0a --supports--> v0e\n- v0d --refutes--> v08\n- v11 --supports--> v12\n- v09 --refutes--> v00\n- v08 --refutes--> v14\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9bdf6c9c63b6790a1ee4943ca2c26b72b0e9fb35bbc2a5faa8852eeee8aa5740","schema":"livegraph.boolean-propagation.v1","spec_hash":"a49548dbfd5173e5b161cd0a8e0babf0187b36d288dcc7d2eba43f8559315798","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":14,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-efa4d408b63a3610c26f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --refutes--> v13\n- v0c --supports--> v0f\n- v10 --supports--> v0b\n- v0e --refutes--> v07\n- v15 --supports--> v05\n- v12 --refutes--> v0d\n- v14 --supports--> v11\n- v11 --supports--> v01\n- v10 --supports--> v06\n- v02 --supports--> v14\n- v07 --supports--> v0a\n- v05 --supports--> v09\n- v0c --refutes--> v12\n- v0d --refutes--> v10\n- v07 --refutes--> v02\n- v06 --supports--> v08\n- v05 --supports--> v04\n- v06 --supports--> v15\n- v07 --refutes--> v03\n- v01 --refutes--> v00\n- v00 --supports--> v0c\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b0fb97e1fd2bdf2203f65e465c6f88c7541fd58e73a135c66554f729e7c7453f","schema":"livegraph.boolean-propagation.v1","spec_hash":"efa4d408b63a3610c26f3eb93d369d95f371fcff2de736f1998628a0b8ff6efb","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-3c2184994ff88fd888fd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v11\n- v04 --supports--> v06\n- v02 --supports--> v07\n- v0d --supports--> v05\n- v0c --refutes--> v0b\n- v13 --refutes--> v0d\n- v02 --supports--> v04\n- v12 --supports--> v15\n- v0e --refutes--> v02\n- v0a --supports--> v01\n- v0a --refutes--> v10\n- v0f --refutes--> v14\n- v04 --supports--> v0c\n- v0b --supports--> v0f\n- v09 --supports--> v0a\n- v0f --supports--> v13\n- v0b --supports--> v00\n- v12 --refutes--> v08\n- v15 --supports--> v0e\n- v01 --refutes--> v03\n- v01 --supports--> v12\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"595280707eb84fb06f47dc0b8ecfc24d64a045d02dd035ae529abd39df04cfda","schema":"livegraph.boolean-propagation.v1","spec_hash":"3c2184994ff88fd888fd6c9c4729bcf997cfdd2f963a181b6f900feddf6db3a4","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-a800942487b7e22fef59","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v06\n- v09 --refutes--> v02\n- v15 --refutes--> v0a\n- v0b --refutes--> v04\n- v15 --refutes--> v0e\n- v0d --refutes--> v15\n- v09 --refutes--> v0c\n- v0b --refutes--> v11\n- v07 --supports--> v13\n- v09 --supports--> v10\n- v11 --supports--> v09\n- v02 --supports--> v0d\n- v14 --refutes--> v07\n- v13 --supports--> v05\n- v0f --refutes--> v03\n- v05 --refutes--> v0f\n- v08 --supports--> v12\n- v0f --supports--> v08\n- v01 --refutes--> v0b\n- v0f --supports--> v00\n- v08 --supports--> v01\n\n[VERIFIED ROOT]\nv14 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2e4608c2623b1cb0ec4abbc4227d15cdc7fbfd8a01df2cb83075da733896609e","schema":"livegraph.boolean-propagation.v1","spec_hash":"a800942487b7e22fef5965ce7dc08e04f564a446d3b3d1d69f1ed4a4eed24799","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v14\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-0da758997ddcd9912f7f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v05\n- v0a --supports--> v02\n- v0b --supports--> v03\n- v03 --refutes--> v13\n- v0d --supports--> v0f\n- v12 --refutes--> v04\n- v0a --supports--> v07\n- v14 --supports--> v01\n- v12 --refutes--> v0e\n- v04 --supports--> v06\n- v13 --supports--> v0d\n- v00 --refutes--> v14\n- v05 --refutes--> v00\n- v01 --supports--> v0a\n- v01 --refutes--> v11\n- v0f --refutes--> v08\n- v0c --supports--> v09\n- v06 --refutes--> v0c\n- v06 --supports--> v15\n- v0c --supports--> v0b\n- v05 --refutes--> v10\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3862481a9def6022e455d29ce29a285a2558b8885b804bdb4992132b66e86f6f","schema":"livegraph.boolean-propagation.v1","spec_hash":"0da758997ddcd9912f7f55a103ba8aa67ad8fea85ba12be0da0ff4dac589bb84","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-8edad85ec0f8fdc9e00d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v12\n- v08 --supports--> v02\n- v0e --supports--> v03\n- v0a --supports--> v05\n- v00 --refutes--> v06\n- v0d --supports--> v0b\n- v12 --refutes--> v00\n- v08 --refutes--> v07\n- v03 --refutes--> v11\n- v11 --refutes--> v08\n- v10 --supports--> v09\n- v08 --refutes--> v15\n- v13 --refutes--> v01\n- v00 --refutes--> v14\n- v13 --supports--> v04\n- v10 --supports--> v13\n- v00 --refutes--> v0d\n- v0d --refutes--> v0a\n- v15 --refutes--> v0c\n- v0c --supports--> v0f\n- v05 --refutes--> v0e\n\n[VERIFIED ROOT]\nv10 = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9e9e38246b8f0d32761f62483efc1390ad2721fe0b313217c636894295e88845","schema":"livegraph.boolean-propagation.v1","spec_hash":"8edad85ec0f8fdc9e00dd944d9ed1da16630ea63b3202b78606d4e41492bb102","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-eba5b0cb39e464df2e76","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v0d\n- v14 --supports--> v0a\n- v00 --supports--> v06\n- v07 --refutes--> v0f\n- v12 --supports--> v14\n- v10 --refutes--> v07\n- v00 --refutes--> v08\n- v13 --refutes--> v00\n- v07 --supports--> v12\n- v0c --supports--> v15\n- v14 --refutes--> v11\n- v09 --refutes--> v03\n- v08 --refutes--> v09\n- v10 --supports--> v02\n- v09 --supports--> v05\n- v03 --refutes--> v10\n- v15 --supports--> v04\n- v0a --refutes--> v0b\n- v04 --refutes--> v13\n- v00 --supports--> v01\n- v15 --supports--> v0e\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e5b00a317b8d1607b67ca64f58e6688f926d9c8be1e4ad072596cb75178dee92","schema":"livegraph.boolean-propagation.v1","spec_hash":"eba5b0cb39e464df2e762ff8068366829134f8c809853b5f3f85351b9eeb8807","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-8c6afec3c53eccdc3169","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v0b\n- v13 --supports--> v06\n- v0c --supports--> v14\n- v0c --refutes--> v13\n- v03 --refutes--> v02\n- v0a --supports--> v0e\n- v00 --refutes--> v04\n- v0b --refutes--> v12\n- v10 --supports--> v11\n- v02 --supports--> v07\n- v04 --supports--> v10\n- v0e --supports--> v0c\n- v12 --refutes--> v09\n- v02 --supports--> v0f\n- v11 --refutes--> v08\n- v13 --supports--> v03\n- v09 --refutes--> v00\n- v11 --refutes--> v01\n- v10 --supports--> v0d\n- v10 --supports--> v05\n- v0e --supports--> v15\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"21e419f09df4ec6ec918edb514d49594f3db8952f84e11737f94167a29ef0d8f","schema":"livegraph.boolean-propagation.v1","spec_hash":"8c6afec3c53eccdc31690080746a5ffe71af85a1e88eb296b1b039e4a237062b","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":14,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-fdd4349c971b837e2020","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v0d\n- v04 --refutes--> v0a\n- v04 --refutes--> v0c\n- v01 --supports--> v08\n- v11 --refutes--> v12\n- v09 --refutes--> v00\n- v00 --refutes--> v01\n- v11 --supports--> v15\n- v0f --supports--> v09\n- v04 --supports--> v14\n- v11 --supports--> v0b\n- v13 --supports--> v10\n- v08 --refutes--> v03\n- v0b --refutes--> v13\n- v07 --supports--> v02\n- v10 --supports--> v05\n- v0c --refutes--> v0e\n- v07 --refutes--> v0f\n- v08 --refutes--> v11\n- v06 --refutes--> v04\n- v0e --supports--> v07\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5650e8d43aba708b88dacd9cec45fd9dbc27cb3e9365be8cd43c65a4b4eaf291","schema":"livegraph.boolean-propagation.v1","spec_hash":"fdd4349c971b837e20204bd81105bc4093ea5fa81f40baeb590070386f0d75d0","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":14,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-95eb10d78aa720eeb042","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v02\n- v03 --refutes--> v0a\n- v01 --refutes--> v13\n- v02 --supports--> v15\n- v14 --supports--> v05\n- v02 --supports--> v03\n- v07 --refutes--> v0c\n- v0e --refutes--> v11\n- v0f --refutes--> v12\n- v10 --refutes--> v07\n- v06 --refutes--> v14\n- v00 --refutes--> v0e\n- v0a --supports--> v09\n- v11 --supports--> v01\n- v09 --supports--> v10\n- v05 --supports--> v0f\n- v0e --refutes--> v0b\n- v0a --supports--> v04\n- v0f --supports--> v0d\n- v0a --supports--> v08\n- v07 --refutes--> v06\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0b2dd01bd7319a137f6bf78cbe6c497da77fe292c58c200e8d93db1a9ac1175e","schema":"livegraph.boolean-propagation.v1","spec_hash":"95eb10d78aa720eeb042b1eb94e217295264d83f7a1418ec14c7647c2ea8cd09","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-d4f74bb61eb749d17599","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v12\n- v10 --supports--> v0d\n- v0f --refutes--> v0b\n- v0d --supports--> v11\n- v08 --refutes--> v15\n- v0d --supports--> v14\n- v0d --refutes--> v07\n- v02 --supports--> v13\n- v11 --refutes--> v0f\n- v09 --supports--> v05\n- v13 --refutes--> v0c\n- v06 --refutes--> v03\n- v0a --refutes--> v02\n- v15 --refutes--> v09\n- v12 --refutes--> v0e\n- v05 --supports--> v00\n- v10 --supports--> v04\n- v13 --supports--> v10\n- v05 --refutes--> v06\n- v0f --supports--> v08\n- v0f --refutes--> v01\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2c180bb573e82a22eb25cf6b4b6d1dc7b966fa1896a1cd5bcfad351d33414d09","schema":"livegraph.boolean-propagation.v1","spec_hash":"d4f74bb61eb749d17599a3b55eac24a33e1f6198a3c361e5f58204212755b79a","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-d29a9c99b30f4baec9a4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --refutes--> v08\n- v0b --supports--> v12\n- v10 --supports--> v0b\n- v0d --refutes--> v07\n- v02 --supports--> v0a\n- v0d --supports--> v15\n- v05 --supports--> v0f\n- v06 --refutes--> v10\n- v11 --refutes--> v05\n- v02 --supports--> v0c\n- v08 --supports--> v00\n- v12 --refutes--> v13\n- v04 --supports--> v02\n- v07 --refutes--> v0e\n- v05 --supports--> v03\n- v0f --supports--> v01\n- v02 --supports--> v06\n- v0f --refutes--> v04\n- v08 --supports--> v09\n- v09 --supports--> v0d\n- v12 --refutes--> v14\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"69318af5d72d972006c17b07a810e8962b5bde74d7fdb659b8bbd2f31ea5a92f","schema":"livegraph.boolean-propagation.v1","spec_hash":"d29a9c99b30f4baec9a464e2efa88089ad6c08be1f3a534b98b8783526a7aa07","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-9e658ca82d7c716a8e9e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v08\n- v14 --refutes--> v10\n- v0a --supports--> v00\n- v12 --refutes--> v03\n- v0a --supports--> v13\n- v0d --refutes--> v0a\n- v0f --refutes--> v01\n- v0b --supports--> v0c\n- v01 --supports--> v0b\n- v03 --supports--> v0e\n- v0c --refutes--> v05\n- v00 --refutes--> v14\n- v0e --refutes--> v0d\n- v07 --supports--> v0f\n- v03 --supports--> v09\n- v10 --supports--> v04\n- v05 --supports--> v12\n- v07 --supports--> v11\n- v0c --refutes--> v02\n- v0f --supports--> v15\n- v00 --supports--> v06\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ade0be8b3bedba92ae987f5b281110cab062111909e4ec01a82f469487190d17","schema":"livegraph.boolean-propagation.v1","spec_hash":"9e658ca82d7c716a8e9e71a940036177476c04ed73aaa1bd50f619b81b4bd480","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-71721e53c1d8fc97f750","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v06\n- v08 --refutes--> v00\n- v08 --refutes--> v05\n- v10 --supports--> v02\n- v08 --supports--> v0e\n- v0a --supports--> v11\n- v0d --supports--> v04\n- v0d --refutes--> v13\n- v06 --supports--> v0c\n- v13 --supports--> v10\n- v13 --supports--> v0f\n- v03 --refutes--> v01\n- v01 --refutes--> v15\n- v12 --refutes--> v09\n- v01 --supports--> v0a\n- v0c --supports--> v08\n- v0d --refutes--> v07\n- v11 --supports--> v12\n- v10 --refutes--> v14\n- v0e --supports--> v03\n- v0b --refutes--> v0d\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"db979d247077e3e8dcd9f8621d89dde6da98708cb447e314b20c698fe60b25d3","schema":"livegraph.boolean-propagation.v1","spec_hash":"71721e53c1d8fc97f750e3d9d631a65a8f6b71616f759aa8dda9a9e1b3dbae49","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":14,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-f7fd35d4ec6cf614bfcf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v14\n- v13 --supports--> v11\n- v0b --refutes--> v00\n- v06 --refutes--> v0e\n- v0b --refutes--> v07\n- v01 --refutes--> v05\n- v0b --supports--> v12\n- v02 --supports--> v04\n- v09 --refutes--> v03\n- v15 --supports--> v09\n- v11 --supports--> v01\n- v0f --supports--> v13\n- v0f --supports--> v08\n- v05 --supports--> v02\n- v03 --supports--> v0d\n- v03 --supports--> v06\n- v11 --supports--> v10\n- v14 --supports--> v0f\n- v13 --refutes--> v0c\n- v02 --supports--> v0a\n- v04 --refutes--> v0b\n\n[VERIFIED ROOT]\nv15 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2470634e79a829dbb3378e8286536eed9710ba8c396c1e2cd8071ed5c3765170","schema":"livegraph.boolean-propagation.v1","spec_hash":"f7fd35d4ec6cf614bfcf6020cf401661fbf5881391f5cc06c8265bcfc4fdfcc6","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v15\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-8030ba706ac59766eb62","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --supports--> v10\n- v14 --supports--> v04\n- v04 --refutes--> v0a\n- v12 --supports--> v13\n- v05 --refutes--> v09\n- v13 --supports--> v06\n- v14 --refutes--> v07\n- v0b --supports--> v0e\n- v05 --supports--> v0d\n- v11 --supports--> v14\n- v0f --supports--> v00\n- v0b --supports--> v0c\n- v08 --refutes--> v11\n- v00 --refutes--> v0b\n- v02 --refutes--> v05\n- v04 --supports--> v01\n- v01 --supports--> v02\n- v10 --refutes--> v0f\n- v0b --refutes--> v03\n- v12 --refutes--> v15\n- v0e --refutes--> v08\n\n[VERIFIED ROOT]\nv12 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3037c728805595e13c208238af4cff57e9711df7f79cf3931d48dd4c93332228","schema":"livegraph.boolean-propagation.v1","spec_hash":"8030ba706ac59766eb62030d736187761e8cd58afe08f851a4bf5f6034c3de69","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v12\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-f1bff067fa9f206fcb33","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v01\n- v0c --supports--> v06\n- v06 --supports--> v02\n- v13 --supports--> v14\n- v14 --supports--> v09\n- v0a --supports--> v13\n- v0a --refutes--> v03\n- v10 --refutes--> v07\n- v00 --refutes--> v0c\n- v09 --refutes--> v10\n- v0e --refutes--> v05\n- v00 --refutes--> v04\n- v09 --supports--> v15\n- v10 --refutes--> v11\n- v10 --refutes--> v12\n- v05 --refutes--> v00\n- v13 --refutes--> v0b\n- v0d --supports--> v0e\n- v10 --supports--> v0f\n- v07 --supports--> v0d\n- v02 --refutes--> v08\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4578bb946f47df74d706ea22a483f85299f107d52ea6731eebd4981794f66272","schema":"livegraph.boolean-propagation.v1","spec_hash":"f1bff067fa9f206fcb3333a0713088d3a29670ccb81df29fd609c91fa9afdddf","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-f56a9c8d08f7649025fa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --refutes--> v12\n- v02 --supports--> v01\n- v01 --refutes--> v0c\n- v15 --supports--> v0a\n- v03 --refutes--> v09\n- v0a --supports--> v0b\n- v02 --supports--> v11\n- v0c --refutes--> v0f\n- v10 --supports--> v13\n- v0a --supports--> v00\n- v14 --refutes--> v05\n- v0f --refutes--> v14\n- v0e --supports--> v02\n- v10 --refutes--> v04\n- v0c --supports--> v0d\n- v06 --refutes--> v0e\n- v15 --refutes--> v07\n- v13 --supports--> v06\n- v12 --supports--> v03\n- v02 --refutes--> v08\n- v09 --refutes--> v15\n\n[VERIFIED ROOT]\nv10 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bf72a0330dbe5f183311a6f51785f82277f968f65516c863b5e2817d6cabd004","schema":"livegraph.boolean-propagation.v1","spec_hash":"f56a9c8d08f7649025fac662825b5d793061255d67fd9baa3ec7392359ae0d75","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-d345302e2e12d52e4329","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v0a\n- v0f --refutes--> v09\n- v02 --refutes--> v0b\n- v05 --supports--> v00\n- v0e --supports--> v0f\n- v01 --supports--> v0e\n- v07 --refutes--> v12\n- v11 --refutes--> v14\n- v07 --supports--> v06\n- v07 --supports--> v08\n- v00 --supports--> v07\n- v0c --supports--> v02\n- v10 --refutes--> v04\n- v10 --refutes--> v11\n- v0b --refutes--> v05\n- v14 --refutes--> v0d\n- v01 --supports--> v15\n- v09 --refutes--> v10\n- v08 --refutes--> v01\n- v01 --refutes--> v03\n- v09 --refutes--> v13\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f094b7b61438a596a702cad22fff41b7b34b6efcbbf3018d4115b1522e54f8d7","schema":"livegraph.boolean-propagation.v1","spec_hash":"d345302e2e12d52e4329371ce4b169bcfd13fb9540c57168313ed31fb85965f3","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-99f89adb07339f0f167f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v05\n- v12 --refutes--> v09\n- v0b --supports--> v02\n- v03 --refutes--> v0f\n- v0b --refutes--> v13\n- v13 --refutes--> v11\n- v13 --supports--> v0e\n- v05 --refutes--> v08\n- v0c --supports--> v12\n- v0d --refutes--> v01\n- v0e --refutes--> v0d\n- v09 --refutes--> v03\n- v0f --refutes--> v04\n- v07 --supports--> v0b\n- v12 --refutes--> v0a\n- v03 --refutes--> v15\n- v0e --supports--> v06\n- v08 --refutes--> v14\n- v0c --refutes--> v00\n- v01 --refutes--> v10\n- v08 --refutes--> v0c\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4a69a231b0d46fbae887f0eb1f40ae0fd848bdf880c1e8efbacbf620b7659a52","schema":"livegraph.boolean-propagation.v1","spec_hash":"99f89adb07339f0f167f4d6ee83d1830c0da5089f4fd6708dd20afcdf61953a9","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":14,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-72ea370c2686d2710bb0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v11\n- v10 --supports--> v06\n- v09 --refutes--> v02\n- v0f --refutes--> v13\n- v03 --refutes--> v15\n- v07 --supports--> v0d\n- v10 --refutes--> v03\n- v04 --refutes--> v10\n- v08 --refutes--> v09\n- v14 --supports--> v12\n- v0e --refutes--> v08\n- v00 --supports--> v07\n- v01 --supports--> v04\n- v11 --supports--> v0c\n- v03 --supports--> v0b\n- v09 --refutes--> v01\n- v0c --refutes--> v00\n- v08 --refutes--> v0a\n- v14 --supports--> v0f\n- v03 --refutes--> v05\n- v0d --supports--> v14\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"30a8f7f875f0f9c0b8b6c706d956064549845c48e94003ac96ac88169cb81b6b","schema":"livegraph.boolean-propagation.v1","spec_hash":"72ea370c2686d2710bb082f91fdbde10a78216f59c81982620963409f731f01e","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-4353081c37f8eadeb490","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v10\n- v0f --supports--> v14\n- v0c --refutes--> v0d\n- v14 --supports--> v0e\n- v06 --supports--> v0f\n- v0c --refutes--> v05\n- v0b --supports--> v04\n- v0e --refutes--> v07\n- v03 --refutes--> v0b\n- v14 --refutes--> v01\n- v0a --supports--> v02\n- v06 --supports--> v00\n- v0a --refutes--> v13\n- v0c --refutes--> v12\n- v12 --refutes--> v15\n- v11 --supports--> v03\n- v04 --refutes--> v09\n- v15 --refutes--> v11\n- v07 --supports--> v0c\n- v06 --refutes--> v08\n- v13 --supports--> v06\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cdbfc59a0c7c76ae8ac8e7b8e25a9a6ce855e7ca3961db0ac8d3bb4372dcb6e9","schema":"livegraph.boolean-propagation.v1","spec_hash":"4353081c37f8eadeb4906ea09f258946800969fee89f2e016d9d54f28c7865c4","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-5de2ba9161c80eb36a2a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v0e\n- v12 --supports--> v0b\n- v0b --refutes--> v02\n- v07 --refutes--> v00\n- v15 --refutes--> v05\n- v13 --refutes--> v0f\n- v01 --refutes--> v04\n- v0d --supports--> v03\n- v0f --supports--> v09\n- v05 --refutes--> v12\n- v08 --refutes--> v10\n- v14 --refutes--> v13\n- v12 --supports--> v0a\n- v01 --refutes--> v0c\n- v13 --refutes--> v11\n- v03 --refutes--> v14\n- v0f --supports--> v01\n- v07 --supports--> v0d\n- v0b --supports--> v07\n- v10 --supports--> v15\n- v06 --refutes--> v08\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dbd9342dffa2ad1c179b2d5ca43edf4e7eecb76a38ce61ca9afa59c29d50739a","schema":"livegraph.boolean-propagation.v1","spec_hash":"5de2ba9161c80eb36a2a6ffac432c388f6740229327c046814d44715f85ed198","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-b2cd2c6212915fbe817a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v0f\n- v0a --supports--> v00\n- v12 --refutes--> v06\n- v05 --refutes--> v15\n- v00 --refutes--> v12\n- v09 --refutes--> v0d\n- v15 --supports--> v07\n- v04 --refutes--> v0e\n- v08 --supports--> v02\n- v07 --refutes--> v0b\n- v03 --refutes--> v08\n- v10 --refutes--> v05\n- v01 --refutes--> v0a\n- v07 --supports--> v0c\n- v07 --supports--> v04\n- v15 --refutes--> v11\n- v02 --supports--> v10\n- v14 --supports--> v03\n- v04 --refutes--> v09\n- v04 --refutes--> v13\n- v09 --supports--> v01\n\n[VERIFIED ROOT]\nv14 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b95e0d0f01115f8e5f2ab89908a80da9a31d0d94a40e6f1d0a8ca11eedf90b59","schema":"livegraph.boolean-propagation.v1","spec_hash":"b2cd2c6212915fbe817a20281e20d27e7a075e309f8deb695c2d4bf4c3988133","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v14\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-3ef83d948eb8741e9e31","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v05\n- v08 --refutes--> v0e\n- v0e --refutes--> v15\n- v07 --refutes--> v04\n- v01 --supports--> v12\n- v0e --supports--> v0c\n- v11 --refutes--> v14\n- v15 --refutes--> v01\n- v07 --refutes--> v09\n- v0f --refutes--> v02\n- v13 --supports--> v08\n- v0d --refutes--> v06\n- v0d --refutes--> v03\n- v11 --refutes--> v0d\n- v05 --refutes--> v13\n- v04 --refutes--> v0f\n- v03 --supports--> v00\n- v00 --refutes--> v07\n- v01 --supports--> v10\n- v0d --supports--> v0b\n- v0d --supports--> v0a\n\n[VERIFIED ROOT]\nv11 = FALSE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1efa6224308fa6425741516dc5eac8840c5b9795c9b9dcc94b5ec6f339fb2d65","schema":"livegraph.boolean-propagation.v1","spec_hash":"3ef83d948eb8741e9e319734134a069b7914682950977e066aa7b84203e728bf","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-0d3199e2fe06bd4e4f00","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v11\n- v0b --refutes--> v14\n- v11 --refutes--> v00\n- v0b --refutes--> v04\n- v0a --refutes--> v02\n- v0d --supports--> v07\n- v01 --supports--> v0d\n- v10 --supports--> v05\n- v01 --supports--> v12\n- v11 --supports--> v09\n- v07 --refutes--> v08\n- v0b --refutes--> v06\n- v08 --supports--> v15\n- v02 --refutes--> v0e\n- v11 --refutes--> v13\n- v13 --refutes--> v0c\n- v14 --refutes--> v10\n- v10 --supports--> v03\n- v03 --supports--> v01\n- v13 --supports--> v0a\n- v15 --supports--> v0f\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f55b391e9bf2c94a639cb8abcc9eff266c6dd58a49e378628aea562025b1ac64","schema":"livegraph.boolean-propagation.v1","spec_hash":"0d3199e2fe06bd4e4f00cb9979f8f89f231ab3e6f9a1075570a2b70d57d28306","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-ca610757d0fa733c7c11","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v03\n- v01 --supports--> v05\n- v12 --supports--> v07\n- v05 --supports--> v13\n- v01 --supports--> v08\n- v03 --refutes--> v01\n- v09 --supports--> v0f\n- v09 --refutes--> v0d\n- v09 --supports--> v0a\n- v07 --refutes--> v06\n- v0a --refutes--> v14\n- v05 --refutes--> v0c\n- v02 --refutes--> v12\n- v15 --refutes--> v11\n- v06 --supports--> v15\n- v02 --refutes--> v0b\n- v0e --supports--> v02\n- v00 --supports--> v0e\n- v0c --refutes--> v10\n- v10 --supports--> v00\n- v15 --supports--> v04\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"089f82599182cd2fe4772aadb9e66d05d6448200efb1aebff34c517dddcf1ed0","schema":"livegraph.boolean-propagation.v1","spec_hash":"ca610757d0fa733c7c11266c9fed0edb1099ed0355fa1d03fb1060d05bd0886a","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-ab15d3bb46c55ef7c47b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v12\n- v15 --supports--> v13\n- v0d --supports--> v01\n- v14 --refutes--> v10\n- v11 --supports--> v0d\n- v0d --supports--> v05\n- v10 --supports--> v08\n- v03 --supports--> v06\n- v00 --refutes--> v09\n- v05 --refutes--> v02\n- v06 --refutes--> v0f\n- v10 --supports--> v07\n- v13 --supports--> v0a\n- v00 --refutes--> v0e\n- v02 --supports--> v00\n- v09 --supports--> v03\n- v04 --supports--> v14\n- v04 --refutes--> v0c\n- v13 --refutes--> v11\n- v02 --refutes--> v0b\n- v07 --refutes--> v15\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"02062f0981cc1515730f31b4fbe3fde7a5382963583236184b8191d4193d1991","schema":"livegraph.boolean-propagation.v1","spec_hash":"ab15d3bb46c55ef7c47b8bcb71c128869ed9fe9e39c607bfbef28bf779da6ff8","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-559ddeea16a1f91d9e7a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v0c\n- v15 --refutes--> v0e\n- v01 --refutes--> v06\n- v02 --supports--> v12\n- v0b --supports--> v01\n- v05 --refutes--> v0d\n- v08 --supports--> v11\n- v13 --refutes--> v15\n- v02 --supports--> v13\n- v0e --supports--> v10\n- v0e --supports--> v08\n- v14 --supports--> v02\n- v0d --refutes--> v0f\n- v11 --supports--> v05\n- v01 --refutes--> v07\n- v0a --supports--> v03\n- v0f --refutes--> v0b\n- v0d --refutes--> v00\n- v06 --supports--> v09\n- v0a --supports--> v14\n- v14 --refutes--> v04\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cf5f09bdd91d5f1e1e49339a73511ab0c20cd7dccd4543d8b356d2441acb6373","schema":"livegraph.boolean-propagation.v1","spec_hash":"559ddeea16a1f91d9e7add159d68840f6482da8af3e4fb5f250c0b70c48cf9d6","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-e333d42ea4d831187964","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v03\n- v03 --supports--> v05\n- v09 --refutes--> v0b\n- v06 --refutes--> v0e\n- v14 --supports--> v01\n- v14 --supports--> v04\n- v11 --refutes--> v0a\n- v12 --refutes--> v00\n- v0b --refutes--> v06\n- v11 --refutes--> v15\n- v01 --refutes--> v0c\n- v05 --supports--> v09\n- v0b --supports--> v08\n- v01 --refutes--> v10\n- v0e --refutes--> v0f\n- v12 --refutes--> v13\n- v15 --refutes--> v07\n- v0b --supports--> v0d\n- v02 --refutes--> v14\n- v0f --refutes--> v12\n- v01 --supports--> v11\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dc556c312f1ff16aad94d7196429f34c2b3fad9884418190b67b7d1671f18ba2","schema":"livegraph.boolean-propagation.v1","spec_hash":"e333d42ea4d831187964c41045e969786e4fa4793b484b9b11c9da183eabe065","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-3bf05c3f67442a76822d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v0d\n- v15 --supports--> v11\n- v0e --supports--> v14\n- v13 --refutes--> v0c\n- v04 --refutes--> v10\n- v02 --refutes--> v03\n- v05 --supports--> v13\n- v06 --supports--> v08\n- v0f --supports--> v02\n- v15 --supports--> v01\n- v04 --supports--> v15\n- v02 --refutes--> v05\n- v02 --supports--> v12\n- v01 --supports--> v0a\n- v13 --refutes--> v00\n- v0c --supports--> v06\n- v14 --refutes--> v09\n- v07 --refutes--> v0f\n- v01 --supports--> v07\n- v0d --supports--> v04\n- v01 --refutes--> v0b\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d81704a8a56e534d9c7d518ee110247b46cd89462189077b9608f59be9b8b44e","schema":"livegraph.boolean-propagation.v1","spec_hash":"3bf05c3f67442a76822d8af2d620afa11fa7173466667d1f26b22e6feba10271","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-31eaa893ef24dd99887f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v04\n- v0d --supports--> v08\n- v05 --supports--> v0f\n- v11 --refutes--> v10\n- v12 --supports--> v14\n- v05 --supports--> v03\n- v03 --supports--> v06\n- v06 --refutes--> v02\n- v10 --supports--> v05\n- v00 --supports--> v0b\n- v15 --supports--> v0e\n- v08 --supports--> v11\n- v15 --refutes--> v0d\n- v03 --supports--> v01\n- v14 --refutes--> v15\n- v14 --refutes--> v13\n- v12 --supports--> v07\n- v00 --supports--> v09\n- v0c --supports--> v00\n- v0d --refutes--> v0a\n- v04 --refutes--> v0c\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b9f50360942115cbe4f113f4ed61cd8ec0645567a06db562ff517a5bdc4d6d2b","schema":"livegraph.boolean-propagation.v1","spec_hash":"31eaa893ef24dd99887fa1d33cfad150f2cc6ff44a5f85ca88999de265161a9c","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-1aec0325927aa6ecde04","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v0d\n- v07 --supports--> v09\n- v0e --supports--> v11\n- v05 --refutes--> v0c\n- v0e --refutes--> v15\n- v0f --supports--> v04\n- v0d --refutes--> v0f\n- v08 --supports--> v12\n- v04 --refutes--> v03\n- v00 --refutes--> v07\n- v06 --supports--> v10\n- v00 --supports--> v13\n- v02 --supports--> v00\n- v15 --supports--> v06\n- v03 --refutes--> v01\n- v07 --supports--> v05\n- v01 --supports--> v0a\n- v0d --refutes--> v14\n- v12 --supports--> v02\n- v01 --supports--> v0e\n- v08 --supports--> v0b\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0b88ac3f2d1774a3a60542991e00ed6607b37cda10a2aef266cba9aa20d14a09","schema":"livegraph.boolean-propagation.v1","spec_hash":"1aec0325927aa6ecde0429a58e76730c2520234f768d5543eb62af4f21768865","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-0fb1112b3590c5327b99","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v05\n- v09 --supports--> v01\n- v03 --refutes--> v13\n- v02 --refutes--> v12\n- v00 --supports--> v10\n- v0f --refutes--> v0e\n- v05 --supports--> v0d\n- v08 --refutes--> v0b\n- v15 --supports--> v09\n- v12 --supports--> v07\n- v11 --supports--> v04\n- v12 --supports--> v11\n- v0d --refutes--> v03\n- v00 --refutes--> v0c\n- v00 --refutes--> v0f\n- v0d --refutes--> v14\n- v00 --supports--> v0a\n- v01 --refutes--> v02\n- v15 --refutes--> v06\n- v13 --refutes--> v15\n- v11 --supports--> v08\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2efae0d15e8b792d22517f46d6a5a0af2f4cdda93bc9528a78267cbb44352bdd","schema":"livegraph.boolean-propagation.v1","spec_hash":"0fb1112b3590c5327b99c34d34af6c0dcc06f7663df65e0b0acc335c149a39ad","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-57ffa48dfd8e7e240237","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v06\n- v00 --refutes--> v14\n- v07 --refutes--> v0a\n- v00 --supports--> v08\n- v07 --refutes--> v05\n- v02 --refutes--> v03\n- v15 --supports--> v11\n- v13 --refutes--> v01\n- v05 --supports--> v09\n- v04 --supports--> v0e\n- v09 --supports--> v0f\n- v0d --supports--> v15\n- v0b --supports--> v00\n- v08 --supports--> v0d\n- v15 --supports--> v02\n- v0e --refutes--> v07\n- v00 --supports--> v10\n- v01 --refutes--> v04\n- v0b --supports--> v0c\n- v0f --supports--> v0b\n- v09 --supports--> v12\n\n[VERIFIED ROOT]\nv13 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1f21fba01fa3a0cf01a9d6ee0c71677d7a62947833c15ca165b3ccf338003e28","schema":"livegraph.boolean-propagation.v1","spec_hash":"57ffa48dfd8e7e24023709a0eaae355cb7d760f59d69ef6c80ec1bb3011228de","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v13\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-1e059d438bbdb58e6411","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v0c\n- v0a --refutes--> v07\n- v0a --refutes--> v0d\n- v05 --supports--> v00\n- v0d --refutes--> v0f\n- v03 --supports--> v14\n- v0f --refutes--> v11\n- v14 --refutes--> v04\n- v15 --supports--> v10\n- v13 --refutes--> v05\n- v06 --refutes--> v0a\n- v14 --supports--> v02\n- v01 --supports--> v0e\n- v13 --supports--> v0b\n- v00 --refutes--> v06\n- v08 --supports--> v15\n- v11 --refutes--> v08\n- v01 --supports--> v03\n- v01 --supports--> v09\n- v04 --supports--> v13\n- v08 --supports--> v12\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a9dc0fd68a4d7759c6d590ecb9affcc2c2cd0aa85e0a225a07d2aa56d6731517","schema":"livegraph.boolean-propagation.v1","spec_hash":"1e059d438bbdb58e641185f37554f3594eee965616742c780890433abd1bbf2c","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-ece2a61b61e96a6bc407","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v00\n- v04 --supports--> v12\n- v14 --refutes--> v05\n- v00 --supports--> v0a\n- v14 --supports--> v03\n- v0a --refutes--> v06\n- v0b --refutes--> v13\n- v12 --supports--> v0b\n- v0c --supports--> v0f\n- v0a --refutes--> v08\n- v0f --refutes--> v14\n- v02 --supports--> v0d\n- v14 --supports--> v15\n- v08 --refutes--> v0e\n- v13 --supports--> v0c\n- v13 --refutes--> v09\n- v0a --supports--> v01\n- v02 --supports--> v04\n- v0e --refutes--> v10\n- v00 --refutes--> v11\n- v03 --supports--> v07\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c80ea8f1d2654eab538c8cb6a2a1d956216e157bbb323a2b6f1da800f41a09c4","schema":"livegraph.boolean-propagation.v1","spec_hash":"ece2a61b61e96a6bc4079de125f2ddd86a62bfb852ba54b33dd3d98cf556ff08","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-76a77bc512f67dacf6b4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v01\n- v0a --supports--> v0e\n- v0b --supports--> v0f\n- v04 --refutes--> v08\n- v00 --supports--> v11\n- v01 --supports--> v05\n- v01 --supports--> v0d\n- v11 --refutes--> v04\n- v0f --refutes--> v0c\n- v0d --supports--> v12\n- v14 --refutes--> v00\n- v14 --supports--> v07\n- v0d --supports--> v0b\n- v0c --refutes--> v03\n- v04 --refutes--> v15\n- v0c --supports--> v13\n- v08 --refutes--> v0a\n- v09 --refutes--> v02\n- v09 --refutes--> v14\n- v0b --supports--> v10\n- v03 --supports--> v06\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1342d0688be6d548f1b4d5dc11480cd8302d34907b055b74ac1256adaa7f9199","schema":"livegraph.boolean-propagation.v1","spec_hash":"76a77bc512f67dacf6b43e560d1653d03c56c8ff77f105de9923c6e0e690764b","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-61834fb98ea28a0e76ca","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v0d\n- v0c --supports--> v08\n- v02 --supports--> v00\n- v15 --supports--> v02\n- v05 --refutes--> v0c\n- v08 --supports--> v09\n- v06 --supports--> v01\n- v10 --refutes--> v0e\n- v0a --supports--> v11\n- v08 --refutes--> v14\n- v0a --supports--> v12\n- v0a --refutes--> v0b\n- v0c --supports--> v04\n- v07 --refutes--> v0a\n- v11 --supports--> v10\n- v03 --supports--> v06\n- v00 --supports--> v03\n- v0e --refutes--> v15\n- v0e --supports--> v13\n- v0c --supports--> v0f\n- v01 --supports--> v05\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv14\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7f82a0a6c2097c6ed3cf0dab002b4e286db3baf9427ef683a480cf272713979e","schema":"livegraph.boolean-propagation.v1","spec_hash":"61834fb98ea28a0e76ca31904977d524553120c0ba2a8088ffc369fd01d7d066","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-f6cea4d65e127aa96b4b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v13\n- v0e --supports--> v05\n- v05 --refutes--> v14\n- v13 --supports--> v03\n- v08 --supports--> v06\n- v0f --refutes--> v0d\n- v0d --supports--> v07\n- v08 --refutes--> v04\n- v0c --supports--> v0e\n- v0c --supports--> v09\n- v0e --refutes--> v0a\n- v12 --refutes--> v15\n- v01 --refutes--> v0f\n- v15 --supports--> v02\n- v10 --refutes--> v12\n- v0c --supports--> v11\n- v07 --refutes--> v0b\n- v0d --refutes--> v00\n- v0b --supports--> v10\n- v05 --supports--> v08\n- v15 --refutes--> v0c\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e63dd86467fe6efa1b249fff5916a92c0951a7257102c0362c855b7ec6bbd8af","schema":"livegraph.boolean-propagation.v1","spec_hash":"f6cea4d65e127aa96b4b80722557a0e1a55249583e346fc9977670cb1a2278f0","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-2a32adfc3b3957c0efa2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v0e\n- v0a --supports--> v0d\n- v00 --supports--> v11\n- v07 --supports--> v03\n- v07 --supports--> v09\n- v08 --refutes--> v13\n- v0c --supports--> v00\n- v02 --refutes--> v0f\n- v09 --supports--> v08\n- v0b --refutes--> v02\n- v01 --supports--> v0c\n- v11 --refutes--> v15\n- v15 --refutes--> v07\n- v00 --supports--> v04\n- v00 --supports--> v05\n- v14 --supports--> v0a\n- v0a --supports--> v10\n- v0f --supports--> v12\n- v0d --supports--> v0b\n- v0b --supports--> v06\n- v0f --refutes--> v01\n\n[VERIFIED ROOT]\nv14 = FALSE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8c78a3626b39eab1b690d7091168ab561ef8a61cfbf77c87ead6281fec5a1cd2","schema":"livegraph.boolean-propagation.v1","spec_hash":"2a32adfc3b3957c0efa2aa11665aa5f30c18412ead3c7ad67ce82cf3ac72d96a","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v14\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-d2e1f9e9aebb7db81e2c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v02\n- v10 --supports--> v09\n- v09 --supports--> v0e\n- v09 --supports--> v12\n- v02 --supports--> v10\n- v11 --supports--> v0b\n- v14 --refutes--> v07\n- v07 --refutes--> v15\n- v06 --supports--> v08\n- v06 --refutes--> v14\n- v05 --supports--> v00\n- v0c --supports--> v04\n- v05 --refutes--> v11\n- v0e --refutes--> v0a\n- v0b --supports--> v06\n- v11 --supports--> v0f\n- v0b --supports--> v0d\n- v0b --supports--> v01\n- v15 --refutes--> v13\n- v0c --refutes--> v05\n- v13 --refutes--> v03\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9bd210a3136db938f04e3b6889927517107910982c4e772bc4a1504f7d685624","schema":"livegraph.boolean-propagation.v1","spec_hash":"d2e1f9e9aebb7db81e2c969f16bf4f027b788f38fffe746a9742d830dc39dae9","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-f477d9088206de3eafc9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v08\n- v04 --refutes--> v0a\n- v13 --supports--> v02\n- v0d --supports--> v11\n- v12 --refutes--> v15\n- v00 --supports--> v10\n- v10 --refutes--> v0e\n- v12 --refutes--> v05\n- v13 --supports--> v03\n- v08 --refutes--> v0b\n- v06 --refutes--> v09\n- v0f --refutes--> v06\n- v09 --supports--> v0d\n- v06 --refutes--> v0c\n- v10 --supports--> v14\n- v0a --supports--> v12\n- v0b --refutes--> v04\n- v10 --supports--> v13\n- v0a --refutes--> v07\n- v02 --supports--> v01\n- v01 --refutes--> v0f\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"32d5249807136025b20c7efcf41024348071dbcc370a1d06937041835d6e6e9b","schema":"livegraph.boolean-propagation.v1","spec_hash":"f477d9088206de3eafc9d6014732971133cd3d489c9cb3054f4307ae49642f3f","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":14,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-eabb5e1d2f5793211918","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v15\n- v05 --supports--> v0e\n- v05 --refutes--> v03\n- v01 --refutes--> v0a\n- v14 --supports--> v0d\n- v11 --refutes--> v05\n- v12 --supports--> v10\n- v04 --supports--> v09\n- v00 --refutes--> v08\n- v01 --supports--> v02\n- v15 --refutes--> v06\n- v0b --supports--> v00\n- v10 --supports--> v04\n- v08 --refutes--> v12\n- v11 --supports--> v0f\n- v04 --refutes--> v07\n- v09 --supports--> v14\n- v0b --supports--> v13\n- v15 --refutes--> v11\n- v03 --refutes--> v01\n- v0d --refutes--> v0c\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"44506d400dc6f80fb3caf5ad8e85cfdbdbb45f150f611b39c221ee4cc1b50813","schema":"livegraph.boolean-propagation.v1","spec_hash":"eabb5e1d2f5793211918d26cf21ff1a86f51470d88eccd018e0055a969eb50fa","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-b33e8c620db32aa8bd02","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --supports--> v09\n- v12 --refutes--> v0f\n- v04 --supports--> v0e\n- v04 --refutes--> v0c\n- v0b --supports--> v15\n- v06 --supports--> v00\n- v0e --refutes--> v0d\n- v15 --supports--> v12\n- v0d --refutes--> v05\n- v06 --supports--> v11\n- v02 --supports--> v07\n- v13 --supports--> v10\n- v04 --supports--> v01\n- v0d --refutes--> v06\n- v14 --refutes--> v13\n- v08 --refutes--> v02\n- v13 --supports--> v03\n- v0a --supports--> v04\n- v10 --supports--> v0b\n- v00 --refutes--> v08\n- v02 --supports--> v14\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6e7813e8f32805ef665b52e83bcea38e14a6a8e849d122c89a494a13a7fb16a0","schema":"livegraph.boolean-propagation.v1","spec_hash":"b33e8c620db32aa8bd02e436ec3bae6f18d752873d365ee90522ea488bcc08de","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-a242ea001857ac189e7e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v08\n- v06 --refutes--> v0e\n- v0f --refutes--> v09\n- v0b --refutes--> v02\n- v05 --supports--> v06\n- v05 --supports--> v0c\n- v13 --refutes--> v0a\n- v03 --supports--> v04\n- v14 --supports--> v13\n- v0b --refutes--> v05\n- v09 --refutes--> v0d\n- v0e --supports--> v01\n- v08 --supports--> v0f\n- v0d --supports--> v14\n- v13 --refutes--> v11\n- v0d --supports--> v15\n- v07 --supports--> v03\n- v09 --refutes--> v12\n- v11 --refutes--> v0b\n- v09 --supports--> v10\n- v14 --refutes--> v00\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6c150e854c57b35c43ad1029f0c87a71f94fd30bcba65495a960c62f0982b9a9","schema":"livegraph.boolean-propagation.v1","spec_hash":"a242ea001857ac189e7efe6d390e7441a893f96affb197f9b8c4ae1a7922f604","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-389d33ee16c62448874a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v10\n- v00 --refutes--> v0a\n- v08 --refutes--> v15\n- v13 --supports--> v07\n- v10 --refutes--> v04\n- v0a --refutes--> v01\n- v09 --supports--> v05\n- v0b --refutes--> v03\n- v03 --supports--> v08\n- v11 --refutes--> v0b\n- v0a --supports--> v09\n- v06 --refutes--> v14\n- v13 --refutes--> v00\n- v00 --supports--> v12\n- v09 --refutes--> v11\n- v02 --refutes--> v0d\n- v14 --refutes--> v0e\n- v0e --supports--> v02\n- v0f --refutes--> v13\n- v0a --supports--> v0c\n- v10 --refutes--> v0f\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv15\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1d1a148425cc17e82c168d6d5cec48154e06b9d6fbc6172709228dea7b9d2d93","schema":"livegraph.boolean-propagation.v1","spec_hash":"389d33ee16c62448874ac6710d59a67584e3f3321e23de27cc59f6eb6e1a815b","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-2d735585deb4ab320dbd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v06\n- v0c --supports--> v01\n- v11 --refutes--> v13\n- v0b --refutes--> v03\n- v03 --supports--> v0a\n- v07 --refutes--> v0e\n- v0f --supports--> v14\n- v0c --supports--> v02\n- v03 --refutes--> v12\n- v07 --refutes--> v15\n- v05 --supports--> v0d\n- v0f --supports--> v11\n- v09 --refutes--> v0b\n- v13 --supports--> v0c\n- v06 --supports--> v09\n- v0a --supports--> v07\n- v0e --refutes--> v08\n- v00 --refutes--> v05\n- v0a --supports--> v10\n- v08 --supports--> v00\n- v06 --refutes--> v04\n\n[VERIFIED ROOT]\nv0f = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7ab7d87d83c41146052df7d83042a324a5ca44c8ec2dbf3a3f87bf1fc95e19eb","schema":"livegraph.boolean-propagation.v1","spec_hash":"2d735585deb4ab320dbdbb0139e8944e4f060e90c600bd715104eb7edeee7be1","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":14,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-51c8b39cc1e60ea4766a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --supports--> v0e\n- v0f --refutes--> v01\n- v09 --supports--> v03\n- v05 --supports--> v06\n- v0f --supports--> v12\n- v04 --supports--> v14\n- v13 --refutes--> v0c\n- v0d --refutes--> v07\n- v12 --refutes--> v11\n- v02 --refutes--> v10\n- v05 --supports--> v08\n- v0f --refutes--> v0a\n- v10 --refutes--> v05\n- v04 --supports--> v0d\n- v03 --refutes--> v0f\n- v00 --supports--> v15\n- v0c --supports--> v04\n- v00 --refutes--> v13\n- v04 --supports--> v0b\n- v07 --supports--> v02\n- v06 --refutes--> v09\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b470ba10959bc71e69619507177e6ffb55789463ecaa090502c1676f06f52cea","schema":"livegraph.boolean-propagation.v1","spec_hash":"51c8b39cc1e60ea4766a548ef31a248d0f2db1cde7434a5ddd175540dc43b7be","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":14,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-085f0646644b04cf9ca0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v06\n- v09 --refutes--> v08\n- v13 --refutes--> v00\n- v0f --refutes--> v14\n- v07 --refutes--> v0e\n- v10 --supports--> v11\n- v05 --refutes--> v0a\n- v03 --supports--> v16\n- v15 --refutes--> v02\n- v03 --supports--> v0d\n- v02 --supports--> v07\n- v0a --refutes--> v15\n- v08 --refutes--> v12\n- v03 --refutes--> v0c\n- v00 --supports--> v10\n- v12 --supports--> v03\n- v0f --supports--> v09\n- v02 --supports--> v0b\n- v0f --refutes--> v04\n- v16 --supports--> v05\n- v00 --supports--> v01\n- v11 --refutes--> v0f\n\n[VERIFIED ROOT]\nv13 = FALSE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"73bed68d1579090effb471873dba927d28ef117e2ce36af92e48635897a14ec5","schema":"livegraph.boolean-propagation.v1","spec_hash":"085f0646644b04cf9ca0764a0059a75989817973ca3016e8f76362fca7257ec9","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v13\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-bf3d1eac734972e89f39","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v16\n- v0c --refutes--> v06\n- v0a --supports--> v11\n- v04 --supports--> v02\n- v02 --refutes--> v0c\n- v10 --supports--> v05\n- v0e --refutes--> v15\n- v08 --refutes--> v0a\n- v0e --refutes--> v10\n- v11 --supports--> v00\n- v13 --refutes--> v08\n- v08 --refutes--> v09\n- v10 --refutes--> v0f\n- v01 --refutes--> v0b\n- v16 --supports--> v03\n- v10 --refutes--> v14\n- v06 --supports--> v13\n- v11 --supports--> v07\n- v13 --supports--> v0d\n- v03 --supports--> v12\n- v07 --supports--> v0e\n- v12 --refutes--> v04\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bbb29f8e72f39aff142d30848f4f03afb32df06b43b6a624bdc09b6a67d8ed70","schema":"livegraph.boolean-propagation.v1","spec_hash":"bf3d1eac734972e89f395147ed3e269afab67bf2759a5784bf6d76dd049e0c10","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-76636194880ba1506190","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v06\n- v00 --refutes--> v11\n- v0a --supports--> v0f\n- v14 --refutes--> v00\n- v01 --refutes--> v04\n- v0b --supports--> v16\n- v08 --refutes--> v05\n- v16 --supports--> v0a\n- v0c --supports--> v13\n- v13 --refutes--> v12\n- v08 --refutes--> v03\n- v05 --supports--> v15\n- v0a --supports--> v02\n- v0b --supports--> v07\n- v15 --supports--> v0c\n- v12 --supports--> v09\n- v0a --supports--> v14\n- v10 --supports--> v0b\n- v01 --refutes--> v0e\n- v0b --refutes--> v0d\n- v04 --supports--> v08\n- v11 --supports--> v01\n\n[VERIFIED ROOT]\nv10 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"545199831e4a08d310fe6f964d397ab222a06f8ae5e6fdd57f680ede315a9ade","schema":"livegraph.boolean-propagation.v1","spec_hash":"76636194880ba1506190bde370397b8823a5cb17911ea72593bdc646537fa11d","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-80931a4551c78ccf260c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v0a\n- v0a --supports--> v14\n- v06 --refutes--> v0e\n- v0e --supports--> v09\n- v09 --supports--> v0f\n- v0e --refutes--> v0c\n- v02 --supports--> v07\n- v03 --refutes--> v15\n- v00 --supports--> v10\n- v09 --refutes--> v16\n- v08 --refutes--> v0b\n- v12 --refutes--> v0d\n- v16 --refutes--> v08\n- v00 --refutes--> v04\n- v02 --refutes--> v11\n- v01 --supports--> v02\n- v09 --refutes--> v05\n- v15 --refutes--> v00\n- v14 --supports--> v01\n- v10 --supports--> v12\n- v0d --supports--> v06\n- v0a --supports--> v13\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8eec6215826df99f063f09c05d0deaedeaa365f6baa57efa74fb4c8c98bd2a7b","schema":"livegraph.boolean-propagation.v1","spec_hash":"80931a4551c78ccf260c2e1ad57099a1e2359343b69663437e4aa2cc8c2a3002","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-af31bf3c7ea1224e11d0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v02\n- v12 --supports--> v0f\n- v04 --supports--> v0a\n- v02 --supports--> v05\n- v07 --supports--> v0d\n- v13 --refutes--> v03\n- v05 --supports--> v10\n- v05 --refutes--> v09\n- v07 --refutes--> v13\n- v03 --refutes--> v01\n- v0a --supports--> v11\n- v11 --refutes--> v08\n- v01 --refutes--> v15\n- v01 --supports--> v00\n- v03 --supports--> v16\n- v15 --refutes--> v12\n- v09 --refutes--> v04\n- v06 --refutes--> v0c\n- v0b --supports--> v07\n- v0a --refutes--> v14\n- v07 --refutes--> v0e\n- v08 --refutes--> v0b\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ae822b1e26951608802c9f30e0edcc465d1357c705dade8826ca8802e705e423","schema":"livegraph.boolean-propagation.v1","spec_hash":"af31bf3c7ea1224e11d0ca2cd3828f74113c0d4c19cea400db6df09e2402cfc8","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-7ddb7fe9463b94fb1514","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v08\n- v08 --supports--> v0e\n- v0e --refutes--> v16\n- v0b --supports--> v0d\n- v10 --supports--> v0c\n- v03 --supports--> v09\n- v0d --supports--> v04\n- v00 --supports--> v01\n- v10 --refutes--> v13\n- v0e --supports--> v05\n- v0e --refutes--> v02\n- v12 --refutes--> v15\n- v04 --refutes--> v0f\n- v03 --refutes--> v10\n- v06 --supports--> v07\n- v03 --refutes--> v14\n- v0a --refutes--> v00\n- v13 --supports--> v06\n- v00 --refutes--> v11\n- v0e --refutes--> v0a\n- v0f --refutes--> v03\n- v15 --supports--> v0b\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c583174b06fdb27d399f568085256bdeac15e83ace1de477824a36a22690ea3a","schema":"livegraph.boolean-propagation.v1","spec_hash":"7ddb7fe9463b94fb151444b1de09b2bcdba3206f2de985cce4ff42d12cff1785","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-083a8b4eff62a3569238","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v13\n- v03 --supports--> v07\n- v0f --refutes--> v05\n- v08 --refutes--> v11\n- v0c --supports--> v04\n- v01 --refutes--> v03\n- v0e --refutes--> v0c\n- v08 --supports--> v0d\n- v11 --refutes--> v09\n- v14 --supports--> v00\n- v04 --refutes--> v14\n- v02 --refutes--> v0b\n- v0f --refutes--> v16\n- v06 --refutes--> v0f\n- v13 --supports--> v08\n- v02 --refutes--> v0a\n- v12 --supports--> v10\n- v16 --supports--> v01\n- v12 --supports--> v0e\n- v02 --refutes--> v12\n- v11 --supports--> v02\n- v08 --refutes--> v15\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"643a32dcf540fd7db49dd47fcfdd92d83c1d5512e8b95da5c187cc3e432e5014","schema":"livegraph.boolean-propagation.v1","spec_hash":"083a8b4eff62a3569238f37fee7748b4672b1dfb98a7e6e6937911d1eedc8429","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-02d0bc16eb63bfe495a3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --supports--> v16\n- v0d --supports--> v08\n- v0d --refutes--> v07\n- v0f --refutes--> v0c\n- v00 --supports--> v10\n- v04 --refutes--> v06\n- v0c --supports--> v04\n- v06 --refutes--> v09\n- v01 --supports--> v02\n- v15 --refutes--> v11\n- v05 --supports--> v0e\n- v00 --supports--> v12\n- v12 --supports--> v03\n- v00 --supports--> v14\n- v08 --refutes--> v13\n- v03 --supports--> v0b\n- v03 --refutes--> v15\n- v02 --refutes--> v0f\n- v08 --refutes--> v0a\n- v06 --refutes--> v0d\n- v13 --refutes--> v00\n- v0e --refutes--> v01\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv16\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"797c500fcf42b933989845a9a1abbde02d6844a28d76dbf23cceadcf697445ea","schema":"livegraph.boolean-propagation.v1","spec_hash":"02d0bc16eb63bfe495a314e6336d2612dd7241893e0adef53aa0b6f07ca94ad4","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v16\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-4b7009ad9ea9282d632b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v16 --supports--> v03\n- v04 --refutes--> v10\n- v00 --refutes--> v15\n- v01 --supports--> v07\n- v08 --refutes--> v06\n- v16 --refutes--> v0c\n- v04 --refutes--> v16\n- v11 --supports--> v08\n- v01 --refutes--> v13\n- v05 --supports--> v14\n- v00 --refutes--> v02\n- v13 --supports--> v0d\n- v03 --supports--> v09\n- v12 --refutes--> v00\n- v16 --supports--> v0a\n- v15 --refutes--> v11\n- v0d --refutes--> v0b\n- v0f --supports--> v0e\n- v0b --supports--> v04\n- v0e --supports--> v01\n- v08 --refutes--> v0f\n- v03 --refutes--> v05\n\n[VERIFIED ROOT]\nv12 = FALSE\n\n[TARGET]\nv14\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dc23d81f5209dd6db60e0f90c8fcafc393422cf2707fc09b219b7396afebabeb","schema":"livegraph.boolean-propagation.v1","spec_hash":"4b7009ad9ea9282d632bf248ddb5225d6a71ec04937595ab98ad40d2ab3406bf","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v12\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-2ea2b26677497cb525e0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v0c\n- v0c --refutes--> v0e\n- v0c --refutes--> v09\n- v0d --refutes--> v11\n- v10 --supports--> v0a\n- v0d --refutes--> v01\n- v13 --refutes--> v16\n- v11 --refutes--> v10\n- v05 --refutes--> v00\n- v0d --refutes--> v03\n- v0f --refutes--> v13\n- v10 --refutes--> v14\n- v04 --supports--> v02\n- v16 --refutes--> v04\n- v12 --supports--> v0f\n- v0d --refutes--> v0b\n- v02 --refutes--> v0d\n- v00 --supports--> v12\n- v0d --supports--> v07\n- v0a --refutes--> v08\n- v16 --supports--> v15\n- v08 --refutes--> v06\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9b3f2a4698f1769945b19c3c85f2ec13c51c3c731c292a00802f8e9a1881a98a","schema":"livegraph.boolean-propagation.v1","spec_hash":"2ea2b26677497cb525e06b75b7577da2c55eb097e15330c160bb4b624eca78c7","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-5561992c9aaeefc1df6f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v14\n- v0f --refutes--> v02\n- v12 --supports--> v08\n- v14 --refutes--> v15\n- v03 --supports--> v0b\n- v0a --refutes--> v04\n- v12 --refutes--> v0a\n- v14 --refutes--> v00\n- v10 --supports--> v03\n- v02 --supports--> v0d\n- v03 --refutes--> v16\n- v0d --supports--> v06\n- v02 --supports--> v01\n- v0c --refutes--> v05\n- v0e --supports--> v09\n- v0b --refutes--> v0f\n- v0a --refutes--> v11\n- v14 --supports--> v13\n- v05 --refutes--> v12\n- v09 --refutes--> v10\n- v00 --supports--> v0c\n- v12 --refutes--> v07\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7149ba23df70ed7d5511fc385fa70bedf7c98361b866d4d26b94e0b455556bef","schema":"livegraph.boolean-propagation.v1","spec_hash":"5561992c9aaeefc1df6f6e3348c03ed5e04bce7f7f61de2525efda66b9b84d86","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":15,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-56548943fe0fb95fd94e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v16\n- v01 --supports--> v12\n- v0f --refutes--> v00\n- v04 --supports--> v0b\n- v0b --refutes--> v01\n- v03 --supports--> v02\n- v08 --refutes--> v0e\n- v10 --supports--> v03\n- v0f --supports--> v08\n- v16 --supports--> v06\n- v0e --refutes--> v04\n- v08 --supports--> v13\n- v12 --supports--> v09\n- v05 --supports--> v0a\n- v01 --refutes--> v11\n- v10 --supports--> v15\n- v02 --refutes--> v05\n- v0d --supports--> v14\n- v16 --refutes--> v0d\n- v09 --refutes--> v0c\n- v09 --supports--> v10\n- v0e --supports--> v07\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv14\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d14a023f994eeebe63edcd0cb938aa5762bb04631e96c63ad04d8866dbf344e0","schema":"livegraph.boolean-propagation.v1","spec_hash":"56548943fe0fb95fd94e47aae3e139c100e7e22e50890e79b5e03c4410d3dc38","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-4cbd884a21a0145a6c1d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --refutes--> v01\n- v0f --refutes--> v06\n- v0d --refutes--> v12\n- v01 --refutes--> v15\n- v06 --supports--> v08\n- v11 --refutes--> v09\n- v04 --refutes--> v02\n- v15 --refutes--> v0e\n- v0a --refutes--> v00\n- v0a --refutes--> v03\n- v02 --refutes--> v13\n- v08 --refutes--> v0b\n- v15 --supports--> v16\n- v0b --supports--> v11\n- v00 --supports--> v10\n- v00 --supports--> v07\n- v14 --refutes--> v0f\n- v11 --refutes--> v0a\n- v12 --refutes--> v04\n- v00 --refutes--> v0d\n- v0a --supports--> v0c\n- v13 --refutes--> v05\n\n[VERIFIED ROOT]\nv14 = FALSE\n\n[TARGET]\nv16\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"34734c9b7fdd7ef0750c384a75b691596301e7226fdccdd2ab8e5bcbb3d6208b","schema":"livegraph.boolean-propagation.v1","spec_hash":"4cbd884a21a0145a6c1d31d03b31b62266a104f18d2a872355bcb2afdddcac55","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v14\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v16\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-1f341c4b318ecc4c5d4f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v09\n- v00 --refutes--> v04\n- v16 --refutes--> v0c\n- v00 --supports--> v02\n- v04 --supports--> v10\n- v13 --supports--> v06\n- v0a --supports--> v16\n- v05 --supports--> v07\n- v15 --refutes--> v03\n- v16 --supports--> v11\n- v0d --refutes--> v05\n- v01 --supports--> v0e\n- v11 --refutes--> v0f\n- v11 --supports--> v0d\n- v0e --refutes--> v12\n- v00 --refutes--> v0b\n- v07 --refutes--> v01\n- v10 --refutes--> v13\n- v15 --refutes--> v08\n- v03 --refutes--> v0a\n- v14 --supports--> v15\n- v0e --refutes--> v00\n\n[VERIFIED ROOT]\nv14 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"19ebfc9bff016acc21107c22ceb0cf6bd6f00573944cc17fffd8611d2fa51c11","schema":"livegraph.boolean-propagation.v1","spec_hash":"1f341c4b318ecc4c5d4fa1cbbd678f8c8f26b0a22a3331cf7c43642b12a1c7d4","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v14\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-d11eb94ea736c11e6822","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v10\n- v07 --refutes--> v0e\n- v0e --refutes--> v13\n- v12 --refutes--> v01\n- v05 --supports--> v09\n- v0d --supports--> v15\n- v05 --supports--> v0a\n- v04 --refutes--> v05\n- v0e --refutes--> v0b\n- v13 --supports--> v02\n- v13 --supports--> v04\n- v03 --refutes--> v07\n- v15 --refutes--> v16\n- v15 --refutes--> v14\n- v0c --refutes--> v12\n- v01 --refutes--> v0d\n- v0f --supports--> v06\n- v0d --supports--> v08\n- v14 --refutes--> v00\n- v06 --refutes--> v03\n- v13 --refutes--> v11\n- v00 --supports--> v0f\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"232d717d660e8a0cfa8f558e9297e9ef7ca205a4e0be04570c6bf7a68f679162","schema":"livegraph.boolean-propagation.v1","spec_hash":"d11eb94ea736c11e6822283c83a7507629c060dedce4e333887743aaa5c4538a","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-c6265c7d74c642305beb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v16\n- v09 --supports--> v07\n- v0c --refutes--> v0b\n- v02 --supports--> v05\n- v09 --refutes--> v01\n- v00 --supports--> v0d\n- v0d --supports--> v0f\n- v16 --refutes--> v0c\n- v05 --supports--> v14\n- v01 --supports--> v0e\n- v14 --refutes--> v15\n- v03 --refutes--> v11\n- v14 --refutes--> v09\n- v0d --supports--> v03\n- v0c --refutes--> v10\n- v0e --refutes--> v04\n- v12 --refutes--> v00\n- v01 --supports--> v13\n- v10 --refutes--> v08\n- v0c --refutes--> v0a\n- v08 --refutes--> v02\n- v10 --supports--> v06\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"391d156bfbb218cb95b8fdfe98555783b0771ea70123005b22cde97498e72d8c","schema":"livegraph.boolean-propagation.v1","spec_hash":"c6265c7d74c642305bebc29585f582b669de0a57716ea6c92d29accbb00483cb","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-7111ce0dc2375cea94fb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v08\n- v16 --refutes--> v0f\n- v15 --refutes--> v13\n- v07 --supports--> v12\n- v0b --supports--> v0e\n- v0f --refutes--> v01\n- v09 --supports--> v0d\n- v0c --supports--> v16\n- v0f --refutes--> v15\n- v09 --refutes--> v03\n- v13 --supports--> v06\n- v05 --refutes--> v00\n- v05 --refutes--> v02\n- v05 --refutes--> v04\n- v12 --refutes--> v09\n- v00 --refutes--> v07\n- v06 --refutes--> v05\n- v11 --supports--> v10\n- v08 --supports--> v0c\n- v08 --refutes--> v0a\n- v0e --refutes--> v11\n- v0c --refutes--> v14\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"32999662f0a90c577c8687c6b23184b95e8c9c7d7ca6743a0649726879ad5b02","schema":"livegraph.boolean-propagation.v1","spec_hash":"7111ce0dc2375cea94fb32ae3ef6a4cbbb1f7f21cfe03f11b27187a40aff7a9e","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-35e04658d5a8cd7e7d84","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --refutes--> v01\n- v10 --refutes--> v02\n- v04 --refutes--> v08\n- v01 --supports--> v11\n- v0b --supports--> v16\n- v01 --refutes--> v15\n- v0f --refutes--> v07\n- v05 --supports--> v0b\n- v0b --refutes--> v12\n- v0d --supports--> v10\n- v15 --refutes--> v0c\n- v16 --supports--> v0a\n- v05 --supports--> v00\n- v15 --refutes--> v03\n- v02 --supports--> v09\n- v05 --refutes--> v06\n- v07 --supports--> v0d\n- v09 --supports--> v04\n- v14 --refutes--> v13\n- v0e --refutes--> v14\n- v16 --refutes--> v0f\n- v08 --supports--> v0e\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cefc3b8581027474397b34d653b657ba2a86925a92bb2d9830c76c70e7228af9","schema":"livegraph.boolean-propagation.v1","spec_hash":"35e04658d5a8cd7e7d84fe90c18cef8898e93aea0690808f0599484aeb0ff867","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-c8f2cb8233f452782833","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --refutes--> v0f\n- v0d --refutes--> v09\n- v12 --refutes--> v07\n- v15 --refutes--> v05\n- v13 --supports--> v0d\n- v0b --refutes--> v14\n- v11 --supports--> v0b\n- v0f --refutes--> v0c\n- v0c --refutes--> v06\n- v0b --supports--> v01\n- v06 --supports--> v03\n- v0a --supports--> v10\n- v03 --supports--> v0e\n- v11 --refutes--> v00\n- v0a --refutes--> v04\n- v04 --refutes--> v12\n- v14 --supports--> v08\n- v09 --supports--> v15\n- v12 --supports--> v16\n- v02 --refutes--> v11\n- v03 --supports--> v0a\n- v12 --refutes--> v13\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9039facc31a4dda0fca01bd97658b0d56ab44352eb239abef76b3127e1db6b22","schema":"livegraph.boolean-propagation.v1","spec_hash":"c8f2cb8233f452782833eb09b380c95cbda3f4b14f99e29afae6f5c83b177376","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-2b1e611ad0e34dd63d8d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v02\n- v12 --supports--> v0e\n- v06 --supports--> v05\n- v06 --supports--> v09\n- v0c --refutes--> v06\n- v10 --supports--> v00\n- v16 --supports--> v13\n- v0b --supports--> v0f\n- v14 --supports--> v0d\n- v04 --refutes--> v14\n- v0b --supports--> v08\n- v06 --supports--> v01\n- v02 --refutes--> v16\n- v07 --supports--> v04\n- v0b --supports--> v11\n- v01 --supports--> v07\n- v0e --refutes--> v03\n- v00 --refutes--> v0a\n- v13 --refutes--> v0b\n- v0e --supports--> v0c\n- v0c --refutes--> v15\n- v14 --refutes--> v10\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"27764126684429f5a62e1f04a5740898168f7d3884fb463efd97ea030040f42a","schema":"livegraph.boolean-propagation.v1","spec_hash":"2b1e611ad0e34dd63d8df8da6a3e1c6ae1fb8abed8fafd57bf122314ebacae1a","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-98d46e4b0d89091dcc06","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v09\n- v0e --refutes--> v04\n- v14 --refutes--> v12\n- v02 --refutes--> v03\n- v0c --supports--> v01\n- v11 --refutes--> v14\n- v15 --refutes--> v0f\n- v0e --refutes--> v00\n- v0d --supports--> v05\n- v03 --refutes--> v06\n- v0a --refutes--> v02\n- v08 --refutes--> v0b\n- v08 --supports--> v0a\n- v12 --refutes--> v10\n- v01 --refutes--> v0d\n- v0d --supports--> v11\n- v13 --refutes--> v15\n- v03 --supports--> v0c\n- v0f --refutes--> v0e\n- v03 --supports--> v16\n- v04 --refutes--> v08\n- v12 --refutes--> v07\n\n[VERIFIED ROOT]\nv13 = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"acfc3f4943a3cc5810fe51370ff8ddf0bc11dcbb87600165019880a6414e0610","schema":"livegraph.boolean-propagation.v1","spec_hash":"98d46e4b0d89091dcc06a5b16b1bf6b1237842a688b6c4f0ef442b515898754b","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-4a940be7e06995fa685c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v02\n- v14 --supports--> v01\n- v03 --supports--> v16\n- v0e --refutes--> v13\n- v0d --supports--> v03\n- v10 --refutes--> v07\n- v13 --supports--> v04\n- v05 --supports--> v08\n- v11 --refutes--> v0d\n- v0b --refutes--> v0c\n- v03 --supports--> v06\n- v09 --supports--> v10\n- v0b --supports--> v11\n- v0f --refutes--> v05\n- v09 --supports--> v00\n- v08 --supports--> v0e\n- v0e --supports--> v0a\n- v14 --refutes--> v0b\n- v13 --supports--> v14\n- v07 --supports--> v12\n- v15 --refutes--> v0f\n- v12 --supports--> v15\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv16\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7dfd5b0ce175ee6ee63f1fcc5250741afd487d2ab09cb0542f8ed976d6b8c9c1","schema":"livegraph.boolean-propagation.v1","spec_hash":"4a940be7e06995fa685c8334ccb968184ff50bdc964aed8269ebcb50dbc0dc23","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v16\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-7f32a5617559442d3560","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v09\n- v0e --refutes--> v04\n- v15 --refutes--> v0c\n- v0e --refutes--> v0a\n- v02 --supports--> v0f\n- v05 --refutes--> v00\n- v06 --supports--> v10\n- v07 --refutes--> v11\n- v05 --refutes--> v0b\n- v0d --refutes--> v13\n- v0a --supports--> v15\n- v00 --refutes--> v07\n- v11 --refutes--> v02\n- v10 --refutes--> v05\n- v15 --refutes--> v08\n- v01 --supports--> v03\n- v0f --refutes--> v0e\n- v15 --refutes--> v12\n- v03 --supports--> v06\n- v00 --supports--> v14\n- v08 --supports--> v0d\n- v0f --supports--> v16\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7a0191ca5ace1e9e3a9d4461e8464baa81612e01269a63421f495cadfb7a8e78","schema":"livegraph.boolean-propagation.v1","spec_hash":"7f32a5617559442d356090854918603e99abe105b81c57d7c2541190fad92ade","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-72c492a851a1fa8c811a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v04\n- v03 --supports--> v13\n- v10 --supports--> v0e\n- v05 --supports--> v0a\n- v12 --refutes--> v0f\n- v13 --supports--> v14\n- v13 --supports--> v10\n- v04 --refutes--> v00\n- v11 --refutes--> v01\n- v11 --supports--> v06\n- v00 --supports--> v0d\n- v00 --refutes--> v16\n- v00 --supports--> v12\n- v05 --refutes--> v07\n- v10 --supports--> v08\n- v06 --supports--> v05\n- v08 --supports--> v0b\n- v09 --supports--> v03\n- v0c --refutes--> v15\n- v0f --refutes--> v11\n- v0b --refutes--> v02\n- v07 --supports--> v0c\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv15\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"53987602afc30924aeed84d2136468733af50b685b026d1abecc1b037760cd02","schema":"livegraph.boolean-propagation.v1","spec_hash":"72c492a851a1fa8c811a2c1112e05f5760ff3650b01d0e421a69b0c13c62764c","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":15,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-c4e6f5c9c0a28d2a61aa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v16 --refutes--> v00\n- v09 --refutes--> v07\n- v12 --refutes--> v13\n- v06 --supports--> v09\n- v0a --supports--> v15\n- v00 --supports--> v06\n- v11 --supports--> v12\n- v01 --refutes--> v04\n- v0d --supports--> v05\n- v04 --refutes--> v0a\n- v09 --refutes--> v0c\n- v0d --supports--> v01\n- v11 --supports--> v10\n- v14 --refutes--> v0d\n- v0a --refutes--> v03\n- v13 --supports--> v0b\n- v00 --supports--> v0e\n- v11 --supports--> v08\n- v0b --supports--> v14\n- v14 --supports--> v02\n- v0c --refutes--> v0f\n- v0f --refutes--> v11\n\n[VERIFIED ROOT]\nv16 = FALSE\n\n[TARGET]\nv15\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"980965cadc965fb792e0c6a083bc87b45f11258a635c18be0b7bb9d63dba57da","schema":"livegraph.boolean-propagation.v1","spec_hash":"c4e6f5c9c0a28d2a61aa4d896365e9391519b21738ec8316b3e9b62c8f1a9e1a","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v16\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-34d59880c80a2b452903","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v12\n- v02 --supports--> v14\n- v0c --refutes--> v03\n- v0e --supports--> v07\n- v0c --supports--> v06\n- v14 --supports--> v15\n- v07 --supports--> v13\n- v15 --supports--> v0e\n- v12 --supports--> v01\n- v10 --refutes--> v00\n- v02 --supports--> v11\n- v12 --refutes--> v0d\n- v0c --supports--> v08\n- v06 --refutes--> v0b\n- v0c --supports--> v05\n- v0f --supports--> v02\n- v06 --supports--> v16\n- v13 --refutes--> v0c\n- v0b --supports--> v10\n- v13 --supports--> v0a\n- v09 --refutes--> v0f\n- v04 --refutes--> v09\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"19458137cd571b5ad2550b4dbc39ce683d393972d33b167ec67d9eb562aa1033","schema":"livegraph.boolean-propagation.v1","spec_hash":"34d59880c80a2b452903574b46809228a77ccaed4a3da15a9db50d0d87b0cf1d","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-5b11a1cefcdf49480f0f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v08\n- v0f --refutes--> v14\n- v05 --supports--> v0e\n- v0a --supports--> v13\n- v10 --refutes--> v07\n- v13 --refutes--> v11\n- v04 --refutes--> v15\n- v01 --refutes--> v0d\n- v14 --supports--> v00\n- v10 --refutes--> v02\n- v00 --refutes--> v06\n- v0c --supports--> v10\n- v0e --supports--> v16\n- v10 --refutes--> v0a\n- v0d --refutes--> v0f\n- v15 --refutes--> v01\n- v16 --supports--> v04\n- v0a --supports--> v0b\n- v00 --refutes--> v12\n- v00 --refutes--> v03\n- v01 --refutes--> v09\n- v03 --supports--> v0c\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8dc13ed0da2220d9cba06a663cd6248399b9978e22c609db2b868d1aa5dbf2c1","schema":"livegraph.boolean-propagation.v1","spec_hash":"5b11a1cefcdf49480f0f44bb7587da7b186ac6612f7ef2a91e065534b2fb86d2","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-9801c5a08210b2a26842","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --supports--> v00\n- v14 --supports--> v10\n- v09 --supports--> v0b\n- v12 --refutes--> v0f\n- v0a --supports--> v03\n- v0f --refutes--> v09\n- v09 --refutes--> v04\n- v16 --refutes--> v13\n- v06 --supports--> v02\n- v03 --refutes--> v12\n- v13 --supports--> v14\n- v00 --refutes--> v06\n- v01 --supports--> v05\n- v0c --refutes--> v07\n- v07 --supports--> v0e\n- v0c --refutes--> v0d\n- v00 --refutes--> v08\n- v06 --supports--> v11\n- v0e --supports--> v16\n- v04 --supports--> v01\n- v06 --refutes--> v0a\n- v0e --refutes--> v15\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d410d59159fdadb3dd793165e9bae58d94582bee9579af8e71ce0bf8618e8551","schema":"livegraph.boolean-propagation.v1","spec_hash":"9801c5a08210b2a26842587ccf60ef2dc395695faceb826b9f8294c0ee861100","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-2cc1ff8cb5db1a32c97a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v05\n- v05 --supports--> v10\n- v0a --supports--> v0c\n- v14 --refutes--> v12\n- v07 --supports--> v0f\n- v0c --refutes--> v01\n- v05 --supports--> v0d\n- v13 --supports--> v08\n- v0c --supports--> v06\n- v03 --refutes--> v09\n- v0d --supports--> v15\n- v0b --refutes--> v0e\n- v09 --refutes--> v11\n- v0d --supports--> v0a\n- v0f --supports--> v14\n- v03 --supports--> v04\n- v16 --supports--> v02\n- v09 --supports--> v00\n- v13 --refutes--> v03\n- v12 --supports--> v16\n- v06 --supports--> v07\n- v0b --refutes--> v13\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"82088e9153138e7be68cf1761815cc333b0d22e30c4bb13f88f0df3bf5764f77","schema":"livegraph.boolean-propagation.v1","spec_hash":"2cc1ff8cb5db1a32c97a5d5e38d941823618781ec3833f6f59f15c3ef7f4fc89","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-bcd579bdc22c148b1147","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v0d\n- v00 --supports--> v0a\n- v11 --refutes--> v0f\n- v0d --refutes--> v02\n- v09 --refutes--> v01\n- v05 --refutes--> v0e\n- v15 --refutes--> v0c\n- v06 --refutes--> v12\n- v05 --supports--> v07\n- v0e --supports--> v03\n- v02 --refutes--> v10\n- v12 --supports--> v00\n- v0c --refutes--> v0b\n- v13 --supports--> v09\n- v15 --supports--> v14\n- v13 --refutes--> v16\n- v0b --refutes--> v06\n- v09 --refutes--> v04\n- v11 --refutes--> v15\n- v13 --supports--> v08\n- v10 --refutes--> v11\n- v00 --refutes--> v13\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"99f069ec8ab84863a96589f14c615553a850233b4bd27a3d6b27f385b0749db8","schema":"livegraph.boolean-propagation.v1","spec_hash":"bcd579bdc22c148b1147a587619a7d0015b928998cde54273734848fe52455b0","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-d3ba177d4bbec1d73257","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v0e\n- v12 --supports--> v02\n- v01 --supports--> v0f\n- v14 --refutes--> v16\n- v02 --supports--> v14\n- v12 --refutes--> v06\n- v04 --supports--> v11\n- v08 --supports--> v12\n- v07 --supports--> v05\n- v05 --supports--> v10\n- v12 --supports--> v09\n- v00 --refutes--> v01\n- v0a --supports--> v0c\n- v10 --refutes--> v13\n- v03 --refutes--> v0a\n- v14 --refutes--> v0d\n- v0f --refutes--> v03\n- v0c --supports--> v15\n- v15 --supports--> v07\n- v08 --refutes--> v0b\n- v04 --refutes--> v08\n- v10 --supports--> v04\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv16\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ea6d5283bb9ae8efef17c58788ac86c09fcbcbebabf558e8316e75c932632eb9","schema":"livegraph.boolean-propagation.v1","spec_hash":"d3ba177d4bbec1d732576dbdd1b01b9c292d9b6f28f6d4e1705496b7c33bceb4","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v16\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-ebd665cb3cdcfd5d5d2e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --supports--> v09\n- v02 --supports--> v15\n- v16 --supports--> v0e\n- v08 --supports--> v0a\n- v04 --refutes--> v06\n- v07 --refutes--> v13\n- v00 --supports--> v03\n- v12 --refutes--> v05\n- v08 --supports--> v0b\n- v0a --supports--> v00\n- v07 --supports--> v14\n- v0e --refutes--> v07\n- v01 --refutes--> v16\n- v15 --supports--> v12\n- v16 --refutes--> v0f\n- v06 --refutes--> v01\n- v02 --supports--> v11\n- v08 --supports--> v0d\n- v00 --supports--> v10\n- v10 --refutes--> v04\n- v0c --supports--> v02\n- v12 --refutes--> v08\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"64cceb20b78817695daab6e249b5fcc25d685ce2f150a22cde87cfa12a409706","schema":"livegraph.boolean-propagation.v1","spec_hash":"ebd665cb3cdcfd5d5d2e854fa8fc85e20834a79f781de35d2da1a8d185f99ba0","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-20d9e8780c42b7d01437","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v0e\n- v13 --supports--> v16\n- v0d --supports--> v01\n- v08 --supports--> v0c\n- v10 --refutes--> v03\n- v13 --supports--> v0a\n- v0e --refutes--> v14\n- v0c --refutes--> v0d\n- v00 --refutes--> v13\n- v03 --supports--> v02\n- v12 --supports--> v00\n- v0e --refutes--> v06\n- v03 --refutes--> v15\n- v01 --supports--> v0f\n- v04 --refutes--> v0b\n- v0c --supports--> v09\n- v11 --refutes--> v04\n- v01 --supports--> v11\n- v0a --refutes--> v10\n- v15 --refutes--> v08\n- v13 --refutes--> v05\n- v07 --supports--> v12\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ff07977cab6c350dad274696cab15502ece0eea146bdaafe742344513e79ef5d","schema":"livegraph.boolean-propagation.v1","spec_hash":"20d9e8780c42b7d01437abe1b905b7b9477430b2875a032080f559d15aef6ec2","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-b1495ad4e3afc0e947c2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v04\n- v12 --supports--> v13\n- v12 --supports--> v07\n- v06 --refutes--> v03\n- v08 --refutes--> v02\n- v11 --supports--> v05\n- v06 --refutes--> v00\n- v04 --supports--> v16\n- v14 --supports--> v15\n- v10 --supports--> v06\n- v11 --refutes--> v08\n- v0f --supports--> v0d\n- v14 --supports--> v0c\n- v09 --refutes--> v10\n- v06 --refutes--> v14\n- v08 --refutes--> v09\n- v01 --supports--> v11\n- v04 --refutes--> v0e\n- v0d --supports--> v12\n- v07 --supports--> v0b\n- v0e --supports--> v0a\n- v0a --supports--> v01\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv15\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1624244a8ab2786c12017ecd34e1b7f8df207294c80c2e842aa12d4dab8d60bc","schema":"livegraph.boolean-propagation.v1","spec_hash":"b1495ad4e3afc0e947c2648cb99bd0f24989f44ab8251ad2c82c7a196c92dd75","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":15,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-12ac03934f6cb8bbee9a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v06\n- v14 --supports--> v02\n- v11 --refutes--> v14\n- v10 --refutes--> v07\n- v10 --supports--> v11\n- v15 --refutes--> v16\n- v12 --supports--> v0d\n- v03 --supports--> v13\n- v0a --supports--> v03\n- v13 --refutes--> v0f\n- v08 --supports--> v0b\n- v0e --supports--> v15\n- v15 --supports--> v04\n- v0e --refutes--> v09\n- v0b --refutes--> v05\n- v01 --refutes--> v10\n- v04 --refutes--> v0c\n- v02 --refutes--> v08\n- v06 --supports--> v0a\n- v06 --refutes--> v00\n- v0b --supports--> v12\n- v13 --refutes--> v0e\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"529e76af648e710e7cf9cc74a47da6d442d69dd8ea9208ef8f65ee97e458f657","schema":"livegraph.boolean-propagation.v1","spec_hash":"12ac03934f6cb8bbee9a7e76e3450bdca2d8414cf8525bb9dba35d78cdec47b7","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-73fd7763b2b5e009d6ee","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v03\n- v08 --supports--> v16\n- v0d --supports--> v0b\n- v09 --refutes--> v0e\n- v0b --supports--> v02\n- v01 --refutes--> v10\n- v13 --supports--> v05\n- v0f --refutes--> v04\n- v0f --supports--> v14\n- v05 --refutes--> v11\n- v0e --refutes--> v0f\n- v08 --refutes--> v07\n- v0f --supports--> v0a\n- v0c --supports--> v15\n- v11 --refutes--> v01\n- v05 --refutes--> v12\n- v01 --supports--> v09\n- v0e --refutes--> v06\n- v04 --supports--> v00\n- v16 --refutes--> v0c\n- v15 --supports--> v0d\n- v02 --supports--> v13\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"030d434527b224793ef3b7a657110fd79d070577e65389c002bc56ed1c7a2f11","schema":"livegraph.boolean-propagation.v1","spec_hash":"73fd7763b2b5e009d6ee58257826c1c80bf6c5dedfc1cfeed74fa562343ad70c","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-dc5cb94251528bcca2ec","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v07\n- v08 --refutes--> v02\n- v16 --refutes--> v04\n- v08 --refutes--> v0d\n- v15 --refutes--> v0e\n- v01 --supports--> v06\n- v02 --supports--> v0c\n- v09 --refutes--> v12\n- v11 --supports--> v0a\n- v05 --refutes--> v16\n- v0c --refutes--> v01\n- v0e --supports--> v11\n- v04 --supports--> v13\n- v13 --refutes--> v10\n- v0b --refutes--> v09\n- v11 --supports--> v00\n- v15 --supports--> v0f\n- v09 --supports--> v08\n- v04 --refutes--> v03\n- v10 --supports--> v0b\n- v04 --supports--> v14\n- v0a --supports--> v05\n\n[VERIFIED ROOT]\nv15 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"602414c8a24327376e14648b650e3bb3adea1de735f1b161c22d1f556cb6e7e5","schema":"livegraph.boolean-propagation.v1","spec_hash":"dc5cb94251528bcca2ec038b72d019e80adde4a83efb8950feb5323449e2abf3","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v15\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-a307e52ec7a2eb6b9fdf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v16 --refutes--> v08\n- v16 --supports--> v0f\n- v06 --supports--> v01\n- v11 --supports--> v0a\n- v15 --supports--> v14\n- v16 --refutes--> v07\n- v0d --supports--> v11\n- v0b --refutes--> v04\n- v06 --supports--> v16\n- v04 --refutes--> v0e\n- v05 --refutes--> v0c\n- v03 --supports--> v0b\n- v0c --refutes--> v03\n- v04 --refutes--> v13\n- v0f --supports--> v05\n- v0a --supports--> v06\n- v02 --refutes--> v09\n- v13 --refutes--> v15\n- v06 --refutes--> v00\n- v0a --supports--> v10\n- v04 --supports--> v12\n- v14 --refutes--> v02\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bbca450b22cd05f4a751dd232ab905a949c8ab18c19f67f72e7be72d1f2c16d8","schema":"livegraph.boolean-propagation.v1","spec_hash":"a307e52ec7a2eb6b9fdf34c81adadb051904cc6f2c03e4f302c4f6f85326b5c2","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-41fec36d5f7ee2342a72","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --refutes--> v10\n- v0b --refutes--> v07\n- v14 --refutes--> v03\n- v02 --supports--> v13\n- v0f --supports--> v0e\n- v11 --refutes--> v14\n- v0f --refutes--> v15\n- v12 --supports--> v11\n- v01 --supports--> v0a\n- v13 --refutes--> v0f\n- v07 --refutes--> v08\n- v0a --refutes--> v0b\n- v05 --supports--> v16\n- v16 --refutes--> v00\n- v02 --supports--> v04\n- v06 --refutes--> v01\n- v09 --supports--> v12\n- v00 --refutes--> v09\n- v06 --refutes--> v0d\n- v07 --supports--> v02\n- v07 --refutes--> v0c\n- v0f --refutes--> v05\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5119bdcde1eb9421c373422aa1d021dcbe89a06d55c5b3bc716e8af7a10fef50","schema":"livegraph.boolean-propagation.v1","spec_hash":"41fec36d5f7ee2342a72fdac830119512591b035a6a049d51c595ed69dd0640c","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-0ff6c2f53b663828cfaf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v13\n- v01 --supports--> v14\n- v05 --supports--> v0e\n- v13 --supports--> v11\n- v09 --supports--> v0c\n- v0d --supports--> v12\n- v08 --supports--> v03\n- v12 --supports--> v0a\n- v02 --refutes--> v0b\n- v03 --refutes--> v01\n- v14 --refutes--> v10\n- v12 --refutes--> v08\n- v05 --refutes--> v16\n- v0b --supports--> v0d\n- v01 --refutes--> v04\n- v02 --supports--> v00\n- v05 --supports--> v0f\n- v09 --supports--> v15\n- v06 --refutes--> v02\n- v10 --refutes--> v05\n- v0e --supports--> v07\n- v11 --supports--> v06\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"478789b07496af90d43794e70ef423c575c79cccf5bbc9e742f52f8ff0d576b0","schema":"livegraph.boolean-propagation.v1","spec_hash":"0ff6c2f53b663828cfaf099aae53b8b431c7c516f331cb28260c20b238dc239d","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":15,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-03d02739a3dba18c2e92","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v0d\n- v0f --supports--> v10\n- v03 --supports--> v0a\n- v03 --supports--> v13\n- v15 --supports--> v0e\n- v08 --supports--> v16\n- v08 --refutes--> v11\n- v00 --supports--> v02\n- v12 --supports--> v00\n- v0e --supports--> v01\n- v0c --supports--> v12\n- v10 --refutes--> v05\n- v0c --refutes--> v14\n- v04 --refutes--> v0f\n- v01 --refutes--> v08\n- v16 --refutes--> v09\n- v0b --supports--> v03\n- v13 --refutes--> v0c\n- v00 --refutes--> v15\n- v10 --supports--> v0b\n- v09 --supports--> v07\n- v15 --refutes--> v06\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bbb5dac1e9da0287a9aa6c9f9569eda3d31f14a860a0dde0e5748e1d4dfbfcb4","schema":"livegraph.boolean-propagation.v1","spec_hash":"03d02739a3dba18c2e9228f6016069f65ddd1509350c1178da90c9f5dd728ffe","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-d50cf79f52c5c19ba02f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v13\n- v0f --refutes--> v01\n- v15 --supports--> v11\n- v13 --refutes--> v15\n- v05 --refutes--> v00\n- v03 --supports--> v0d\n- v08 --refutes--> v16\n- v0d --refutes--> v06\n- v09 --refutes--> v12\n- v14 --refutes--> v04\n- v14 --supports--> v07\n- v00 --refutes--> v09\n- v0b --refutes--> v0c\n- v08 --refutes--> v05\n- v10 --supports--> v02\n- v0f --refutes--> v0a\n- v06 --refutes--> v0f\n- v09 --supports--> v14\n- v15 --supports--> v08\n- v0b --supports--> v10\n- v15 --supports--> v0e\n- v04 --refutes--> v03\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7c409922325817035de99439fbac35299447ec56fb4881fb41e6c8b19bcfa509","schema":"livegraph.boolean-propagation.v1","spec_hash":"d50cf79f52c5c19ba02f04e0b0d585dceaee6acd1e86a6ae48e523fa95820ff2","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-e362f1e89a15f4fbcd76","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v05\n- v03 --supports--> v0f\n- v03 --supports--> v0e\n- v10 --supports--> v08\n- v01 --refutes--> v0c\n- v08 --supports--> v07\n- v02 --supports--> v0b\n- v06 --supports--> v04\n- v14 --refutes--> v0d\n- v11 --refutes--> v12\n- v0b --refutes--> v11\n- v01 --refutes--> v0a\n- v0e --supports--> v16\n- v12 --supports--> v15\n- v0e --refutes--> v09\n- v15 --refutes--> v10\n- v06 --supports--> v03\n- v0d --supports--> v02\n- v09 --refutes--> v14\n- v02 --refutes--> v00\n- v11 --supports--> v13\n- v05 --refutes--> v01\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"27cebcff0265fd3d62f0f203775bc0579756ce41d33c0795c46ca43fa7ef40d7","schema":"livegraph.boolean-propagation.v1","spec_hash":"e362f1e89a15f4fbcd767dfefa602fdc27232be3793e4f1da66c02d4e232ac1b","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-c0ae349e3b1815270df0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v12\n- v03 --supports--> v08\n- v09 --refutes--> v16\n- v03 --refutes--> v0f\n- v0c --supports--> v0b\n- v07 --refutes--> v10\n- v13 --supports--> v05\n- v11 --supports--> v04\n- v05 --refutes--> v01\n- v01 --refutes--> v11\n- v08 --supports--> v09\n- v16 --refutes--> v15\n- v12 --refutes--> v14\n- v08 --refutes--> v0a\n- v16 --supports--> v07\n- v0b --supports--> v00\n- v0d --supports--> v0c\n- v07 --refutes--> v13\n- v00 --supports--> v02\n- v03 --refutes--> v0e\n- v00 --refutes--> v03\n- v12 --refutes--> v0d\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b6084978171a4790cbc9a752ab7840bf184edcce1c6c2488b1007b1e103bf545","schema":"livegraph.boolean-propagation.v1","spec_hash":"c0ae349e3b1815270df00627e2e9773c971f8575330daf84bc02ad0fe1169fda","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-649bc6c981d932f91a41","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v15\n- v10 --supports--> v02\n- v12 --supports--> v07\n- v16 --refutes--> v01\n- v0c --supports--> v08\n- v16 --supports--> v05\n- v05 --refutes--> v06\n- v02 --refutes--> v0d\n- v07 --supports--> v00\n- v15 --supports--> v0f\n- v10 --refutes--> v03\n- v0f --refutes--> v13\n- v0a --refutes--> v0c\n- v0a --refutes--> v0b\n- v00 --supports--> v14\n- v12 --refutes--> v04\n- v02 --supports--> v11\n- v0c --supports--> v12\n- v13 --refutes--> v0a\n- v10 --refutes--> v09\n- v14 --refutes--> v10\n- v0e --supports--> v16\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dac8b7ab3cfe51384d3748f8ef3651e3375470ca1b2a128b5d435a94f80aa5c0","schema":"livegraph.boolean-propagation.v1","spec_hash":"649bc6c981d932f91a41e8a78c7f5b1343be00374d5b44790a094788d0fe1f7d","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-6a68c27af4e078930a7c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v0d\n- v06 --refutes--> v15\n- v0a --supports--> v02\n- v04 --supports--> v16\n- v0c --supports--> v13\n- v07 --refutes--> v0a\n- v14 --refutes--> v10\n- v0e --supports--> v06\n- v05 --refutes--> v14\n- v08 --refutes--> v0e\n- v12 --supports--> v01\n- v08 --supports--> v0f\n- v09 --supports--> v07\n- v06 --refutes--> v0c\n- v03 --refutes--> v09\n- v02 --supports--> v05\n- v05 --supports--> v0b\n- v04 --supports--> v03\n- v02 --supports--> v00\n- v10 --refutes--> v08\n- v0d --supports--> v04\n- v02 --refutes--> v11\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c9459e9fbba90601f798ce218694db8c7910d82314429dc749805b579912792e","schema":"livegraph.boolean-propagation.v1","spec_hash":"6a68c27af4e078930a7cf42cc3035059fafdfa006679002b9b6431d4afc6f3b0","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":15,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-d0f5cc77243156e6d31d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v14\n- v01 --refutes--> v16\n- v0a --refutes--> v0c\n- v0c --refutes--> v03\n- v05 --refutes--> v07\n- v03 --refutes--> v0b\n- v0f --refutes--> v02\n- v14 --refutes--> v10\n- v08 --supports--> v09\n- v05 --supports--> v12\n- v16 --supports--> v05\n- v06 --refutes--> v04\n- v0b --refutes--> v11\n- v13 --supports--> v01\n- v0f --supports--> v08\n- v0d --refutes--> v06\n- v04 --refutes--> v0a\n- v0d --supports--> v0e\n- v14 --refutes--> v0d\n- v0c --supports--> v15\n- v11 --refutes--> v13\n- v08 --refutes--> v00\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"228dbbefa5787876c2bc0140477a30d10b49177b18f164f933dbd11b8144b29b","schema":"livegraph.boolean-propagation.v1","spec_hash":"d0f5cc77243156e6d31d23a354901d2dd222f6495f566673ef1e6a71da18ddea","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-680dce791d68f0f9ef36","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v04\n- v0a --supports--> v07\n- v02 --refutes--> v15\n- v03 --refutes--> v13\n- v0b --refutes--> v06\n- v0d --refutes--> v0b\n- v11 --supports--> v0d\n- v15 --supports--> v16\n- v15 --supports--> v05\n- v0f --supports--> v10\n- v15 --refutes--> v0e\n- v04 --refutes--> v0f\n- v15 --supports--> v14\n- v13 --refutes--> v09\n- v0c --refutes--> v0a\n- v02 --refutes--> v00\n- v16 --refutes--> v0c\n- v01 --refutes--> v11\n- v06 --supports--> v02\n- v13 --refutes--> v12\n- v10 --refutes--> v03\n- v0f --supports--> v08\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"72cce2c1f3663eb31b7b6ee74cc691b53a7627ddabaf155c78b8b35a08186033","schema":"livegraph.boolean-propagation.v1","spec_hash":"680dce791d68f0f9ef36a5f26834683cb1c8b31fdf3fe7cdf43a9bff27ed4fb3","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-e2e47de3383b57c57e42","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v0c\n- v07 --refutes--> v08\n- v13 --supports--> v10\n- v02 --supports--> v16\n- v16 --refutes--> v14\n- v12 --refutes--> v0d\n- v09 --supports--> v05\n- v00 --refutes--> v04\n- v16 --refutes--> v00\n- v08 --supports--> v0a\n- v0f --supports--> v11\n- v0c --refutes--> v15\n- v0b --supports--> v12\n- v06 --refutes--> v0f\n- v09 --supports--> v07\n- v10 --refutes--> v09\n- v11 --supports--> v13\n- v09 --refutes--> v01\n- v0c --refutes--> v0b\n- v12 --refutes--> v02\n- v08 --refutes--> v03\n- v16 --supports--> v0e\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"76b25d25b3818b3c7822fac5ef6f8c38d5c0ceaa2c8088cc5263af0813bc7cf9","schema":"livegraph.boolean-propagation.v1","spec_hash":"e2e47de3383b57c57e42214ec518502a7036c4f2d47ee92a50b264bc5ec31ce3","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-9d3eba26ef7ef9b7ebf7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --supports--> v15\n- v06 --supports--> v0d\n- v04 --refutes--> v08\n- v14 --refutes--> v10\n- v13 --supports--> v0a\n- v05 --refutes--> v01\n- v0e --supports--> v13\n- v12 --refutes--> v0e\n- v0e --refutes--> v09\n- v0f --supports--> v06\n- v10 --supports--> v00\n- v02 --supports--> v0f\n- v0b --supports--> v12\n- v06 --refutes--> v14\n- v03 --refutes--> v07\n- v0c --refutes--> v03\n- v0a --supports--> v02\n- v12 --supports--> v16\n- v05 --supports--> v0c\n- v05 --supports--> v11\n- v00 --supports--> v05\n- v04 --refutes--> v0b\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"26d3bc4397eb3defde0b898e39ebe699772784e6607ca9b86648af3d5e18f8e6","schema":"livegraph.boolean-propagation.v1","spec_hash":"9d3eba26ef7ef9b7ebf752b0f9331b3efb0e0216c4acce769490dbb991a1c3be","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":15,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-0e9b4124c00677250219","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --supports--> v01\n- v0b --refutes--> v04\n- v0e --supports--> v0a\n- v15 --supports--> v05\n- v01 --supports--> v0d\n- v0c --supports--> v10\n- v02 --supports--> v13\n- v10 --refutes--> v16\n- v0b --refutes--> v00\n- v15 --supports--> v0f\n- v05 --supports--> v03\n- v02 --supports--> v06\n- v04 --supports--> v0c\n- v06 --supports--> v0b\n- v0b --refutes--> v08\n- v12 --supports--> v02\n- v10 --refutes--> v07\n- v01 --refutes--> v11\n- v09 --refutes--> v15\n- v16 --supports--> v14\n- v03 --refutes--> v12\n- v0e --supports--> v09\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"22e22f414d61eb38efb66b48f20fa87a2a0e9d4bf2add99050ec8037021496ee","schema":"livegraph.boolean-propagation.v1","spec_hash":"0e9b4124c00677250219a35d5cade38aa6e5c213bdcc6cbe09b5033390af5fc1","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-3de562df75eb922a45b6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v11\n- v16 --refutes--> v06\n- v00 --refutes--> v10\n- v01 --refutes--> v0c\n- v03 --supports--> v0a\n- v0c --refutes--> v16\n- v12 --supports--> v0e\n- v14 --refutes--> v05\n- v0a --supports--> v09\n- v02 --supports--> v0d\n- v0d --refutes--> v12\n- v03 --refutes--> v13\n- v07 --refutes--> v02\n- v03 --supports--> v0f\n- v12 --supports--> v15\n- v05 --supports--> v03\n- v10 --refutes--> v07\n- v12 --refutes--> v0b\n- v06 --refutes--> v04\n- v0a --supports--> v00\n- v04 --refutes--> v14\n- v12 --supports--> v08\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"713eda2242638b20321a0a8b835f61bc705bf5826f272916bdadafd203a94305","schema":"livegraph.boolean-propagation.v1","spec_hash":"3de562df75eb922a45b6617d50083d59264028c7dde44f7b713ae0da7e762ec5","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-46e5ea5db1fdf2a45c89","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v0c\n- v08 --supports--> v06\n- v0a --refutes--> v05\n- v05 --supports--> v00\n- v0b --supports--> v03\n- v16 --refutes--> v02\n- v11 --refutes--> v10\n- v10 --refutes--> v16\n- v0d --supports--> v0f\n- v00 --supports--> v12\n- v11 --supports--> v09\n- v14 --supports--> v0a\n- v03 --refutes--> v15\n- v02 --supports--> v08\n- v03 --refutes--> v14\n- v0b --supports--> v04\n- v00 --refutes--> v0d\n- v06 --supports--> v0e\n- v03 --supports--> v13\n- v02 --supports--> v01\n- v0f --supports--> v11\n- v07 --refutes--> v0b\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5980f2d0eecd3c6285de2ec89c3a8a9dfa41f83c4bbc071f6dd3a06580660b21","schema":"livegraph.boolean-propagation.v1","spec_hash":"46e5ea5db1fdf2a45c89029b7cdaef6add3f66a50d9526999593f896d41273ab","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-9f9a44d4f476b3be13ec","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v03\n- v11 --supports--> v0e\n- v01 --refutes--> v0f\n- v0e --refutes--> v09\n- v0f --supports--> v08\n- v07 --refutes--> v0a\n- v00 --refutes--> v01\n- v0e --supports--> v14\n- v14 --refutes--> v02\n- v05 --refutes--> v0d\n- v05 --refutes--> v00\n- v0c --refutes--> v07\n- v08 --supports--> v15\n- v0b --refutes--> v11\n- v15 --supports--> v06\n- v07 --supports--> v13\n- v15 --supports--> v0b\n- v03 --refutes--> v12\n- v14 --refutes--> v0c\n- v05 --refutes--> v16\n- v12 --refutes--> v10\n- v12 --refutes--> v04\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8925b2ee8abe1460f2aa41c79b03a48b013903fa7a92bdd91335c9c7789d7397","schema":"livegraph.boolean-propagation.v1","spec_hash":"9f9a44d4f476b3be13ec7c29f0087a7bad97e35974c9e58dbbb7064e380f3d33","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-2ec8c37a78319d01b468","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v06\n- v01 --supports--> v0f\n- v0e --supports--> v0a\n- v0b --refutes--> v12\n- v00 --supports--> v08\n- v0a --refutes--> v0b\n- v05 --refutes--> v15\n- v0c --supports--> v03\n- v11 --supports--> v05\n- v02 --refutes--> v10\n- v15 --refutes--> v0c\n- v06 --supports--> v16\n- v0f --refutes--> v00\n- v02 --refutes--> v0e\n- v09 --supports--> v01\n- v03 --supports--> v09\n- v0c --refutes--> v0d\n- v0e --supports--> v04\n- v02 --supports--> v07\n- v0b --supports--> v13\n- v16 --supports--> v02\n- v03 --supports--> v14\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d18dfc0923e8a654f1f11a06a7b1259638d07232d473fa435eca70237d54fd9f","schema":"livegraph.boolean-propagation.v1","spec_hash":"2ec8c37a78319d01b46896a2417e4b1545f60cdfffa8687921b82c8d0b1175f7","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-e1f80e72357ebb36d0e8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v0c\n- v15 --supports--> v0b\n- v0c --supports--> v08\n- v01 --supports--> v11\n- v0a --supports--> v10\n- v05 --supports--> v02\n- v10 --supports--> v0f\n- v0f --refutes--> v0d\n- v15 --refutes--> v07\n- v02 --refutes--> v01\n- v01 --supports--> v0a\n- v12 --refutes--> v03\n- v16 --refutes--> v09\n- v07 --refutes--> v16\n- v04 --supports--> v15\n- v04 --supports--> v0e\n- v04 --refutes--> v14\n- v13 --supports--> v04\n- v03 --supports--> v05\n- v0f --refutes--> v00\n- v0c --refutes--> v12\n- v09 --supports--> v06\n\n[VERIFIED ROOT]\nv13 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a06213bfbcd612136b3a121ec8618859b35ec96c7da26e5ba31951696ba9a22c","schema":"livegraph.boolean-propagation.v1","spec_hash":"e1f80e72357ebb36d0e8a098d7bc841d3539b6f594d7782d6ae0d4285cba1904","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-f3fdad9d9384c1cc8e0e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v0c\n- v05 --supports--> v0d\n- v0a --refutes--> v0e\n- v04 --refutes--> v07\n- v0b --refutes--> v0a\n- v12 --refutes--> v04\n- v0a --refutes--> v03\n- v08 --supports--> v00\n- v04 --refutes--> v10\n- v16 --supports--> v09\n- v08 --supports--> v12\n- v08 --supports--> v14\n- v0c --supports--> v0b\n- v03 --refutes--> v01\n- v0d --supports--> v16\n- v13 --refutes--> v08\n- v0f --refutes--> v15\n- v11 --supports--> v13\n- v03 --refutes--> v11\n- v07 --supports--> v05\n- v0f --refutes--> v02\n- v0b --supports--> v06\n\n[VERIFIED ROOT]\nv0f = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2c983caaf5d122959312b8a5eebcbb1a5a8b4605d758500d67d3f85c9240eeb5","schema":"livegraph.boolean-propagation.v1","spec_hash":"f3fdad9d9384c1cc8e0e5f211bfa6d7cec9cadc6a92524ecf152c6cd10dcf83b","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-69e4a47a822cd889bb56","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --supports--> v02\n- v05 --refutes--> v06\n- v0f --refutes--> v16\n- v16 --supports--> v13\n- v01 --refutes--> v07\n- v08 --refutes--> v14\n- v05 --supports--> v0d\n- v04 --refutes--> v11\n- v14 --supports--> v09\n- v11 --supports--> v0e\n- v0f --refutes--> v00\n- v12 --refutes--> v0c\n- v01 --refutes--> v0f\n- v02 --supports--> v04\n- v12 --refutes--> v01\n- v02 --refutes--> v03\n- v0a --refutes--> v12\n- v15 --refutes--> v05\n- v0d --supports--> v08\n- v04 --supports--> v10\n- v04 --supports--> v0b\n- v09 --refutes--> v0a\n\n[VERIFIED ROOT]\nv15 = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"12cba9cc23364f539c7333e47115249a78bc61ba7695982320bcb131b3f7d1a6","schema":"livegraph.boolean-propagation.v1","spec_hash":"69e4a47a822cd889bb56997d53714bb75fada3886ffccede474e106df3db6b2a","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v15\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-81c78d385625c45d3ca5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v14\n- v04 --refutes--> v0c\n- v01 --supports--> v0e\n- v0f --refutes--> v10\n- v07 --supports--> v02\n- v12 --refutes--> v01\n- v11 --refutes--> v0b\n- v11 --supports--> v0a\n- v0e --supports--> v0f\n- v06 --supports--> v04\n- v0a --supports--> v09\n- v10 --refutes--> v05\n- v11 --refutes--> v13\n- v00 --supports--> v11\n- v14 --refutes--> v00\n- v11 --refutes--> v15\n- v01 --supports--> v0d\n- v0c --refutes--> v16\n- v0a --refutes--> v12\n- v06 --refutes--> v03\n- v05 --refutes--> v08\n- v08 --refutes--> v06\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv16\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dd5bc125f357ac96afdacc480a6b010fa666d2a29afe925ce4d1cfb79bb30818","schema":"livegraph.boolean-propagation.v1","spec_hash":"81c78d385625c45d3ca503b7c247d1a7fcb38fe0851c069892a322b65ba45c52","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":15,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v16\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-af355de6990d99fa53a0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --supports--> v00\n- v0f --refutes--> v06\n- v0a --supports--> v04\n- v11 --supports--> v09\n- v0b --refutes--> v11\n- v0d --refutes--> v0b\n- v0c --supports--> v05\n- v0e --supports--> v12\n- v03 --refutes--> v0c\n- v16 --refutes--> v13\n- v04 --supports--> v01\n- v04 --refutes--> v0f\n- v05 --refutes--> v16\n- v07 --refutes--> v0d\n- v15 --supports--> v0a\n- v16 --refutes--> v07\n- v0e --refutes--> v02\n- v05 --refutes--> v08\n- v0f --supports--> v0e\n- v05 --supports--> v14\n- v12 --refutes--> v10\n- v09 --supports--> v15\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"172159c18005239c52cc64aea3d3e38c9ecb63c68352317903fda2cdb12099ae","schema":"livegraph.boolean-propagation.v1","spec_hash":"af355de6990d99fa53a0feec76bd3b95d6592b3501c60b8f1a9d6d4b3883ac12","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-1ac092c0e9f35e8300b4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --supports--> v0e\n- v00 --supports--> v08\n- v0e --supports--> v03\n- v0d --supports--> v09\n- v11 --supports--> v16\n- v03 --supports--> v02\n- v0f --supports--> v0a\n- v04 --refutes--> v05\n- v03 --refutes--> v15\n- v0c --supports--> v0d\n- v11 --refutes--> v10\n- v11 --supports--> v00\n- v0f --supports--> v06\n- v0e --refutes--> v12\n- v0d --supports--> v14\n- v13 --refutes--> v04\n- v00 --refutes--> v01\n- v07 --supports--> v0f\n- v02 --supports--> v0b\n- v0a --supports--> v13\n- v01 --refutes--> v0c\n- v05 --supports--> v11\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"82c568c1b0f90729ce3a198eefdb8732c4db11f995f0f6b9b66d3721749d8e2d","schema":"livegraph.boolean-propagation.v1","spec_hash":"1ac092c0e9f35e8300b4b5ed4f8b9be25826374e42de1b6032c16ef17f8cb425","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":15,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-3cf4d85457c50d8afcc0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v0f\n- v10 --supports--> v0c\n- v03 --supports--> v00\n- v12 --refutes--> v02\n- v09 --refutes--> v10\n- v09 --supports--> v16\n- v0b --supports--> v12\n- v06 --supports--> v13\n- v15 --supports--> v07\n- v06 --refutes--> v04\n- v06 --refutes--> v14\n- v01 --refutes--> v0a\n- v03 --supports--> v05\n- v02 --supports--> v11\n- v05 --supports--> v08\n- v08 --refutes--> v06\n- v01 --supports--> v09\n- v0c --supports--> v0e\n- v15 --supports--> v03\n- v0f --supports--> v15\n- v04 --supports--> v0b\n- v02 --refutes--> v01\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e9815a93619cc0bbcc47f564fbfbad21419ef0dd59e1f2678aeb9f5d29f7e15b","schema":"livegraph.boolean-propagation.v1","spec_hash":"3cf4d85457c50d8afcc06ff3cbe976fdfdfe8e1bd96836c2e29c89e86dbc8495","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-add642da0b711a814c20","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v05\n- v0c --supports--> v12\n- v0c --supports--> v10\n- v0f --supports--> v15\n- v06 --refutes--> v11\n- v03 --refutes--> v02\n- v15 --supports--> v06\n- v05 --refutes--> v09\n- v0b --refutes--> v0a\n- v01 --supports--> v0e\n- v16 --refutes--> v0f\n- v11 --refutes--> v07\n- v01 --supports--> v00\n- v05 --refutes--> v16\n- v0e --supports--> v08\n- v11 --refutes--> v14\n- v11 --supports--> v01\n- v0c --supports--> v13\n- v08 --refutes--> v0c\n- v13 --refutes--> v04\n- v02 --supports--> v0b\n- v03 --refutes--> v0d\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9bb1d5d999042e0332097f4f3776b936ef9e9754c8f95dca60bfa476609d7d0a","schema":"livegraph.boolean-propagation.v1","spec_hash":"add642da0b711a814c20c8e8fa4efe4ee5b89539d2d83e59996ffecfcdb2ad83","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-2f9116db2c5af8f034c7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v04\n- v16 --supports--> v06\n- v0f --supports--> v15\n- v00 --refutes--> v02\n- v05 --refutes--> v0b\n- v03 --refutes--> v12\n- v08 --supports--> v01\n- v02 --supports--> v0a\n- v15 --supports--> v10\n- v0b --refutes--> v08\n- v12 --refutes--> v11\n- v10 --supports--> v03\n- v07 --supports--> v0d\n- v12 --refutes--> v00\n- v02 --supports--> v16\n- v06 --supports--> v07\n- v08 --supports--> v09\n- v0d --supports--> v05\n- v00 --refutes--> v0e\n- v02 --refutes--> v13\n- v00 --refutes--> v0c\n- v14 --supports--> v0f\n\n[VERIFIED ROOT]\nv14 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f4313b3fb515fdf86917c38307808da6d91d3559248036b4a2a813a7a59bd5cc","schema":"livegraph.boolean-propagation.v1","spec_hash":"2f9116db2c5af8f034c76d7c294295181371e8af2224cfb0403530f912152d33","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v14\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-833386bb8db69f7a8033","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v08\n- v07 --refutes--> v0b\n- v0b --refutes--> v0f\n- v0e --supports--> v00\n- v08 --refutes--> v11\n- v00 --supports--> v10\n- v10 --supports--> v05\n- v08 --refutes--> v04\n- v0d --refutes--> v01\n- v0c --supports--> v09\n- v01 --refutes--> v02\n- v0f --supports--> v0e\n- v03 --refutes--> v16\n- v14 --refutes--> v0d\n- v03 --refutes--> v12\n- v03 --refutes--> v0a\n- v03 --supports--> v14\n- v01 --refutes--> v0c\n- v13 --supports--> v07\n- v05 --refutes--> v03\n- v07 --refutes--> v15\n- v0b --refutes--> v06\n\n[VERIFIED ROOT]\nv13 = FALSE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f0dcf0b75075f36708c97b38f0d7f6835be1b0a31bb8abe5cf7d0822a8eaf9c6","schema":"livegraph.boolean-propagation.v1","spec_hash":"833386bb8db69f7a803349550402d726f6e49ae4ba9a9c9a4b99d164c757e843","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":15,\"root\":\"v13\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-5e6761c6969e230bdce8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v08\n- v01 --refutes--> v16\n- v0c --supports--> v14\n- v14 --supports--> v04\n- v14 --refutes--> v0e\n- v07 --refutes--> v0c\n- v14 --refutes--> v12\n- v10 --refutes--> v0d\n- v02 --supports--> v09\n- v0a --supports--> v02\n- v0f --supports--> v10\n- v0f --supports--> v05\n- v08 --refutes--> v07\n- v13 --refutes--> v0a\n- v0a --refutes--> v00\n- v09 --refutes--> v06\n- v16 --refutes--> v0f\n- v16 --supports--> v15\n- v12 --supports--> v03\n- v03 --supports--> v13\n- v01 --refutes--> v0b\n- v09 --supports--> v11\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"eb876d1d0ac05a63a1930dd373d50e87076d12b51536fe13ef507c6c69b1b938","schema":"livegraph.boolean-propagation.v1","spec_hash":"5e6761c6969e230bdce8f9ff836a80b2a853567ec8ee561ce0d35e4eec0a9116","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-46e60f650f326030f351","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --refutes--> v0e\n- v10 --supports--> v11\n- v16 --supports--> v0c\n- v0b --supports--> v04\n- v0d --refutes--> v10\n- v11 --refutes--> v06\n- v08 --refutes--> v14\n- v06 --supports--> v15\n- v02 --refutes--> v03\n- v05 --refutes--> v08\n- v02 --refutes--> v13\n- v0f --refutes--> v16\n- v16 --refutes--> v01\n- v01 --supports--> v07\n- v14 --refutes--> v00\n- v01 --refutes--> v0b\n- v09 --supports--> v0d\n- v0b --refutes--> v05\n- v0e --supports--> v02\n- v06 --refutes--> v0a\n- v09 --supports--> v12\n- v0a --supports--> v0f\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7578f7102a531301e20d800401c46bb827c71015fb803bc828852b2b501e3be2","schema":"livegraph.boolean-propagation.v1","spec_hash":"46e60f650f326030f3512c52d26642a3f5d010ad0ae100cdc2b06101492a1034","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-5b8970ff770c67f50b7a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v15\n- v11 --supports--> v00\n- v11 --supports--> v10\n- v13 --refutes--> v12\n- v0f --supports--> v06\n- v10 --supports--> v13\n- v0e --supports--> v14\n- v16 --refutes--> v11\n- v10 --refutes--> v01\n- v0d --refutes--> v16\n- v05 --supports--> v09\n- v0b --supports--> v0d\n- v05 --refutes--> v04\n- v09 --supports--> v0c\n- v09 --refutes--> v0a\n- v0a --refutes--> v03\n- v0e --refutes--> v08\n- v03 --refutes--> v0f\n- v13 --supports--> v0e\n- v07 --supports--> v05\n- v11 --refutes--> v02\n- v06 --refutes--> v0b\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv15\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bc8751ecf6eec024d0ebd627d7da75e72770dcff28cc642a231aa0beb303eacc","schema":"livegraph.boolean-propagation.v1","spec_hash":"5b8970ff770c67f50b7ae5a70503d054b6495800bb746fd3dcfc87ae0f65ebc9","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-aa6f69c1e66cc585a768","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v0d\n- v08 --refutes--> v16\n- v07 --supports--> v02\n- v09 --refutes--> v04\n- v07 --supports--> v01\n- v03 --supports--> v00\n- v12 --supports--> v07\n- v16 --supports--> v05\n- v0c --refutes--> v09\n- v09 --refutes--> v13\n- v07 --supports--> v0c\n- v03 --supports--> v0f\n- v15 --supports--> v08\n- v0e --refutes--> v06\n- v0b --supports--> v15\n- v10 --supports--> v0e\n- v04 --supports--> v03\n- v0d --refutes--> v12\n- v0c --supports--> v14\n- v16 --supports--> v11\n- v11 --refutes--> v10\n- v0e --refutes--> v0a\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4bd83d9fd470ffaa9a86abb477a0015ff2186e782d1a2edf08d12b10ed74d000","schema":"livegraph.boolean-propagation.v1","spec_hash":"aa6f69c1e66cc585a7686fa1bc524ac1469df32b375ae605ab5dc9b66f94a941","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-9c1373b716ada87c3526","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v16 --refutes--> v02\n- v06 --refutes--> v0d\n- v09 --refutes--> v0e\n- v00 --refutes--> v12\n- v06 --supports--> v03\n- v08 --refutes--> v10\n- v09 --refutes--> v01\n- v0b --supports--> v0a\n- v00 --refutes--> v04\n- v03 --refutes--> v05\n- v12 --supports--> v06\n- v14 --refutes--> v00\n- v0c --supports--> v07\n- v15 --refutes--> v16\n- v07 --refutes--> v08\n- v03 --refutes--> v09\n- v08 --supports--> v13\n- v0a --refutes--> v0f\n- v0f --supports--> v15\n- v08 --supports--> v11\n- v13 --supports--> v0b\n- v16 --refutes--> v14\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c1615e6b4479ab00ea6997396a31873f67a2a84acf385d1835468b3438c48e71","schema":"livegraph.boolean-propagation.v1","spec_hash":"9c1373b716ada87c3526e72b54709583a26f07f04370341cb807e429a2ac7eb1","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-cabe80f83c7ceadee876","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v09\n- v05 --refutes--> v11\n- v0c --refutes--> v0a\n- v15 --refutes--> v0b\n- v06 --refutes--> v12\n- v0f --supports--> v15\n- v15 --refutes--> v06\n- v04 --supports--> v0c\n- v05 --supports--> v0f\n- v16 --supports--> v14\n- v0e --supports--> v07\n- v00 --refutes--> v16\n- v03 --refutes--> v05\n- v13 --supports--> v0d\n- v16 --supports--> v10\n- v0d --supports--> v01\n- v07 --supports--> v00\n- v0a --refutes--> v03\n- v00 --supports--> v02\n- v01 --refutes--> v04\n- v16 --refutes--> v08\n- v06 --supports--> v0e\n\n[VERIFIED ROOT]\nv13 = FALSE\n\n[TARGET]\nv14\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b1cfbe51452217505dc2a6129bc953f2c2d61a13c20e9cce05a9e92b5edc4aa6","schema":"livegraph.boolean-propagation.v1","spec_hash":"cabe80f83c7ceadee876245e6b3b2f247d8e8e08c8d324d51bdd1d4f505e68c9","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v13\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-7f5601b38b0a15c0bf55","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v14\n- v16 --refutes--> v01\n- v0f --refutes--> v03\n- v08 --refutes--> v05\n- v08 --refutes--> v07\n- v12 --supports--> v04\n- v11 --refutes--> v13\n- v03 --refutes--> v11\n- v05 --refutes--> v09\n- v0a --refutes--> v10\n- v0a --supports--> v06\n- v02 --refutes--> v0d\n- v12 --supports--> v0c\n- v04 --refutes--> v16\n- v03 --supports--> v0e\n- v09 --refutes--> v0f\n- v0d --supports--> v0b\n- v05 --refutes--> v15\n- v06 --supports--> v02\n- v0b --refutes--> v08\n- v12 --refutes--> v00\n- v13 --refutes--> v12\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b21a759dcc51c13377db9614a3b0d13fce1583d18a4cbd599cc985377700ae07","schema":"livegraph.boolean-propagation.v1","spec_hash":"7f5601b38b0a15c0bf55b042ea14fbc5392d6c9bf4633d421aaae8770f4f2e04","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-220123ac4388e0ccaa8b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v12\n- v14 --refutes--> v01\n- v01 --supports--> v09\n- v07 --supports--> v00\n- v06 --supports--> v14\n- v05 --refutes--> v11\n- v0d --refutes--> v06\n- v12 --supports--> v13\n- v10 --refutes--> v0d\n- v0a --supports--> v0e\n- v05 --supports--> v15\n- v01 --refutes--> v0c\n- v06 --refutes--> v02\n- v07 --supports--> v08\n- v0e --supports--> v05\n- v12 --supports--> v04\n- v0c --supports--> v07\n- v08 --refutes--> v03\n- v04 --supports--> v16\n- v16 --refutes--> v0b\n- v16 --refutes--> v10\n- v04 --refutes--> v0f\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ac344825aca2e224603b62618c76bd4ff2f41f2d7f9e187b616b8300863946b0","schema":"livegraph.boolean-propagation.v1","spec_hash":"220123ac4388e0ccaa8b39e417cbd56314113914d4e0ce84f55ef52b3f3ac26e","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-8f70ba96e31d2081d4da","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v15\n- v12 --supports--> v09\n- v03 --supports--> v08\n- v07 --refutes--> v11\n- v02 --refutes--> v01\n- v11 --supports--> v0d\n- v01 --supports--> v13\n- v09 --refutes--> v00\n- v11 --refutes--> v0f\n- v11 --refutes--> v04\n- v10 --refutes--> v02\n- v0c --refutes--> v05\n- v03 --refutes--> v14\n- v13 --supports--> v0e\n- v08 --supports--> v10\n- v0d --refutes--> v03\n- v0e --supports--> v12\n- v05 --refutes--> v07\n- v07 --refutes--> v16\n- v00 --supports--> v0b\n- v0e --refutes--> v06\n- v12 --supports--> v0a\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"144feff4137e59ec49459ed9c91fbb0c8c3bbc7aa00e5a67abdaeede76a9e131","schema":"livegraph.boolean-propagation.v1","spec_hash":"8f70ba96e31d2081d4dae676134c0af0cee0688c4c8f23d29ee4eab6ad6b1c39","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":15,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-d1cf3f495a01767c3e6e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v13\n- v09 --refutes--> v14\n- v06 --supports--> v0e\n- v13 --refutes--> v15\n- v06 --supports--> v03\n- v06 --supports--> v11\n- v15 --supports--> v0d\n- v04 --supports--> v07\n- v0a --supports--> v05\n- v0d --supports--> v08\n- v0b --supports--> v04\n- v16 --supports--> v01\n- v03 --supports--> v10\n- v12 --refutes--> v16\n- v05 --refutes--> v0b\n- v0d --supports--> v09\n- v01 --refutes--> v06\n- v05 --supports--> v0f\n- v04 --supports--> v02\n- v0c --refutes--> v00\n- v03 --refutes--> v0a\n- v09 --supports--> v12\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e9f826503b3aef1febd7d188462f3b01d2fe3e6999f4a4d762d59f7ddcbf03af","schema":"livegraph.boolean-propagation.v1","spec_hash":"d1cf3f495a01767c3e6e3511cf5e3b6cc5d09494b0ef5fc255fd7367bc6e277a","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-312f52b1d18abc76a90d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v01\n- v0f --refutes--> v0e\n- v02 --supports--> v04\n- v02 --refutes--> v0b\n- v06 --supports--> v0c\n- v0a --supports--> v08\n- v04 --supports--> v09\n- v01 --refutes--> v06\n- v13 --refutes--> v0f\n- v0f --supports--> v11\n- v12 --refutes--> v05\n- v13 --refutes--> v03\n- v12 --supports--> v14\n- v07 --refutes--> v0d\n- v0c --supports--> v07\n- v0a --refutes--> v02\n- v0e --supports--> v10\n- v14 --refutes--> v0a\n- v0f --supports--> v16\n- v15 --supports--> v13\n- v0d --refutes--> v15\n- v10 --refutes--> v00\n\n[VERIFIED ROOT]\nv12 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3df763b5577cdc3ca36b62704f32264b4068b262ae065ff4b5f7375c7e3e1843","schema":"livegraph.boolean-propagation.v1","spec_hash":"312f52b1d18abc76a90d2dbdf04d9b06d1495a9e5572cd1c619995b8bfeb19d0","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v12\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-ee2b49093b4d84fe5185","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v08\n- v03 --supports--> v15\n- v0d --refutes--> v01\n- v06 --refutes--> v11\n- v02 --refutes--> v14\n- v13 --supports--> v06\n- v16 --supports--> v02\n- v0d --supports--> v09\n- v11 --refutes--> v0e\n- v0b --supports--> v0f\n- v0f --refutes--> v16\n- v05 --supports--> v12\n- v0e --supports--> v0b\n- v05 --refutes--> v10\n- v0b --refutes--> v0a\n- v02 --supports--> v04\n- v00 --supports--> v0d\n- v0d --supports--> v07\n- v0c --refutes--> v13\n- v03 --supports--> v0c\n- v04 --refutes--> v00\n- v09 --supports--> v05\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"83a1f2a8c0c0c8123f9fbe942b2a8d0c574f1d10fe8f15ea14e0a27c1cb82ae8","schema":"livegraph.boolean-propagation.v1","spec_hash":"ee2b49093b4d84fe5185ff5cc69e8e72a7d5ad4ca474c1cc79ba73f7f6b7dedd","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-c017e7ca84f461e2fbb7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --supports--> v11\n- v16 --supports--> v06\n- v01 --refutes--> v04\n- v10 --supports--> v02\n- v0c --refutes--> v07\n- v04 --supports--> v0c\n- v0a --supports--> v03\n- v0f --supports--> v0b\n- v0f --refutes--> v10\n- v11 --refutes--> v08\n- v01 --refutes--> v05\n- v12 --refutes--> v13\n- v08 --refutes--> v0a\n- v13 --supports--> v09\n- v16 --supports--> v15\n- v14 --supports--> v00\n- v13 --refutes--> v14\n- v03 --supports--> v01\n- v02 --refutes--> v16\n- v08 --refutes--> v0d\n- v01 --supports--> v0e\n- v0c --refutes--> v12\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f4002374da2ae89025f131aa3e5e4120577c27f1418a55b597344836bcb0e843","schema":"livegraph.boolean-propagation.v1","spec_hash":"c017e7ca84f461e2fbb78dbfdc818f70cfe07a834e0385e35d95a736f38d08fc","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-d2e6d22f965eaf97b820","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v0e\n- v14 --refutes--> v08\n- v0a --supports--> v12\n- v04 --refutes--> v02\n- v03 --supports--> v04\n- v02 --supports--> v09\n- v0c --refutes--> v0a\n- v0b --refutes--> v01\n- v02 --supports--> v11\n- v01 --refutes--> v0d\n- v01 --refutes--> v16\n- v01 --supports--> v0f\n- v07 --supports--> v0b\n- v14 --supports--> v00\n- v13 --refutes--> v14\n- v05 --supports--> v13\n- v10 --supports--> v07\n- v02 --supports--> v15\n- v11 --supports--> v10\n- v0f --supports--> v0c\n- v0a --refutes--> v05\n- v06 --refutes--> v03\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"560a5a95c82df1dae4d702c7aa242830522bc7b7f751fa46e56504e6ee4a1afd","schema":"livegraph.boolean-propagation.v1","spec_hash":"d2e6d22f965eaf97b820a239f157954a355bdc72da0c22dc7c0f726e5e98f7b2","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":15,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-f8862d2badb0115b4926","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v05\n- v02 --supports--> v16\n- v14 --supports--> v13\n- v08 --supports--> v0e\n- v07 --supports--> v0b\n- v16 --supports--> v0f\n- v10 --supports--> v08\n- v14 --refutes--> v00\n- v0c --refutes--> v09\n- v00 --supports--> v06\n- v03 --supports--> v10\n- v0b --refutes--> v04\n- v12 --refutes--> v0c\n- v09 --refutes--> v03\n- v07 --supports--> v01\n- v04 --supports--> v12\n- v08 --supports--> v15\n- v06 --supports--> v02\n- v09 --supports--> v0a\n- v0f --refutes--> v0d\n- v03 --refutes--> v11\n- v08 --refutes--> v14\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5a578d1c7f3c24acef24583fda7aaca997ec05eefc30157d401bd22e5005355a","schema":"livegraph.boolean-propagation.v1","spec_hash":"f8862d2badb0115b49266b8a3f1682ddfd316c1e5aa24767a3ac2a405441da60","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-ab8fb3bc7531ee96c6b6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v01\n- v16 --refutes--> v13\n- v11 --refutes--> v0c\n- v02 --refutes--> v0d\n- v0c --refutes--> v0e\n- v0b --supports--> v15\n- v00 --refutes--> v14\n- v04 --supports--> v0f\n- v0c --refutes--> v16\n- v0c --refutes--> v06\n- v12 --refutes--> v05\n- v02 --supports--> v12\n- v03 --refutes--> v00\n- v05 --refutes--> v10\n- v15 --supports--> v0a\n- v00 --refutes--> v08\n- v00 --supports--> v0b\n- v05 --refutes--> v03\n- v0a --refutes--> v11\n- v15 --supports--> v07\n- v09 --refutes--> v04\n- v01 --refutes--> v02\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"df7d6ccd1fa627f6b41ae0b106cab51e9df03e427a54ee64f6490a5df4330702","schema":"livegraph.boolean-propagation.v1","spec_hash":"ab8fb3bc7531ee96c6b6590659a871dea343a02f72067d36594b054089dbf7d2","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-a6cf6bc340454b737008","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --supports--> v05\n- v03 --supports--> v04\n- v04 --supports--> v01\n- v09 --refutes--> v11\n- v09 --supports--> v07\n- v16 --refutes--> v0d\n- v02 --refutes--> v0b\n- v0b --refutes--> v15\n- v0a --supports--> v02\n- v01 --supports--> v13\n- v14 --refutes--> v0a\n- v06 --supports--> v14\n- v04 --refutes--> v10\n- v02 --supports--> v0c\n- v04 --refutes--> v0f\n- v05 --supports--> v16\n- v16 --supports--> v00\n- v11 --supports--> v06\n- v00 --supports--> v03\n- v16 --refutes--> v08\n- v04 --supports--> v0e\n- v12 --supports--> v09\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e42e413de9339d1e697e10b18d376dbfbb489735de632be651216c99c1b9b2f7","schema":"livegraph.boolean-propagation.v1","spec_hash":"a6cf6bc340454b737008c5499ce1a38290b74b87a14ebb728f201c70c8a88cad","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-1c998d195f6ab83b18f9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v16 --supports--> v0e\n- v02 --refutes--> v06\n- v03 --refutes--> v05\n- v14 --supports--> v07\n- v16 --refutes--> v14\n- v06 --refutes--> v15\n- v05 --supports--> v11\n- v16 --supports--> v0a\n- v15 --supports--> v04\n- v07 --refutes--> v12\n- v11 --refutes--> v01\n- v00 --refutes--> v0d\n- v09 --refutes--> v02\n- v15 --refutes--> v13\n- v07 --supports--> v03\n- v11 --refutes--> v00\n- v0c --refutes--> v09\n- v13 --supports--> v0f\n- v0f --supports--> v10\n- v0f --supports--> v08\n- v16 --refutes--> v0b\n- v0d --supports--> v0c\n\n[VERIFIED ROOT]\nv16 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bdd489b80cfeb6643805fa1d6236fbe8ce15fb1cce127a1f8ce44e46f9178eb8","schema":"livegraph.boolean-propagation.v1","spec_hash":"1c998d195f6ab83b18f9c9c4ca5d29661195c4c458003062462239e142b4e12b","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v16\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-da2f2cab4abb7fb3909b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v05\n- v11 --refutes--> v01\n- v02 --supports--> v08\n- v04 --refutes--> v12\n- v07 --supports--> v09\n- v04 --refutes--> v0a\n- v14 --supports--> v07\n- v11 --supports--> v06\n- v03 --refutes--> v0e\n- v14 --refutes--> v0b\n- v01 --supports--> v15\n- v03 --refutes--> v14\n- v10 --supports--> v0c\n- v00 --refutes--> v03\n- v05 --supports--> v11\n- v13 --refutes--> v04\n- v11 --supports--> v16\n- v07 --refutes--> v02\n- v15 --refutes--> v13\n- v02 --supports--> v0d\n- v0c --supports--> v0f\n- v0d --refutes--> v10\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"30fb5c839a58ff65665cf37e9b71f14b9099d4eeaf177c05c844b468a764581b","schema":"livegraph.boolean-propagation.v1","spec_hash":"da2f2cab4abb7fb3909b4724e48bfe1df4653dab0acb11e694135b7005f1980d","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-335524724d9483acc1d2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --refutes--> v0c\n- v06 --refutes--> v0e\n- v13 --supports--> v12\n- v0f --refutes--> v06\n- v06 --refutes--> v00\n- v09 --supports--> v02\n- v07 --refutes--> v13\n- v08 --refutes--> v03\n- v0e --refutes--> v08\n- v16 --refutes--> v05\n- v08 --refutes--> v04\n- v13 --supports--> v0a\n- v0c --refutes--> v07\n- v02 --refutes--> v16\n- v0e --refutes--> v0d\n- v04 --refutes--> v15\n- v13 --supports--> v0b\n- v08 --refutes--> v01\n- v14 --supports--> v0f\n- v05 --refutes--> v10\n- v09 --supports--> v11\n- v10 --supports--> v14\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1c9bf569ba100014e5d2dcd12ecd75ef0f4a3777b00d9cd96c1132997a3b3f47","schema":"livegraph.boolean-propagation.v1","spec_hash":"335524724d9483acc1d2ac2185ec9b3dbd0c2a2c477b062a7293c20e8d443eb8","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-b56f8c5d60779b480fd2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v12\n- v02 --supports--> v0d\n- v03 --supports--> v09\n- v00 --supports--> v13\n- v11 --refutes--> v05\n- v0b --refutes--> v16\n- v13 --supports--> v11\n- v0d --supports--> v01\n- v0c --supports--> v02\n- v04 --refutes--> v10\n- v0f --supports--> v08\n- v04 --supports--> v06\n- v11 --supports--> v0f\n- v0f --supports--> v14\n- v03 --supports--> v0e\n- v09 --supports--> v04\n- v16 --refutes--> v03\n- v01 --supports--> v00\n- v01 --supports--> v15\n- v0a --refutes--> v0b\n- v0b --supports--> v07\n- v06 --refutes--> v0c\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"054a625b325841545324cad0c3975d9eba0643a4f49f8a040ce3b1ec7cdbc302","schema":"livegraph.boolean-propagation.v1","spec_hash":"b56f8c5d60779b480fd2fe6d4e739110ca166b01a6d5c63b5ac0fde369891c77","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-71ac547bdab9b71cd8fc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v0c\n- v06 --supports--> v15\n- v0a --supports--> v13\n- v10 --refutes--> v0a\n- v05 --supports--> v06\n- v05 --refutes--> v16\n- v05 --refutes--> v0f\n- v06 --supports--> v04\n- v09 --supports--> v14\n- v15 --refutes--> v07\n- v02 --refutes--> v05\n- v08 --refutes--> v0d\n- v01 --refutes--> v09\n- v14 --refutes--> v0b\n- v00 --refutes--> v08\n- v12 --refutes--> v10\n- v0a --refutes--> v02\n- v0b --refutes--> v0e\n- v15 --supports--> v00\n- v0e --refutes--> v11\n- v0d --refutes--> v01\n- v02 --supports--> v03\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"578f65783631116b2203278129ae184e1bf208f9a20cba954ac4fdb47593da87","schema":"livegraph.boolean-propagation.v1","spec_hash":"71ac547bdab9b71cd8fc6e978515e3bd5a102b49e196f01e1d7ac3c99d31a64d","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-42eca509f797bfa35474","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v0e\n- v0f --supports--> v0a\n- v05 --refutes--> v00\n- v01 --supports--> v15\n- v02 --supports--> v04\n- v10 --refutes--> v11\n- v11 --refutes--> v13\n- v01 --refutes--> v07\n- v0c --supports--> v14\n- v14 --refutes--> v0f\n- v10 --refutes--> v12\n- v10 --refutes--> v09\n- v06 --refutes--> v0b\n- v0a --refutes--> v05\n- v13 --supports--> v0c\n- v06 --supports--> v16\n- v16 --refutes--> v08\n- v15 --supports--> v02\n- v0d --refutes--> v10\n- v02 --supports--> v03\n- v00 --supports--> v06\n- v02 --supports--> v0d\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"145e7b3d07869f41c12e6e2a575d0d0778371e548020d4f2292ca0194f0c5b6e","schema":"livegraph.boolean-propagation.v1","spec_hash":"42eca509f797bfa354743ff3bd302a3e1bef3a1a232641ea6aa4f96c27d99093","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-e7097748c1f5930106dc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --refutes--> v06\n- v06 --supports--> v16\n- v0c --refutes--> v01\n- v09 --supports--> v12\n- v00 --refutes--> v02\n- v00 --supports--> v05\n- v15 --supports--> v08\n- v16 --supports--> v0c\n- v0a --refutes--> v13\n- v07 --supports--> v10\n- v08 --supports--> v0e\n- v08 --supports--> v0f\n- v0e --supports--> v00\n- v05 --supports--> v04\n- v04 --refutes--> v0b\n- v00 --refutes--> v0d\n- v09 --supports--> v11\n- v04 --refutes--> v14\n- v14 --supports--> v07\n- v10 --refutes--> v09\n- v05 --supports--> v03\n- v0c --refutes--> v15\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"210f1073bdd2d6daa3c465812806619330e953cc4d82e3337505d45b120ecb2f","schema":"livegraph.boolean-propagation.v1","spec_hash":"e7097748c1f5930106dc78e4246e6d6d2553e7f4983fae2adc0b5d7554927877","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-ac56570107ded2afcd5b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v0d\n- v03 --supports--> v14\n- v0f --supports--> v11\n- v03 --supports--> v0e\n- v12 --refutes--> v01\n- v0d --refutes--> v12\n- v14 --supports--> v02\n- v06 --supports--> v09\n- v13 --supports--> v0c\n- v10 --supports--> v0b\n- v04 --refutes--> v13\n- v16 --refutes--> v03\n- v09 --supports--> v00\n- v13 --refutes--> v06\n- v0b --supports--> v05\n- v05 --supports--> v07\n- v02 --refutes--> v10\n- v12 --supports--> v16\n- v00 --refutes--> v0f\n- v10 --supports--> v15\n- v0d --refutes--> v08\n- v05 --refutes--> v0a\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6ede9671b40d2d200913a52ad49ff050c22ba0c473803efda6f1d4329d743013","schema":"livegraph.boolean-propagation.v1","spec_hash":"ac56570107ded2afcd5b30f184ee15deb162d4c97509016bf479b1e920bee4e6","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-f6f1fd157e335cf0c2f5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v11\n- v07 --supports--> v04\n- v11 --refutes--> v09\n- v14 --supports--> v06\n- v05 --supports--> v0c\n- v09 --supports--> v0f\n- v0e --refutes--> v05\n- v06 --refutes--> v12\n- v04 --supports--> v0a\n- v02 --refutes--> v07\n- v10 --supports--> v0d\n- v09 --supports--> v15\n- v09 --refutes--> v00\n- v03 --refutes--> v13\n- v09 --refutes--> v16\n- v0b --supports--> v02\n- v16 --refutes--> v14\n- v0d --supports--> v03\n- v0c --supports--> v10\n- v04 --supports--> v0e\n- v0e --supports--> v08\n- v09 --refutes--> v01\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8e791d48a660333474643f726922c32faf4b6dcf2557da2d3b2c5e6bfa8df991","schema":"livegraph.boolean-propagation.v1","spec_hash":"f6f1fd157e335cf0c2f51f239a5f9b4df888d9a93b74724631574bdd09f7a616","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-84ed3aed32c1b4aaf26f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v14\n- v07 --refutes--> v09\n- v10 --supports--> v0f\n- v05 --supports--> v04\n- v0e --supports--> v0b\n- v15 --supports--> v06\n- v15 --supports--> v0c\n- v12 --supports--> v05\n- v03 --supports--> v16\n- v12 --supports--> v08\n- v0f --refutes--> v07\n- v0c --supports--> v0e\n- v03 --supports--> v0a\n- v01 --supports--> v03\n- v09 --refutes--> v02\n- v0d --refutes--> v15\n- v16 --refutes--> v00\n- v11 --refutes--> v0d\n- v16 --refutes--> v13\n- v09 --supports--> v11\n- v0b --refutes--> v01\n- v05 --refutes--> v10\n\n[VERIFIED ROOT]\nv12 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5fe0427bf65f6f493b4e44909e7f3f928e1aaf944bfb9a190112d3f7dbabb841","schema":"livegraph.boolean-propagation.v1","spec_hash":"84ed3aed32c1b4aaf26f37d6434247a4cf821a07fecc1956d665e92d14f6ec95","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v12\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-fbfb9c85f13d1aea8fb9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v0b\n- v0e --supports--> v0a\n- v0d --supports--> v11\n- v10 --refutes--> v14\n- v15 --supports--> v12\n- v08 --refutes--> v07\n- v07 --refutes--> v04\n- v10 --supports--> v0c\n- v00 --supports--> v0e\n- v01 --supports--> v15\n- v10 --supports--> v09\n- v13 --refutes--> v10\n- v02 --supports--> v00\n- v11 --supports--> v13\n- v07 --supports--> v0f\n- v10 --supports--> v05\n- v0a --refutes--> v03\n- v0f --refutes--> v0d\n- v15 --supports--> v02\n- v16 --supports--> v08\n- v0a --refutes--> v16\n- v06 --supports--> v01\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7109e2acde68fa417e1a7ffadc17836ee9b79de93aae4809fc2caa26e0f0605b","schema":"livegraph.boolean-propagation.v1","spec_hash":"fbfb9c85f13d1aea8fb94790f83f6ed59b80205d766edc0318d1b2b6dd2eb8e6","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-8f6126db0ed0b4ad0398","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v05\n- v00 --refutes--> v11\n- v16 --refutes--> v07\n- v01 --supports--> v15\n- v07 --refutes--> v0c\n- v11 --supports--> v02\n- v13 --refutes--> v10\n- v12 --supports--> v0e\n- v06 --refutes--> v08\n- v0f --refutes--> v00\n- v05 --refutes--> v0d\n- v0c --supports--> v04\n- v06 --supports--> v13\n- v00 --supports--> v0a\n- v02 --supports--> v03\n- v11 --refutes--> v0b\n- v0e --supports--> v01\n- v05 --refutes--> v06\n- v14 --supports--> v12\n- v0c --supports--> v0f\n- v01 --refutes--> v16\n- v0e --supports--> v09\n\n[VERIFIED ROOT]\nv14 = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4e15a0cf4ce4fff637a6cc1fdcaa51abe87deaf11ae07efa073ce5a6e0ee41e2","schema":"livegraph.boolean-propagation.v1","spec_hash":"8f6126db0ed0b4ad0398e9d8b090e589300fd2bd34b697d2f692b6c1f46ee34d","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v14\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-ec3fb241dab10c61cce7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v0b\n- v05 --refutes--> v03\n- v09 --refutes--> v0f\n- v12 --refutes--> v0c\n- v04 --refutes--> v09\n- v06 --refutes--> v13\n- v0c --supports--> v0a\n- v10 --supports--> v15\n- v10 --supports--> v04\n- v10 --refutes--> v01\n- v13 --supports--> v08\n- v03 --supports--> v02\n- v02 --refutes--> v12\n- v07 --supports--> v06\n- v0a --refutes--> v16\n- v14 --refutes--> v07\n- v0a --supports--> v0d\n- v10 --supports--> v11\n- v0f --refutes--> v05\n- v00 --supports--> v0e\n- v13 --refutes--> v00\n- v0a --supports--> v14\n\n[VERIFIED ROOT]\nv10 = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"62d2d7ee803607485b80f04e72221652bc50afe03f84c5e2a1a486302d893f6f","schema":"livegraph.boolean-propagation.v1","spec_hash":"ec3fb241dab10c61cce78d19b6592951d6f5813b881363736d964cfa51e92bf9","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":15,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-8f824373d09fcf41a612","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --supports--> v0b\n- v00 --refutes--> v12\n- v01 --supports--> v0c\n- v16 --supports--> v0e\n- v0e --supports--> v03\n- v02 --refutes--> v13\n- v06 --refutes--> v10\n- v15 --refutes--> v05\n- v04 --supports--> v14\n- v08 --supports--> v0d\n- v11 --supports--> v00\n- v10 --refutes--> v11\n- v14 --refutes--> v02\n- v0b --supports--> v01\n- v15 --supports--> v06\n- v10 --refutes--> v07\n- v06 --refutes--> v0a\n- v08 --supports--> v0f\n- v00 --refutes--> v09\n- v0e --supports--> v08\n- v0d --supports--> v15\n- v01 --refutes--> v16\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c09408158ab140986124936821d23563fa1b8634fb24f4e360af643d0c6ef641","schema":"livegraph.boolean-propagation.v1","spec_hash":"8f824373d09fcf41a612c4b18be70a3684c0f060f904326fe11f0e37b24e2215","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-bab520b673f783e40081","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v12\n- v16 --supports--> v0d\n- v03 --refutes--> v07\n- v13 --supports--> v15\n- v13 --refutes--> v09\n- v04 --supports--> v13\n- v0c --supports--> v00\n- v0b --supports--> v05\n- v0a --refutes--> v08\n- v0c --refutes--> v16\n- v15 --refutes--> v06\n- v01 --supports--> v04\n- v02 --supports--> v01\n- v16 --refutes--> v10\n- v08 --refutes--> v0c\n- v0a --supports--> v11\n- v16 --supports--> v02\n- v06 --supports--> v0e\n- v15 --refutes--> v14\n- v05 --refutes--> v0f\n- v0f --supports--> v0a\n- v0e --refutes--> v03\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"104b169f383721d6dd45c1e5f251f0ef2e7e8214382e6b587e5be29060873aa9","schema":"livegraph.boolean-propagation.v1","spec_hash":"bab520b673f783e4008119b4d9204788f1898f883958455c023e1b2445d33233","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-42cf0c57daaf6fa376b8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v10\n- v03 --supports--> v14\n- v0c --refutes--> v0e\n- v0c --supports--> v13\n- v16 --refutes--> v0d\n- v00 --refutes--> v03\n- v0e --refutes--> v0b\n- v0a --refutes--> v06\n- v0c --refutes--> v0f\n- v04 --refutes--> v16\n- v10 --refutes--> v01\n- v14 --refutes--> v04\n- v0b --supports--> v02\n- v10 --refutes--> v08\n- v11 --supports--> v12\n- v0d --refutes--> v09\n- v0a --supports--> v15\n- v09 --refutes--> v0c\n- v11 --supports--> v07\n- v10 --refutes--> v00\n- v15 --supports--> v11\n- v04 --refutes--> v05\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0671d70404cb57e8975ad0858fe0174b60365e9edb60e7c4daee4f51f1a4ef25","schema":"livegraph.boolean-propagation.v1","spec_hash":"42cf0c57daaf6fa376b8a49df11545da23b661ab97e9081a42d327a21681c01a","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-aee0697b3b0c7fd1aefa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v16\n- v08 --supports--> v0c\n- v01 --refutes--> v0e\n- v13 --supports--> v10\n- v0f --supports--> v07\n- v0f --refutes--> v00\n- v05 --supports--> v02\n- v06 --refutes--> v03\n- v0d --supports--> v01\n- v05 --refutes--> v13\n- v03 --refutes--> v05\n- v13 --refutes--> v0d\n- v15 --refutes--> v0b\n- v0e --refutes--> v11\n- v11 --supports--> v08\n- v06 --refutes--> v09\n- v00 --supports--> v12\n- v08 --supports--> v0a\n- v04 --supports--> v14\n- v12 --supports--> v04\n- v04 --refutes--> v06\n- v0b --refutes--> v0f\n\n[VERIFIED ROOT]\nv15 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0bf8d391404bcb52a81da21a105ddbd59e6fe04dc57206a9e4dcdec0f0551706","schema":"livegraph.boolean-propagation.v1","spec_hash":"aee0697b3b0c7fd1aefac90d96205ff6c471ad93dc137b2db81e817adc6a1050","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v15\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-72a891617fa0d7c43078","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v0e\n- v0f --supports--> v14\n- v0d --refutes--> v0f\n- v09 --supports--> v0d\n- v03 --refutes--> v00\n- v08 --supports--> v12\n- v14 --supports--> v0b\n- v04 --refutes--> v0c\n- v07 --supports--> v11\n- v0d --supports--> v13\n- v10 --supports--> v03\n- v14 --supports--> v02\n- v0e --refutes--> v09\n- v11 --supports--> v10\n- v0b --refutes--> v06\n- v03 --supports--> v01\n- v08 --refutes--> v05\n- v16 --supports--> v04\n- v14 --supports--> v0a\n- v06 --refutes--> v16\n- v16 --refutes--> v15\n- v08 --refutes--> v07\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"64236b87446e59d3590d081abfe620551fa5c6131716e56793616be87730b159","schema":"livegraph.boolean-propagation.v1","spec_hash":"72a891617fa0d7c430784e165cdc5d3a5497f1f7c815ea98ed8ca8f9d82da1a5","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-fbaea36b3940c39db355","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v11\n- v03 --supports--> v00\n- v15 --supports--> v0e\n- v11 --supports--> v08\n- v12 --supports--> v0c\n- v16 --supports--> v13\n- v15 --refutes--> v0f\n- v09 --supports--> v0a\n- v16 --supports--> v01\n- v14 --supports--> v05\n- v0f --supports--> v02\n- v03 --supports--> v04\n- v06 --refutes--> v12\n- v01 --refutes--> v15\n- v05 --supports--> v16\n- v03 --supports--> v06\n- v0d --refutes--> v0b\n- v0a --supports--> v14\n- v0b --supports--> v03\n- v11 --supports--> v09\n- v0f --supports--> v0d\n- v01 --refutes--> v10\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d6ed661e1d85d6b5ffb2c34b5441c68c71733b163c5807f6c50df5cee5e44847","schema":"livegraph.boolean-propagation.v1","spec_hash":"fbaea36b3940c39db3555d7db1852cf502e85b3e8140c5dd561a8c4791ecd2aa","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-8f1fddf8e485699b9af5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v03\n- v15 --refutes--> v05\n- v00 --refutes--> v02\n- v12 --supports--> v06\n- v08 --refutes--> v0f\n- v16 --supports--> v0c\n- v0c --supports--> v04\n- v11 --supports--> v12\n- v03 --supports--> v10\n- v13 --refutes--> v0a\n- v11 --refutes--> v0e\n- v04 --refutes--> v00\n- v0a --supports--> v08\n- v08 --supports--> v0b\n- v02 --supports--> v14\n- v08 --refutes--> v01\n- v05 --supports--> v11\n- v14 --supports--> v13\n- v12 --supports--> v16\n- v0c --supports--> v07\n- v15 --refutes--> v0d\n- v02 --refutes--> v09\n\n[VERIFIED ROOT]\nv15 = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"358a384b9c9dd240b0909f70c377c5f4eca86ad7c218d0ec9bb1bffa53f00ebb","schema":"livegraph.boolean-propagation.v1","spec_hash":"8f1fddf8e485699b9af5a2af6daaf412d91a61068b98ecb91a34f0587d63a53a","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v15\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-dd33e6a2c923b7583e58","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v05\n- v0e --supports--> v01\n- v0f --refutes--> v16\n- v14 --refutes--> v0c\n- v0c --supports--> v08\n- v0a --refutes--> v10\n- v07 --refutes--> v02\n- v0b --refutes--> v0e\n- v0f --refutes--> v09\n- v10 --refutes--> v15\n- v06 --refutes--> v14\n- v09 --refutes--> v12\n- v07 --refutes--> v00\n- v04 --supports--> v0f\n- v11 --supports--> v13\n- v01 --supports--> v06\n- v11 --supports--> v0a\n- v10 --refutes--> v0b\n- v12 --refutes--> v03\n- v0c --refutes--> v07\n- v0d --supports--> v11\n- v12 --supports--> v0d\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bae49a0ad7b38e5ed68c52f9998acc20120c50a21db31eaa84bce1c33ab565c6","schema":"livegraph.boolean-propagation.v1","spec_hash":"dd33e6a2c923b7583e580c6fd9ed8e2e755d5429ba1223319f960203f6e096ef","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":15,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-df20818f162b136aa52c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v06\n- v09 --refutes--> v0a\n- v16 --refutes--> v14\n- v0f --refutes--> v15\n- v06 --supports--> v08\n- v0c --refutes--> v01\n- v13 --supports--> v00\n- v00 --supports--> v16\n- v0c --supports--> v09\n- v0e --refutes--> v0d\n- v13 --refutes--> v07\n- v10 --supports--> v0e\n- v05 --supports--> v11\n- v0e --refutes--> v0f\n- v10 --refutes--> v03\n- v0b --supports--> v02\n- v0a --supports--> v0b\n- v08 --supports--> v04\n- v14 --refutes--> v05\n- v0a --supports--> v12\n- v04 --supports--> v13\n- v0b --refutes--> v10\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5b134877314e6347e2b8265ba1b83a76b78f50c96367d4e23b1a609ef29bf113","schema":"livegraph.boolean-propagation.v1","spec_hash":"df20818f162b136aa52c8026eda6bc4228dd52203593e038fd8e4f67a49a98ca","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-2df0dafd50ee266598c8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v01\n- v03 --refutes--> v08\n- v02 --refutes--> v04\n- v0c --supports--> v00\n- v0a --refutes--> v12\n- v15 --supports--> v09\n- v0f --refutes--> v0b\n- v0f --supports--> v0e\n- v0c --supports--> v11\n- v08 --refutes--> v06\n- v02 --refutes--> v14\n- v07 --refutes--> v0c\n- v10 --supports--> v13\n- v01 --refutes--> v07\n- v0e --refutes--> v10\n- v0a --supports--> v05\n- v06 --supports--> v15\n- v07 --refutes--> v16\n- v05 --refutes--> v0f\n- v13 --supports--> v03\n- v15 --supports--> v0d\n- v00 --refutes--> v0a\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e21fc1ebb9e2507958249dfa272e447bb1d507996f50004705eeb0d3b6fbe61e","schema":"livegraph.boolean-propagation.v1","spec_hash":"2df0dafd50ee266598c87cf2fd3982eb34e66e138df88d27e85601b5e0d66e50","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-bda874e40f3645e52951","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v14\n- v10 --supports--> v01\n- v0a --supports--> v09\n- v0b --refutes--> v0a\n- v0e --supports--> v0d\n- v16 --supports--> v04\n- v16 --refutes--> v0e\n- v01 --supports--> v07\n- v03 --supports--> v00\n- v0f --supports--> v15\n- v00 --refutes--> v13\n- v12 --supports--> v03\n- v0c --supports--> v0f\n- v0f --supports--> v12\n- v13 --refutes--> v16\n- v03 --refutes--> v02\n- v0e --supports--> v11\n- v08 --supports--> v10\n- v0d --supports--> v08\n- v09 --refutes--> v0c\n- v0b --refutes--> v05\n- v10 --refutes--> v06\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4bbf5a9d439222cc69762f70ee19622fec2a0c8caa82ebc3315b7fafd731ff92","schema":"livegraph.boolean-propagation.v1","spec_hash":"bda874e40f3645e529516f1d4c5b6776ad5e0d58d0333c0d27b7ec947f635ef9","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-a8c634cf2a2f59d99bcd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --supports--> v0d\n- v0e --refutes--> v02\n- v0d --refutes--> v00\n- v05 --supports--> v01\n- v01 --supports--> v0f\n- v15 --refutes--> v16\n- v0c --supports--> v15\n- v11 --supports--> v06\n- v0f --refutes--> v10\n- v0f --refutes--> v03\n- v00 --refutes--> v04\n- v09 --supports--> v0b\n- v11 --refutes--> v09\n- v07 --supports--> v0c\n- v08 --supports--> v12\n- v09 --refutes--> v08\n- v02 --refutes--> v07\n- v0f --supports--> v0e\n- v01 --supports--> v13\n- v04 --refutes--> v05\n- v01 --supports--> v0a\n- v08 --supports--> v14\n\n[VERIFIED ROOT]\nv11 = FALSE\n\n[TARGET]\nv16\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a5e0dbc325137f54037b2db75e20f843edc5cdea9dc2aeaf2e61d79d60265535","schema":"livegraph.boolean-propagation.v1","spec_hash":"a8c634cf2a2f59d99bcdf41cb180b081527eb68b75a254ed01806ff78dc6b668","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v16\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-340e3345cd42bc57fd0d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --refutes--> v16\n- v12 --refutes--> v03\n- v10 --refutes--> v09\n- v11 --refutes--> v15\n- v00 --supports--> v14\n- v14 --refutes--> v0c\n- v15 --supports--> v07\n- v15 --refutes--> v0f\n- v10 --refutes--> v05\n- v13 --refutes--> v02\n- v0e --refutes--> v11\n- v13 --supports--> v08\n- v06 --refutes--> v00\n- v16 --refutes--> v0a\n- v04 --refutes--> v06\n- v08 --refutes--> v04\n- v03 --supports--> v0e\n- v01 --refutes--> v13\n- v06 --refutes--> v0d\n- v15 --refutes--> v01\n- v10 --refutes--> v0b\n- v0a --refutes--> v10\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c488ba7f3bb45128495c1a963a3fcbcf89b227e5e3633243ba42df329dbe49d0","schema":"livegraph.boolean-propagation.v1","spec_hash":"340e3345cd42bc57fd0d8bc90bae006b3648cf5325ec7c3a7502edf79223c1d5","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-6bcc2a0c16feb762f91b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --supports--> v11\n- v0f --refutes--> v01\n- v0a --refutes--> v14\n- v13 --refutes--> v00\n- v07 --supports--> v0e\n- v05 --refutes--> v04\n- v13 --supports--> v09\n- v14 --supports--> v13\n- v0f --refutes--> v16\n- v10 --supports--> v12\n- v09 --refutes--> v03\n- v12 --supports--> v0b\n- v04 --supports--> v07\n- v09 --supports--> v0f\n- v06 --supports--> v0a\n- v13 --supports--> v08\n- v0b --refutes--> v0c\n- v0e --refutes--> v15\n- v0e --refutes--> v10\n- v0d --supports--> v06\n- v0b --supports--> v0d\n- v02 --refutes--> v05\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1eb968335fc064c896c51fc6621b83da2d0097efe08a884435df351d52f11ce0","schema":"livegraph.boolean-propagation.v1","spec_hash":"6bcc2a0c16feb762f91b5ee722e4c6c152304f61fc79e673fad1a2fd11f2147e","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":15,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-8191406739cc03778944","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v0c\n- v06 --refutes--> v14\n- v0c --supports--> v0a\n- v01 --supports--> v0e\n- v0c --supports--> v03\n- v07 --refutes--> v11\n- v10 --refutes--> v06\n- v0b --supports--> v15\n- v06 --supports--> v08\n- v13 --supports--> v12\n- v00 --supports--> v0b\n- v08 --refutes--> v02\n- v12 --supports--> v00\n- v02 --refutes--> v16\n- v0c --refutes--> v13\n- v16 --supports--> v0d\n- v08 --supports--> v0f\n- v0b --supports--> v05\n- v15 --supports--> v10\n- v11 --supports--> v01\n- v07 --supports--> v04\n- v13 --refutes--> v09\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b299b5bcaecefd197958cb6d4a0197cb97f9d22ff307419a2c9afa2f1ead954a","schema":"livegraph.boolean-propagation.v1","spec_hash":"8191406739cc03778944381fa64ec3464aa9de2a36c8f14c476a3ad204eee7a5","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-ba2453b093cba626b4b5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v04\n- v0a --supports--> v14\n- v0f --refutes--> v03\n- v10 --refutes--> v0b\n- v07 --supports--> v06\n- v13 --supports--> v0c\n- v04 --supports--> v12\n- v14 --supports--> v0e\n- v0d --supports--> v13\n- v08 --supports--> v07\n- v04 --refutes--> v0d\n- v02 --supports--> v11\n- v09 --refutes--> v02\n- v13 --refutes--> v01\n- v04 --refutes--> v00\n- v16 --supports--> v08\n- v05 --supports--> v09\n- v0e --refutes--> v10\n- v07 --supports--> v0f\n- v11 --supports--> v16\n- v11 --supports--> v15\n- v10 --supports--> v05\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4993df3a3db1b3c511e8bc7498df27d93dcd1097668fe269313bcdadf7cc9e3c","schema":"livegraph.boolean-propagation.v1","spec_hash":"ba2453b093cba626b4b5a42d5780451cc2e693ad73e77ee2e4b1a8821c8ac23c","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-9ed11a1bb73dd93407a8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v08\n- v01 --refutes--> v00\n- v11 --refutes--> v0d\n- v03 --supports--> v0f\n- v09 --refutes--> v06\n- v12 --supports--> v13\n- v07 --refutes--> v12\n- v0b --refutes--> v15\n- v0f --supports--> v0e\n- v09 --supports--> v04\n- v12 --refutes--> v01\n- v0a --refutes--> v14\n- v07 --refutes--> v0c\n- v02 --supports--> v0a\n- v05 --supports--> v16\n- v16 --refutes--> v0b\n- v06 --refutes--> v03\n- v15 --supports--> v09\n- v0a --supports--> v07\n- v10 --refutes--> v05\n- v11 --refutes--> v10\n- v0f --refutes--> v02\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8e21f42381d5bb06aa71634a9de2aac84a1c52b32e3684afc21abed0fcbe6b0d","schema":"livegraph.boolean-propagation.v1","spec_hash":"9ed11a1bb73dd93407a80f300b7181132742095fd74d400460c4a1e6079e95eb","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-16683bad31773a2a78a8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v00\n- v00 --refutes--> v16\n- v06 --refutes--> v08\n- v0d --refutes--> v04\n- v06 --supports--> v0a\n- v16 --refutes--> v0e\n- v10 --supports--> v03\n- v06 --refutes--> v09\n- v14 --refutes--> v11\n- v0b --supports--> v10\n- v03 --refutes--> v15\n- v15 --refutes--> v14\n- v06 --refutes--> v0c\n- v04 --supports--> v02\n- v11 --refutes--> v12\n- v0b --refutes--> v07\n- v04 --supports--> v0b\n- v13 --supports--> v0d\n- v0e --supports--> v13\n- v0c --refutes--> v01\n- v03 --supports--> v0f\n- v01 --supports--> v05\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a850df94455cfa764cad64526917ab0affd18d96e60c9feab4dcbe052f28a100","schema":"livegraph.boolean-propagation.v1","spec_hash":"16683bad31773a2a78a8a02c3e0283f6d7335602ca370dc29e13f31d2ab1eaed","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-e5e3b8ef2c47c8d7d633","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v11\n- v04 --refutes--> v0c\n- v06 --supports--> v00\n- v02 --supports--> v06\n- v06 --refutes--> v15\n- v10 --refutes--> v0f\n- v06 --refutes--> v08\n- v05 --supports--> v07\n- v07 --supports--> v13\n- v06 --supports--> v01\n- v03 --refutes--> v05\n- v0a --refutes--> v09\n- v0e --refutes--> v14\n- v0f --supports--> v0a\n- v08 --refutes--> v0d\n- v14 --refutes--> v02\n- v08 --supports--> v16\n- v16 --refutes--> v10\n- v07 --refutes--> v04\n- v05 --supports--> v0b\n- v0c --supports--> v0e\n- v12 --supports--> v03\n\n[VERIFIED ROOT]\nv12 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"39857d95b21ba21f99688521d4e5c3214d861958b8afb3f78acd8a31fc1be423","schema":"livegraph.boolean-propagation.v1","spec_hash":"e5e3b8ef2c47c8d7d633e55fef72d700569d3f361af083da8654ecb489d805af","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v12\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-4a0b397a716e0687b32a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v0d\n- v08 --refutes--> v02\n- v00 --refutes--> v05\n- v10 --supports--> v0c\n- v0f --refutes--> v06\n- v16 --supports--> v13\n- v0c --supports--> v12\n- v00 --refutes--> v03\n- v03 --refutes--> v0e\n- v0d --supports--> v15\n- v13 --supports--> v11\n- v0d --supports--> v14\n- v02 --refutes--> v0f\n- v14 --refutes--> v10\n- v06 --refutes--> v01\n- v04 --supports--> v09\n- v08 --supports--> v0b\n- v09 --refutes--> v08\n- v01 --refutes--> v00\n- v00 --supports--> v0a\n- v00 --supports--> v07\n- v13 --supports--> v04\n\n[VERIFIED ROOT]\nv16 = FALSE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"11380e8b73052d879da4fc4a1b9bdfd2115c3ef3e9d9baa64c72d004469a2163","schema":"livegraph.boolean-propagation.v1","spec_hash":"4a0b397a716e0687b32ac2845d8893d848a3755874cb7048f2c349cfde5e0048","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v16\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-82a5200c617c2e0cf9e9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v16 --supports--> v0d\n- v05 --supports--> v0a\n- v04 --refutes--> v06\n- v04 --refutes--> v11\n- v0b --refutes--> v02\n- v02 --supports--> v04\n- v15 --refutes--> v00\n- v00 --supports--> v0c\n- v11 --supports--> v16\n- v13 --refutes--> v08\n- v0a --supports--> v14\n- v03 --supports--> v13\n- v07 --supports--> v01\n- v14 --refutes--> v07\n- v07 --supports--> v03\n- v16 --supports--> v09\n- v00 --refutes--> v0e\n- v02 --supports--> v12\n- v08 --supports--> v0f\n- v08 --supports--> v0b\n- v0b --refutes--> v10\n- v0e --refutes--> v05\n\n[VERIFIED ROOT]\nv15 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"503817dde8ff178c67caab931e1cdfc39f0b880dc5cba00ccf73e33cb48173a1","schema":"livegraph.boolean-propagation.v1","spec_hash":"82a5200c617c2e0cf9e9ea7d8baea7afc18c0bfb645c72e6bd300360d9c08a24","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v15\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-f7a4766f6d27c4c1171b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v04\n- v04 --refutes--> v03\n- v0b --supports--> v08\n- v04 --refutes--> v14\n- v15 --supports--> v01\n- v02 --supports--> v11\n- v16 --refutes--> v12\n- v02 --supports--> v0b\n- v08 --refutes--> v15\n- v0c --refutes--> v07\n- v09 --refutes--> v0a\n- v09 --refutes--> v05\n- v01 --refutes--> v00\n- v0f --refutes--> v0c\n- v14 --refutes--> v0e\n- v09 --supports--> v06\n- v10 --supports--> v09\n- v01 --supports--> v0d\n- v13 --refutes--> v02\n- v0d --refutes--> v0f\n- v0e --supports--> v16\n- v16 --refutes--> v10\n\n[VERIFIED ROOT]\nv13 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"72362798d482a9c4a40c4edcbaecac20efb43f84bbd06a3cb3e26bfffae06240","schema":"livegraph.boolean-propagation.v1","spec_hash":"f7a4766f6d27c4c1171b02176474d6068e8a4c1db9c3434186c33dcb6c646412","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v13\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-60ac14eee2bc5425d396","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v08\n- v16 --refutes--> v0a\n- v12 --supports--> v16\n- v0e --refutes--> v03\n- v09 --refutes--> v04\n- v0d --supports--> v11\n- v10 --supports--> v15\n- v0b --supports--> v02\n- v13 --refutes--> v05\n- v04 --refutes--> v0f\n- v13 --refutes--> v12\n- v11 --supports--> v13\n- v04 --supports--> v0d\n- v08 --supports--> v01\n- v01 --supports--> v14\n- v0b --supports--> v09\n- v0a --supports--> v10\n- v01 --supports--> v00\n- v16 --supports--> v07\n- v11 --supports--> v0c\n- v14 --supports--> v06\n- v06 --refutes--> v0b\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv15\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7b1574a8f346c11e0539c8bad133952c4bf96c83885991060260654a1fb7b4be","schema":"livegraph.boolean-propagation.v1","spec_hash":"60ac14eee2bc5425d396f16fdba83294102fc4921647a8be7b9898d565ac17fc","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-4081baff1fb15d094a3f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v00\n- v02 --refutes--> v14\n- v16 --supports--> v0d\n- v0b --supports--> v10\n- v14 --supports--> v0c\n- v07 --refutes--> v08\n- v02 --supports--> v04\n- v14 --supports--> v12\n- v00 --supports--> v0f\n- v0e --refutes--> v02\n- v08 --supports--> v16\n- v06 --refutes--> v13\n- v01 --refutes--> v05\n- v0f --refutes--> v0a\n- v0e --supports--> v11\n- v0b --supports--> v15\n- v07 --refutes--> v09\n- v15 --refutes--> v01\n- v0c --refutes--> v0b\n- v0f --refutes--> v06\n- v00 --supports--> v03\n- v05 --supports--> v07\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6b5cd9516e42fd864ab1dd097ea31153c55e0c2a725f5bebc8fa1c14e90d802e","schema":"livegraph.boolean-propagation.v1","spec_hash":"4081baff1fb15d094a3f33e4343713d34e6f4cc6a0c10b4ca69045a16560074d","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-4b28d8ffb3c3d82ee50c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v01\n- v11 --refutes--> v0b\n- v00 --refutes--> v07\n- v0f --refutes--> v0c\n- v05 --refutes--> v09\n- v02 --supports--> v04\n- v06 --supports--> v0f\n- v03 --supports--> v0e\n- v10 --supports--> v12\n- v10 --supports--> v14\n- v09 --supports--> v13\n- v0c --refutes--> v0a\n- v07 --refutes--> v03\n- v16 --refutes--> v05\n- v03 --refutes--> v11\n- v11 --supports--> v06\n- v0c --refutes--> v0d\n- v0c --refutes--> v15\n- v08 --supports--> v00\n- v12 --refutes--> v02\n- v01 --refutes--> v16\n- v09 --supports--> v08\n\n[VERIFIED ROOT]\nv10 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8b1d32daf2375eaf64ef72020643bbe881ab37631d1113f73a0f368b7279c100","schema":"livegraph.boolean-propagation.v1","spec_hash":"4b28d8ffb3c3d82ee50cd802de035956c58ac6369b9987deb74f75319700c7b1","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-79d30237a9421b7a6818","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v0d\n- v0e --supports--> v15\n- v06 --supports--> v14\n- v12 --supports--> v06\n- v09 --refutes--> v0e\n- v08 --refutes--> v12\n- v14 --refutes--> v07\n- v01 --refutes--> v0a\n- v01 --refutes--> v16\n- v0c --refutes--> v0b\n- v0c --supports--> v01\n- v16 --supports--> v10\n- v13 --supports--> v05\n- v03 --refutes--> v04\n- v0d --supports--> v11\n- v07 --refutes--> v13\n- v06 --refutes--> v02\n- v0d --refutes--> v00\n- v15 --supports--> v0c\n- v08 --supports--> v0f\n- v13 --supports--> v09\n- v0d --refutes--> v03\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"506f5ae29840cd9e3d221e9c33f7fcabf31680029c00e5f5ac4edbd0f9c48134","schema":"livegraph.boolean-propagation.v1","spec_hash":"79d30237a9421b7a68183ed3632147f0b42af7dab102a90c91346d10ce668c05","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-4b5e0f15fc4a6830c8bd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v01\n- v14 --supports--> v02\n- v07 --refutes--> v14\n- v08 --supports--> v07\n- v04 --supports--> v05\n- v12 --supports--> v0e\n- v00 --supports--> v0b\n- v15 --refutes--> v06\n- v16 --supports--> v00\n- v0a --supports--> v0c\n- v14 --refutes--> v09\n- v03 --refutes--> v0f\n- v06 --supports--> v13\n- v04 --refutes--> v0a\n- v00 --supports--> v0d\n- v0a --supports--> v15\n- v10 --refutes--> v16\n- v12 --supports--> v11\n- v0e --supports--> v03\n- v13 --supports--> v08\n- v0b --supports--> v12\n- v14 --supports--> v10\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2a4c1c3c283018026f3fb2022a91108374bd84d234e05e4bf9f1a1f1221c8ab9","schema":"livegraph.boolean-propagation.v1","spec_hash":"4b5e0f15fc4a6830c8bd005fa5e917d37b3d0a3752e8bc486fed881b9caaf9e3","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-da24450639e8f9f6d5ea","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v06\n- v05 --supports--> v14\n- v0c --refutes--> v16\n- v08 --refutes--> v0b\n- v16 --supports--> v0e\n- v01 --refutes--> v11\n- v12 --refutes--> v10\n- v11 --supports--> v04\n- v0f --supports--> v05\n- v0e --refutes--> v12\n- v12 --refutes--> v0a\n- v08 --supports--> v0f\n- v09 --refutes--> v0c\n- v05 --refutes--> v0d\n- v0f --supports--> v00\n- v03 --refutes--> v02\n- v10 --supports--> v03\n- v10 --refutes--> v07\n- v05 --refutes--> v09\n- v04 --supports--> v08\n- v06 --supports--> v13\n- v10 --supports--> v15\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fd2fc82eaa0d8145c1322d9794c3e497c53a098610edcf494fc65949bbbafade","schema":"livegraph.boolean-propagation.v1","spec_hash":"da24450639e8f9f6d5ea0ec108aa410a577398d6aa80dc1e5a7c3f52dff44fce","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-9ba476dfd5759f3c05f9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v0e\n- v10 --supports--> v0a\n- v15 --refutes--> v02\n- v0b --supports--> v0d\n- v02 --supports--> v01\n- v0a --supports--> v14\n- v05 --supports--> v06\n- v0d --refutes--> v16\n- v04 --refutes--> v10\n- v07 --supports--> v12\n- v11 --supports--> v04\n- v16 --supports--> v0f\n- v02 --supports--> v07\n- v16 --refutes--> v13\n- v12 --supports--> v0b\n- v03 --supports--> v05\n- v0a --supports--> v03\n- v04 --refutes--> v0c\n- v0a --refutes--> v00\n- v0f --supports--> v09\n- v0b --refutes--> v08\n- v06 --supports--> v15\n\n[VERIFIED ROOT]\nv11 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"84a7b189986c2c97138447d7ffa72fc6cf37a7237c0cacba1f60f1e34cf1c75b","schema":"livegraph.boolean-propagation.v1","spec_hash":"9ba476dfd5759f3c05f9b441f2b9c3ef38d2dacf451794fd0c4e9ad57d32beaf","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-9f95d3e6cfd798d444ac","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v05\n- v06 --supports--> v01\n- v00 --refutes--> v11\n- v05 --supports--> v00\n- v03 --refutes--> v0b\n- v13 --refutes--> v15\n- v11 --refutes--> v16\n- v0e --refutes--> v13\n- v15 --refutes--> v0f\n- v04 --supports--> v0a\n- v12 --supports--> v06\n- v00 --refutes--> v0d\n- v16 --supports--> v07\n- v0b --supports--> v12\n- v15 --refutes--> v03\n- v13 --supports--> v0c\n- v09 --refutes--> v0e\n- v00 --refutes--> v14\n- v05 --supports--> v02\n- v07 --supports--> v08\n- v07 --refutes--> v10\n- v01 --refutes--> v04\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ba26447fba31c783dddac3be53046576448c0751d1c2e518ac7d40e3fc0022c0","schema":"livegraph.boolean-propagation.v1","spec_hash":"9f95d3e6cfd798d444acd631ec6aa3a9a8b1dac621f690b491594a5d238b2b7f","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-947d1752c3663d18ab25","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v02\n- v04 --supports--> v0a\n- v0e --supports--> v0c\n- v08 --supports--> v04\n- v0c --refutes--> v0b\n- v0f --refutes--> v0d\n- v09 --supports--> v16\n- v07 --supports--> v01\n- v11 --refutes--> v0e\n- v10 --supports--> v08\n- v0b --supports--> v10\n- v14 --refutes--> v03\n- v09 --refutes--> v11\n- v11 --refutes--> v13\n- v12 --refutes--> v0f\n- v0a --refutes--> v14\n- v14 --refutes--> v06\n- v10 --supports--> v15\n- v0a --refutes--> v00\n- v01 --supports--> v09\n- v0f --refutes--> v07\n- v08 --refutes--> v05\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b6f841912a459dec0bb8eb4d9f6434c5117e51f1469b8f5baf02efd90bd0eeb9","schema":"livegraph.boolean-propagation.v1","spec_hash":"947d1752c3663d18ab25cc66f9756502c82adf42bdc7c97e65360e6040f02b73","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":15,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-d8fe4973e54f8b7009fd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v0c\n- v11 --refutes--> v0e\n- v15 --refutes--> v05\n- v14 --refutes--> v0b\n- v11 --supports--> v12\n- v03 --supports--> v13\n- v0b --supports--> v01\n- v0d --refutes--> v16\n- v0b --refutes--> v11\n- v02 --refutes--> v03\n- v0a --supports--> v08\n- v0a --supports--> v07\n- v03 --supports--> v14\n- v16 --refutes--> v15\n- v04 --supports--> v02\n- v0e --refutes--> v09\n- v0e --refutes--> v0f\n- v05 --refutes--> v0a\n- v09 --supports--> v10\n- v07 --refutes--> v04\n- v11 --supports--> v06\n- v00 --supports--> v0d\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b5a0671f38b31f9edc0d554b1ac7a4889e69d4198a057e5710190e3067361ebb","schema":"livegraph.boolean-propagation.v1","spec_hash":"d8fe4973e54f8b7009fd2020bdad35ebc8646f746e14b2489115d53cfb16443d","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-4c7038cbf86275c9935f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v0c\n- v16 --refutes--> v02\n- v07 --supports--> v11\n- v01 --refutes--> v08\n- v09 --refutes--> v10\n- v0a --supports--> v15\n- v16 --refutes--> v13\n- v0f --refutes--> v14\n- v13 --supports--> v12\n- v07 --supports--> v0a\n- v0d --refutes--> v00\n- v0b --refutes--> v03\n- v06 --supports--> v01\n- v0b --supports--> v04\n- v15 --refutes--> v16\n- v06 --refutes--> v05\n- v03 --supports--> v06\n- v01 --supports--> v09\n- v09 --refutes--> v0f\n- v0e --refutes--> v07\n- v12 --refutes--> v0b\n- v14 --refutes--> v0d\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9ca6977fc6d2e55c0fd465c2efbb87d8a51987bd99f51b711622ef3d0af8f061","schema":"livegraph.boolean-propagation.v1","spec_hash":"4c7038cbf86275c9935f8ae704d42ea370d083070f55c25244d6894a08b065b7","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-e17ded58cbc9595ca995","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v0f\n- v08 --supports--> v04\n- v09 --supports--> v01\n- v14 --refutes--> v0c\n- v13 --refutes--> v06\n- v05 --refutes--> v00\n- v15 --supports--> v13\n- v11 --supports--> v07\n- v14 --refutes--> v08\n- v0a --supports--> v0d\n- v0f --supports--> v0a\n- v0e --refutes--> v09\n- v08 --refutes--> v15\n- v09 --refutes--> v16\n- v01 --refutes--> v10\n- v06 --supports--> v05\n- v0a --refutes--> v0b\n- v13 --supports--> v03\n- v00 --refutes--> v11\n- v09 --supports--> v02\n- v0b --supports--> v0e\n- v07 --refutes--> v12\n\n[VERIFIED ROOT]\nv14 = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"16df1d0b40643b4302bbb2a6503ed86b53e2c51ff411270355e6ce97e3383688","schema":"livegraph.boolean-propagation.v1","spec_hash":"e17ded58cbc9595ca9956aa2acd93e7b1e550dc1731497a60eaffba99948edae","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v14\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-5c01455f164b27ab072a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v11\n- v0e --supports--> v14\n- v0e --refutes--> v04\n- v13 --supports--> v07\n- v14 --supports--> v01\n- v0a --supports--> v12\n- v06 --supports--> v03\n- v16 --refutes--> v02\n- v01 --supports--> v0a\n- v0c --supports--> v06\n- v03 --refutes--> v15\n- v08 --supports--> v0e\n- v09 --refutes--> v10\n- v10 --refutes--> v0c\n- v08 --supports--> v0f\n- v0d --refutes--> v08\n- v03 --refutes--> v0d\n- v16 --refutes--> v00\n- v01 --refutes--> v05\n- v07 --supports--> v16\n- v02 --refutes--> v09\n- v01 --supports--> v0b\n\n[VERIFIED ROOT]\nv13 = TRUE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e25e3c3753a17e91483bf33d82c148e4c71f95b9c05b55b1a4dc5d6e2da1b12b","schema":"livegraph.boolean-propagation.v1","spec_hash":"5c01455f164b27ab072a94b92e5f55e31c3cf01082f5c6140bdf5b3ccd7fefe2","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-683a36dd45bb435226b4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v16\n- v0e --supports--> v10\n- v14 --refutes--> v0c\n- v0c --refutes--> v09\n- v12 --supports--> v15\n- v10 --refutes--> v00\n- v06 --refutes--> v0e\n- v15 --supports--> v02\n- v12 --supports--> v07\n- v0b --supports--> v08\n- v00 --supports--> v12\n- v0f --supports--> v0b\n- v0a --refutes--> v0d\n- v0d --supports--> v11\n- v04 --refutes--> v06\n- v0f --refutes--> v13\n- v05 --supports--> v0a\n- v0b --supports--> v03\n- v01 --refutes--> v14\n- v15 --refutes--> v05\n- v0d --supports--> v0f\n- v0b --refutes--> v01\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d7a3bae73cc240eb205097c272aa5f4bf208bc16b020a40a07268fe4c48efe92","schema":"livegraph.boolean-propagation.v1","spec_hash":"683a36dd45bb435226b4942e28181b06996206425b1d1f3b30d4133f8b73dfe1","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-3a5c4d07ce6be96b9199","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v0b\n- v02 --supports--> v0e\n- v0e --refutes--> v14\n- v08 --refutes--> v04\n- v08 --supports--> v07\n- v10 --supports--> v02\n- v00 --refutes--> v06\n- v10 --supports--> v0d\n- v03 --refutes--> v05\n- v0f --supports--> v16\n- v08 --supports--> v0c\n- v08 --supports--> v0a\n- v05 --refutes--> v12\n- v00 --refutes--> v11\n- v14 --supports--> v01\n- v06 --supports--> v10\n- v13 --supports--> v15\n- v15 --refutes--> v03\n- v09 --refutes--> v13\n- v05 --supports--> v08\n- v16 --refutes--> v09\n- v04 --supports--> v00\n\n[VERIFIED ROOT]\nv0f = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e4315576aa73314926135285c1e27e8c094cce9903e09a49e8159c4b0221b9c2","schema":"livegraph.boolean-propagation.v1","spec_hash":"3a5c4d07ce6be96b91998a3dbe8896590dc4010880d76e84cd2eb0959fb14386","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-c60f019419bc91a1eaa1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v06\n- v15 --refutes--> v01\n- v12 --supports--> v16\n- v07 --refutes--> v0b\n- v10 --supports--> v07\n- v0a --refutes--> v13\n- v15 --supports--> v08\n- v0b --supports--> v0c\n- v01 --supports--> v05\n- v00 --refutes--> v02\n- v05 --supports--> v14\n- v09 --supports--> v10\n- v10 --supports--> v0f\n- v12 --refutes--> v11\n- v00 --supports--> v04\n- v02 --supports--> v0e\n- v03 --supports--> v09\n- v02 --refutes--> v03\n- v05 --refutes--> v12\n- v05 --supports--> v0d\n- v06 --supports--> v0a\n- v16 --supports--> v00\n\n[VERIFIED ROOT]\nv15 = FALSE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c14afbadca37aa7918d93c3b10b2345da988f82533736730280b5da6afb1ebd8","schema":"livegraph.boolean-propagation.v1","spec_hash":"c60f019419bc91a1eaa136b47419a9bcd17aacc3157333c31614a13a585058dd","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v15\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-cb1fb7fa87e60522a315","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --supports--> v0c\n- v13 --supports--> v09\n- v06 --supports--> v0a\n- v04 --refutes--> v0d\n- v10 --refutes--> v15\n- v0d --supports--> v0f\n- v03 --refutes--> v0e\n- v11 --refutes--> v07\n- v03 --refutes--> v16\n- v09 --refutes--> v04\n- v11 --refutes--> v05\n- v08 --supports--> v10\n- v15 --refutes--> v02\n- v0d --supports--> v03\n- v09 --refutes--> v01\n- v02 --refutes--> v06\n- v11 --refutes--> v12\n- v00 --supports--> v11\n- v14 --refutes--> v00\n- v06 --refutes--> v0b\n- v0e --refutes--> v08\n- v12 --refutes--> v13\n\n[VERIFIED ROOT]\nv14 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"54d2be3ff426b24c92e2139ea97d2757fe3646de5cd67aac3e001e3245bd2db5","schema":"livegraph.boolean-propagation.v1","spec_hash":"cb1fb7fa87e60522a315e0e932a99c146f331842900f9198c8f0e7e9919dd1dc","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v14\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-190beac54a986893a9e9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v00\n- v13 --refutes--> v05\n- v05 --refutes--> v09\n- v12 --refutes--> v14\n- v03 --supports--> v0b\n- v14 --supports--> v04\n- v09 --supports--> v12\n- v02 --supports--> v01\n- v0d --refutes--> v0f\n- v07 --supports--> v02\n- v0c --refutes--> v0a\n- v07 --refutes--> v11\n- v01 --refutes--> v03\n- v05 --supports--> v08\n- v15 --supports--> v13\n- v0a --supports--> v0d\n- v0e --refutes--> v0c\n- v0d --supports--> v07\n- v0c --refutes--> v16\n- v15 --supports--> v10\n- v06 --refutes--> v0e\n- v03 --refutes--> v15\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"14bbf39e61703fcdaaf4c72d53679b71ecc8a132c3c1677e690e5818622530f6","schema":"livegraph.boolean-propagation.v1","spec_hash":"190beac54a986893a9e9500fdb0185bbc3d4130e8c042fe766ce1179b716d3d4","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-eb7dac6260fdad5b5949","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v07\n- v09 --refutes--> v00\n- v13 --refutes--> v12\n- v02 --refutes--> v08\n- v02 --supports--> v13\n- v0a --supports--> v0e\n- v12 --supports--> v10\n- v14 --supports--> v02\n- v09 --supports--> v0d\n- v0a --supports--> v0f\n- v09 --supports--> v06\n- v01 --refutes--> v0a\n- v10 --refutes--> v03\n- v03 --supports--> v0b\n- v0e --refutes--> v14\n- v07 --supports--> v05\n- v05 --refutes--> v04\n- v15 --supports--> v01\n- v0b --refutes--> v09\n- v03 --supports--> v0c\n- v16 --refutes--> v15\n- v0e --supports--> v11\n\n[VERIFIED ROOT]\nv16 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a942e94d58eee733d2d044a92bd9df3b7cb515287e0e3d636285091502a5a2ff","schema":"livegraph.boolean-propagation.v1","spec_hash":"eb7dac6260fdad5b5949e0929d7d3e433f2751e62e43e7a1ae2c09a579022141","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v16\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-f26c7d1728bbe2fa8be6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v06\n- v0c --refutes--> v13\n- v0c --supports--> v0d\n- v03 --supports--> v09\n- v0e --refutes--> v15\n- v06 --refutes--> v07\n- v15 --supports--> v14\n- v01 --refutes--> v08\n- v0c --refutes--> v05\n- v0b --supports--> v0e\n- v11 --supports--> v00\n- v14 --supports--> v0c\n- v0c --supports--> v12\n- v13 --supports--> v11\n- v03 --refutes--> v10\n- v07 --supports--> v02\n- v11 --refutes--> v01\n- v16 --supports--> v03\n- v09 --supports--> v0b\n- v03 --refutes--> v0a\n- v01 --supports--> v0f\n- v0f --refutes--> v04\n\n[VERIFIED ROOT]\nv16 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bed20a46c10e033284e662474544fa45ae23376598d4be1a5007f5589bc2d101","schema":"livegraph.boolean-propagation.v1","spec_hash":"f26c7d1728bbe2fa8be674f3b96409ffefd2050a2e97cd6d209155c33f15f01d","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v16\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-b728ddcf1c838b0b2be3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v06\n- v0e --supports--> v0b\n- v0b --refutes--> v05\n- v00 --supports--> v10\n- v02 --refutes--> v08\n- v15 --refutes--> v0d\n- v03 --refutes--> v00\n- v11 --supports--> v0e\n- v13 --refutes--> v07\n- v09 --refutes--> v12\n- v00 --refutes--> v02\n- v13 --supports--> v15\n- v11 --supports--> v0a\n- v08 --refutes--> v14\n- v01 --refutes--> v11\n- v05 --refutes--> v0f\n- v15 --refutes--> v03\n- v16 --refutes--> v09\n- v05 --refutes--> v04\n- v04 --refutes--> v16\n- v14 --refutes--> v01\n- v13 --supports--> v0c\n\n[VERIFIED ROOT]\nv13 = TRUE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e3e8a9c968d0085871d23c6f7cf1174b398b574f505252364027ed63e9d4ac7d","schema":"livegraph.boolean-propagation.v1","spec_hash":"b728ddcf1c838b0b2be3e1577ff1f0cdd2886905ca100dbf7a15b2536ced657b","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-4bcf77ee8f95d2917088","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --supports--> v01\n- v05 --supports--> v13\n- v15 --supports--> v10\n- v0b --refutes--> v0f\n- v01 --supports--> v06\n- v08 --refutes--> v16\n- v0d --supports--> v07\n- v05 --supports--> v04\n- v05 --refutes--> v15\n- v16 --supports--> v0e\n- v01 --supports--> v0a\n- v09 --refutes--> v00\n- v00 --supports--> v0c\n- v11 --supports--> v0b\n- v08 --refutes--> v02\n- v14 --supports--> v09\n- v0b --supports--> v08\n- v12 --supports--> v14\n- v09 --refutes--> v03\n- v00 --refutes--> v05\n- v0e --supports--> v12\n- v0a --supports--> v0d\n\n[VERIFIED ROOT]\nv11 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c588395a61504f1fa9b43a53a9ec5432053086374083a11efbecc0db9b15ded4","schema":"livegraph.boolean-propagation.v1","spec_hash":"4bcf77ee8f95d29170886e582d89df26df292f6412471c8d7f73922f21863afe","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-ee501f9580594c72679e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v13\n- v05 --supports--> v06\n- v0f --refutes--> v04\n- v09 --refutes--> v05\n- v10 --supports--> v15\n- v03 --refutes--> v12\n- v0d --supports--> v14\n- v13 --refutes--> v16\n- v05 --supports--> v01\n- v15 --refutes--> v07\n- v00 --refutes--> v08\n- v0f --refutes--> v09\n- v14 --supports--> v10\n- v05 --supports--> v0a\n- v11 --supports--> v0d\n- v0a --supports--> v0e\n- v16 --refutes--> v0c\n- v15 --refutes--> v02\n- v08 --supports--> v0b\n- v0b --refutes--> v03\n- v15 --refutes--> v00\n- v03 --refutes--> v0f\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6f1abc2688a83282c324e8789c4608022ad555b7d13a72d9ad4c95c9f879f1d1","schema":"livegraph.boolean-propagation.v1","spec_hash":"ee501f9580594c72679e476ad67361c408ab8be915cb18c79198049ec2b0add9","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-a2aa48b2febbd54aa6c4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v03\n- v10 --refutes--> v09\n- v00 --supports--> v13\n- v0e --refutes--> v11\n- v0d --refutes--> v07\n- v04 --refutes--> v0f\n- v0c --refutes--> v0b\n- v0d --supports--> v06\n- v01 --refutes--> v12\n- v14 --refutes--> v0d\n- v0c --refutes--> v0a\n- v0b --refutes--> v02\n- v08 --refutes--> v14\n- v16 --supports--> v04\n- v11 --supports--> v10\n- v12 --refutes--> v16\n- v13 --supports--> v15\n- v10 --supports--> v00\n- v13 --refutes--> v01\n- v14 --supports--> v05\n- v07 --refutes--> v0e\n- v0b --supports--> v08\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"48eda85d5b58e5e4b07c976a2151f6af09b9bd4a7e24b86372185e0fb255df92","schema":"livegraph.boolean-propagation.v1","spec_hash":"a2aa48b2febbd54aa6c454a247e0bf14caabe1fc1b65c0d8e0ffe23f54a7ac55","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-3e5f8583dcd5e4db64d8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v04\n- v07 --supports--> v0d\n- v11 --supports--> v16\n- v00 --supports--> v08\n- v0b --supports--> v13\n- v0c --refutes--> v05\n- v08 --refutes--> v0c\n- v02 --refutes--> v0b\n- v0f --refutes--> v07\n- v12 --supports--> v01\n- v00 --refutes--> v09\n- v0b --refutes--> v15\n- v05 --supports--> v02\n- v0d --supports--> v00\n- v11 --supports--> v03\n- v0a --supports--> v0e\n- v02 --refutes--> v06\n- v03 --supports--> v10\n- v03 --supports--> v0f\n- v13 --supports--> v0a\n- v0e --refutes--> v12\n- v03 --supports--> v14\n\n[VERIFIED ROOT]\nv11 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"68f32adf7e4f7126399e2df7c1a3e76e71ae6899bb6301ab43a890dc2dffcf11","schema":"livegraph.boolean-propagation.v1","spec_hash":"3e5f8583dcd5e4db64d8b4d0b968238c8169feb59f141ebf065254c9ec33f55d","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":15,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-df0d55ba12931cae92d1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --refutes--> v0b\n- v15 --supports--> v01\n- v0d --refutes--> v06\n- v0d --supports--> v00\n- v10 --refutes--> v07\n- v00 --refutes--> v0f\n- v11 --refutes--> v03\n- v01 --supports--> v04\n- v08 --refutes--> v11\n- v0c --refutes--> v05\n- v0b --refutes--> v08\n- v04 --supports--> v0e\n- v03 --supports--> v0d\n- v14 --supports--> v0c\n- v10 --supports--> v13\n- v05 --refutes--> v12\n- v04 --supports--> v14\n- v10 --refutes--> v02\n- v0c --refutes--> v09\n- v0d --refutes--> v16\n- v0a --refutes--> v10\n- v00 --refutes--> v15\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bcd1b887c57aedb940bc923741e0b12376159e61edebc409e59c58a067577a01","schema":"livegraph.boolean-propagation.v1","spec_hash":"df0d55ba12931cae92d19ccfdfabb5132fa90f176f4404cf20a8ce64a991a1bd","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-df235160eace275052e4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v0f\n- v01 --refutes--> v11\n- v15 --supports--> v12\n- v04 --refutes--> v15\n- v0e --supports--> v0d\n- v03 --supports--> v05\n- v08 --supports--> v06\n- v04 --supports--> v02\n- v01 --refutes--> v16\n- v12 --supports--> v08\n- v0d --refutes--> v10\n- v0c --supports--> v14\n- v0d --refutes--> v0a\n- v05 --refutes--> v13\n- v08 --refutes--> v01\n- v10 --refutes--> v0c\n- v07 --refutes--> v00\n- v07 --refutes--> v09\n- v01 --refutes--> v0b\n- v13 --refutes--> v0e\n- v11 --supports--> v07\n- v14 --supports--> v04\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f46b0939ad9780868e81be1d0861693776a6e85d106aa0997bd87a5246358cf5","schema":"livegraph.boolean-propagation.v1","spec_hash":"df235160eace275052e47af532d12160faa651244bf7942dc2cbae91adb7c38d","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-ede2fb9d4944343d5d6e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --supports--> v11\n- v15 --supports--> v13\n- v13 --refutes--> v02\n- v00 --supports--> v0b\n- v0b --refutes--> v0e\n- v02 --refutes--> v01\n- v16 --refutes--> v09\n- v0b --refutes--> v0d\n- v0a --refutes--> v00\n- v03 --refutes--> v16\n- v02 --supports--> v08\n- v03 --supports--> v05\n- v0e --refutes--> v06\n- v09 --refutes--> v04\n- v01 --supports--> v0a\n- v0e --supports--> v10\n- v10 --refutes--> v14\n- v14 --supports--> v03\n- v12 --refutes--> v07\n- v12 --refutes--> v15\n- v0c --supports--> v12\n- v0b --supports--> v0f\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0552f69da6c456aae1047ce1e87169f65bbd0ad838caa5c19be3fe9daf9f50fd","schema":"livegraph.boolean-propagation.v1","spec_hash":"ede2fb9d4944343d5d6e7f74ebb09a2a81b43b1988a36bb3e9fc4e1a3095d207","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-fb59a372ab657471aae0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v04\n- v16 --refutes--> v0f\n- v0b --refutes--> v0a\n- v15 --supports--> v05\n- v13 --supports--> v14\n- v05 --supports--> v0b\n- v0b --supports--> v0d\n- v04 --refutes--> v01\n- v08 --refutes--> v00\n- v10 --refutes--> v08\n- v08 --refutes--> v09\n- v14 --refutes--> v03\n- v07 --refutes--> v12\n- v12 --supports--> v16\n- v10 --refutes--> v0c\n- v0b --supports--> v02\n- v03 --refutes--> v07\n- v03 --refutes--> v06\n- v0b --supports--> v11\n- v02 --refutes--> v10\n- v00 --supports--> v13\n- v0f --supports--> v0e\n\n[VERIFIED ROOT]\nv15 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0295385507be2cebd748d66d1bce305521ae4fb49f85ec2077f98307d1bd9193","schema":"livegraph.boolean-propagation.v1","spec_hash":"fb59a372ab657471aae09255e68a747913ef40aa58e149f303ff1e89a8843a01","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v15\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-e798fadbfa6a2a80c638","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v13\n- v01 --refutes--> v0b\n- v05 --refutes--> v02\n- v16 --refutes--> v07\n- v13 --refutes--> v15\n- v0d --refutes--> v09\n- v02 --supports--> v16\n- v03 --refutes--> v05\n- v11 --refutes--> v06\n- v0c --refutes--> v0f\n- v01 --refutes--> v08\n- v10 --supports--> v14\n- v0a --supports--> v11\n- v05 --refutes--> v04\n- v13 --refutes--> v03\n- v08 --refutes--> v00\n- v12 --supports--> v0e\n- v06 --supports--> v0d\n- v0c --supports--> v01\n- v16 --refutes--> v12\n- v08 --refutes--> v10\n- v09 --refutes--> v0c\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bb945283e4a3c90508c873e6778121ed50cba87f1b177eae5d5ab911c51e84e3","schema":"livegraph.boolean-propagation.v1","spec_hash":"e798fadbfa6a2a80c6388672fa8a3fee8a5dd554e319f8f4bda85a8c3f9c9f6c","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-0deccb3bbe4aec88923f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v0c\n- v0e --refutes--> v07\n- v02 --supports--> v04\n- v11 --refutes--> v13\n- v06 --supports--> v0e\n- v06 --supports--> v08\n- v00 --supports--> v05\n- v07 --supports--> v12\n- v14 --supports--> v06\n- v05 --supports--> v09\n- v10 --supports--> v02\n- v07 --refutes--> v01\n- v14 --supports--> v03\n- v09 --refutes--> v0b\n- v04 --refutes--> v0f\n- v12 --refutes--> v00\n- v0b --refutes--> v11\n- v00 --refutes--> v15\n- v02 --refutes--> v0d\n- v0f --refutes--> v0a\n- v0a --refutes--> v14\n- v0e --refutes--> v16\n\n[VERIFIED ROOT]\nv10 = TRUE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ca8fc6d7031d5b89ce42e7975e930d053a91a8c3741fb4ef4650fa792402f2d7","schema":"livegraph.boolean-propagation.v1","spec_hash":"0deccb3bbe4aec88923f1c0e219a2ba546308e54204c0e28be821828158b41d7","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":15,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-f8b8d2ff2501822a631e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v07\n- v0a --refutes--> v03\n- v0b --supports--> v10\n- v07 --supports--> v09\n- v12 --refutes--> v0c\n- v04 --supports--> v14\n- v05 --refutes--> v0b\n- v10 --supports--> v0d\n- v15 --supports--> v02\n- v0c --refutes--> v08\n- v03 --supports--> v05\n- v13 --refutes--> v15\n- v02 --refutes--> v0a\n- v12 --supports--> v00\n- v10 --supports--> v01\n- v08 --supports--> v04\n- v04 --refutes--> v13\n- v08 --supports--> v0f\n- v0a --refutes--> v0e\n- v09 --supports--> v11\n- v12 --refutes--> v16\n- v04 --refutes--> v06\n\n[VERIFIED ROOT]\nv12 = FALSE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b96b1fa472659fd26841d75d4ad46da4f835b175f51696856a541b83d64ec1cf","schema":"livegraph.boolean-propagation.v1","spec_hash":"f8b8d2ff2501822a631ec14e8137b448dbadc8af144ab662f52ac5db8f2117b7","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v12\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-37d7ec2a4c399f542908","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v03\n- v04 --supports--> v0d\n- v03 --supports--> v15\n- v09 --supports--> v02\n- v0e --refutes--> v09\n- v06 --supports--> v14\n- v09 --refutes--> v0b\n- v12 --supports--> v05\n- v0d --refutes--> v0f\n- v00 --refutes--> v11\n- v11 --supports--> v13\n- v0f --refutes--> v0c\n- v14 --refutes--> v12\n- v03 --refutes--> v0e\n- v0c --refutes--> v16\n- v01 --refutes--> v04\n- v12 --supports--> v0a\n- v14 --supports--> v08\n- v10 --supports--> v00\n- v11 --refutes--> v01\n- v0c --refutes--> v06\n- v14 --refutes--> v07\n\n[VERIFIED ROOT]\nv10 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9c0c418517268b1d432de5c47a2d331b32f2c9e69fc7cde77ed4678e831fcb36","schema":"livegraph.boolean-propagation.v1","spec_hash":"37d7ec2a4c399f542908507f206e4eef2d275c08d1d329c8ac653cfd6e23751f","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":15,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-177be4d729f878524b47","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v05\n- v05 --supports--> v16\n- v0c --refutes--> v08\n- v02 --supports--> v0f\n- v09 --refutes--> v02\n- v16 --refutes--> v14\n- v01 --refutes--> v15\n- v03 --refutes--> v06\n- v14 --refutes--> v03\n- v03 --supports--> v12\n- v0e --refutes--> v09\n- v0a --supports--> v13\n- v09 --supports--> v07\n- v00 --supports--> v01\n- v13 --supports--> v04\n- v13 --supports--> v0d\n- v02 --supports--> v11\n- v0f --supports--> v00\n- v0c --refutes--> v10\n- v13 --supports--> v0e\n- v02 --supports--> v0b\n- v12 --supports--> v0a\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv15\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9988e62c02a32c79db1410652cadc79210d68ecaa90f7140a6ccaafa3d954496","schema":"livegraph.boolean-propagation.v1","spec_hash":"177be4d729f878524b47b0288e5595cca39389934469ac9d4fc5c74934df40a5","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-1b12f2998021718d5258","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v0c\n- v16 --supports--> v03\n- v0b --supports--> v16\n- v0a --supports--> v0d\n- v10 --refutes--> v01\n- v0e --supports--> v11\n- v00 --refutes--> v06\n- v0e --refutes--> v0f\n- v0a --refutes--> v02\n- v03 --refutes--> v13\n- v03 --refutes--> v15\n- v03 --refutes--> v12\n- v09 --refutes--> v08\n- v0c --supports--> v00\n- v14 --supports--> v10\n- v04 --supports--> v07\n- v07 --refutes--> v0e\n- v06 --refutes--> v05\n- v11 --supports--> v0a\n- v08 --refutes--> v14\n- v06 --refutes--> v04\n- v03 --supports--> v09\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f4b037210c88576477d7a47dd393fb7083126375961482432c69effae86edb88","schema":"livegraph.boolean-propagation.v1","spec_hash":"1b12f2998021718d5258f0caeb67276d345d94e1ae7a50fdded74a33657c631a","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-e7b1a0082dc2ccb1703f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v0c\n- v08 --supports--> v06\n- v0f --refutes--> v09\n- v07 --supports--> v0b\n- v07 --refutes--> v0e\n- v15 --refutes--> v16\n- v00 --supports--> v14\n- v10 --refutes--> v11\n- v04 --refutes--> v02\n- v04 --refutes--> v08\n- v13 --supports--> v0a\n- v05 --supports--> v0f\n- v16 --refutes--> v13\n- v0a --refutes--> v01\n- v01 --supports--> v12\n- v14 --refutes--> v05\n- v05 --supports--> v0d\n- v0e --supports--> v00\n- v13 --supports--> v03\n- v08 --refutes--> v10\n- v11 --refutes--> v15\n- v01 --refutes--> v07\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"98cc69f657e3719b46eed9e308075aedeab2a2d8ddfee5f6cca9c72d5159afcd","schema":"livegraph.boolean-propagation.v1","spec_hash":"e7b1a0082dc2ccb1703f9ccb1d66707171397e14256d6998e4de832e92f8833b","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-f5df33514216804fdaf5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --supports--> v0b\n- v01 --supports--> v0c\n- v04 --refutes--> v13\n- v14 --supports--> v08\n- v07 --refutes--> v0f\n- v13 --supports--> v11\n- v0c --refutes--> v14\n- v02 --supports--> v07\n- v09 --supports--> v05\n- v12 --refutes--> v09\n- v0d --supports--> v06\n- v15 --supports--> v0e\n- v07 --supports--> v12\n- v04 --refutes--> v03\n- v0d --supports--> v01\n- v0a --refutes--> v15\n- v04 --supports--> v10\n- v16 --supports--> v04\n- v13 --supports--> v02\n- v08 --supports--> v16\n- v05 --supports--> v00\n- v15 --supports--> v0d\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9f08997ae6e70ec9b9f359c4be3e5b2c3cc50a701d14178640edccb5536cb26b","schema":"livegraph.boolean-propagation.v1","spec_hash":"f5df33514216804fdaf5393108955f2fd799e7a351c49301bef6ad81e921baeb","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-fd18342e94c972a6c7c9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v13\n- v07 --supports--> v0c\n- v05 --supports--> v12\n- v13 --refutes--> v15\n- v0e --supports--> v06\n- v0e --supports--> v16\n- v02 --supports--> v09\n- v04 --refutes--> v0d\n- v0d --supports--> v0e\n- v05 --refutes--> v08\n- v0c --refutes--> v04\n- v15 --supports--> v0a\n- v0d --supports--> v11\n- v02 --refutes--> v10\n- v01 --supports--> v0f\n- v0f --refutes--> v02\n- v08 --supports--> v01\n- v0a --refutes--> v0b\n- v04 --supports--> v03\n- v07 --refutes--> v14\n- v16 --supports--> v05\n- v0b --supports--> v00\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8dbb4d273bb2a8082f03fbe74a7b20e03eb66aba22090b6233f3d29e8e063dfb","schema":"livegraph.boolean-propagation.v1","spec_hash":"fd18342e94c972a6c7c9a2d10216ac08cf24e96b98c1b5c2d8209dd165412f17","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-5bf6e7873ebe770ea698","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v08\n- v0d --supports--> v12\n- v13 --supports--> v00\n- v02 --refutes--> v06\n- v13 --supports--> v04\n- v01 --supports--> v0e\n- v06 --refutes--> v09\n- v15 --refutes--> v0f\n- v12 --refutes--> v15\n- v11 --refutes--> v0b\n- v01 --refutes--> v14\n- v09 --supports--> v0c\n- v16 --refutes--> v0d\n- v02 --refutes--> v05\n- v15 --refutes--> v10\n- v0f --supports--> v13\n- v07 --supports--> v01\n- v0b --refutes--> v16\n- v00 --supports--> v03\n- v00 --supports--> v02\n- v01 --refutes--> v0a\n- v0c --supports--> v07\n\n[VERIFIED ROOT]\nv11 = FALSE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"35fad388a930be8bf0cfa87e681342bf783ff8764b0afae829754ad7b21ecbb8","schema":"livegraph.boolean-propagation.v1","spec_hash":"5bf6e7873ebe770ea69870457d2ca49fdc8f46d8a69d2073a8ce0de120bdd92a","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-94261ad0fedeb7d4cf20","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v02\n- v0d --supports--> v00\n- v12 --supports--> v10\n- v09 --supports--> v0b\n- v13 --refutes--> v0d\n- v10 --supports--> v08\n- v0f --supports--> v0e\n- v05 --supports--> v12\n- v08 --refutes--> v14\n- v0b --supports--> v0a\n- v0c --supports--> v05\n- v16 --supports--> v04\n- v08 --refutes--> v11\n- v04 --supports--> v09\n- v09 --refutes--> v01\n- v0f --refutes--> v15\n- v14 --refutes--> v16\n- v0f --refutes--> v06\n- v0d --refutes--> v0f\n- v02 --refutes--> v03\n- v03 --supports--> v13\n- v0f --supports--> v07\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4e6bbb6c68d785cd7041201b323a3112915623232e38900c480956f86f8390e2","schema":"livegraph.boolean-propagation.v1","spec_hash":"94261ad0fedeb7d4cf20f2e441a6f3a721e4923535c4b181dd609915f2009080","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-a8d8cd70eacdef388e19","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v0e\n- v04 --supports--> v03\n- v09 --refutes--> v01\n- v0e --supports--> v11\n- v14 --refutes--> v13\n- v16 --supports--> v0a\n- v00 --refutes--> v0f\n- v03 --refutes--> v12\n- v00 --refutes--> v02\n- v0a --refutes--> v04\n- v10 --supports--> v09\n- v0d --supports--> v16\n- v05 --refutes--> v14\n- v03 --refutes--> v07\n- v0e --refutes--> v08\n- v00 --supports--> v0c\n- v15 --refutes--> v05\n- v16 --refutes--> v06\n- v14 --refutes--> v00\n- v11 --refutes--> v0d\n- v12 --refutes--> v10\n- v16 --supports--> v0b\n\n[VERIFIED ROOT]\nv15 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"55a8da4ad4e9e668dd14cf5922844da8fbfb71290a40e9a46479b6d174dbba2f","schema":"livegraph.boolean-propagation.v1","spec_hash":"a8d8cd70eacdef388e19924535a2efad6d500aa773f1254b433acf4ca9b93336","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v15\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-2e71e77647008a2da916","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v0d\n- v05 --supports--> v0c\n- v14 --supports--> v0b\n- v13 --supports--> v08\n- v00 --supports--> v10\n- v02 --supports--> v0e\n- v02 --supports--> v13\n- v14 --supports--> v12\n- v08 --supports--> v01\n- v10 --supports--> v05\n- v06 --supports--> v00\n- v0a --refutes--> v06\n- v09 --supports--> v14\n- v11 --refutes--> v03\n- v11 --supports--> v0f\n- v02 --refutes--> v07\n- v11 --refutes--> v04\n- v12 --supports--> v0a\n- v0d --refutes--> v09\n- v00 --supports--> v15\n- v04 --refutes--> v16\n- v04 --refutes--> v02\n\n[VERIFIED ROOT]\nv11 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"928aef88b260126b9a3aafb74c242d4b9a6d282380850775fcc142ec22622e78","schema":"livegraph.boolean-propagation.v1","spec_hash":"2e71e77647008a2da9166cc9859830f1dc09bcaaef97b2cfe19a0fe789d076e7","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":15,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-b9bb2d2e4695a7e1b183","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --refutes--> v0e\n- v14 --supports--> v0a\n- v08 --refutes--> v02\n- v08 --refutes--> v01\n- v11 --supports--> v0f\n- v02 --supports--> v0c\n- v0b --refutes--> v04\n- v0c --supports--> v14\n- v0a --supports--> v03\n- v00 --refutes--> v15\n- v08 --supports--> v12\n- v11 --refutes--> v0b\n- v15 --supports--> v16\n- v07 --refutes--> v11\n- v03 --supports--> v00\n- v0e --supports--> v0d\n- v04 --refutes--> v08\n- v07 --supports--> v06\n- v0d --refutes--> v05\n- v0d --supports--> v13\n- v09 --refutes--> v07\n- v07 --supports--> v10\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e4808781bb8bc58541f87b49653b275e22fc73cd4edd1afa0ac8915a6acefb49","schema":"livegraph.boolean-propagation.v1","spec_hash":"b9bb2d2e4695a7e1b1830c874596f988cbd71fd7eb9150cd13101d976958cd38","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-0c0c938879f5400339d5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v11\n- v11 --refutes--> v14\n- v0a --refutes--> v0e\n- v04 --supports--> v0f\n- v15 --refutes--> v13\n- v0c --refutes--> v02\n- v0a --refutes--> v16\n- v0a --supports--> v05\n- v00 --refutes--> v12\n- v0a --refutes--> v0c\n- v10 --refutes--> v07\n- v14 --supports--> v06\n- v10 --supports--> v03\n- v00 --supports--> v01\n- v01 --supports--> v0b\n- v0f --supports--> v09\n- v09 --refutes--> v10\n- v09 --supports--> v08\n- v14 --supports--> v0d\n- v06 --supports--> v04\n- v0b --refutes--> v15\n- v13 --supports--> v0a\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9696b55c8e628bb6cab361b4d6f867609f9851ec30859a639469a800e8f3db07","schema":"livegraph.boolean-propagation.v1","spec_hash":"0c0c938879f5400339d52aa157ca03acd885e3c94ce8b763d716e2fb7146a72e","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-08d4dc62cb6447e3131d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v04\n- v09 --refutes--> v0b\n- v0b --supports--> v08\n- v06 --refutes--> v0d\n- v0c --refutes--> v0f\n- v13 --refutes--> v16\n- v08 --refutes--> v15\n- v16 --supports--> v0e\n- v08 --refutes--> v11\n- v05 --supports--> v0c\n- v14 --refutes--> v12\n- v05 --supports--> v07\n- v0d --supports--> v10\n- v0c --refutes--> v0a\n- v10 --supports--> v05\n- v16 --refutes--> v03\n- v0c --supports--> v01\n- v03 --refutes--> v06\n- v00 --refutes--> v09\n- v12 --supports--> v02\n- v12 --supports--> v00\n- v0a --supports--> v14\n\n[VERIFIED ROOT]\nv13 = FALSE\n\n[TARGET]\nv15\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c11b9b2df801a82224aa3879a11098cc85ce5a2a86a811ea6fbedc34a4497f45","schema":"livegraph.boolean-propagation.v1","spec_hash":"08d4dc62cb6447e3131d15a823f80a828d0490fe21977755287b73534679edcf","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v13\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-a9219664075a184ff098","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v02\n- v14 --refutes--> v0d\n- v02 --supports--> v0f\n- v09 --supports--> v07\n- v15 --supports--> v0a\n- v11 --refutes--> v16\n- v12 --refutes--> v08\n- v02 --refutes--> v10\n- v0c --supports--> v0e\n- v0c --supports--> v11\n- v00 --refutes--> v05\n- v03 --refutes--> v15\n- v15 --refutes--> v01\n- v07 --supports--> v12\n- v04 --refutes--> v0c\n- v0b --refutes--> v14\n- v15 --refutes--> v09\n- v11 --refutes--> v03\n- v06 --supports--> v00\n- v12 --refutes--> v0b\n- v05 --refutes--> v04\n- v02 --supports--> v13\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"44474d1a3b08ded6962f442df8ab4d96a63480f980e0337f721130892ef4b04b","schema":"livegraph.boolean-propagation.v1","spec_hash":"a9219664075a184ff09809313c43a432e12a3fc9f12d0a61dcf683969ccbeee0","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-76f1170eecc7518d8a17","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v11\n- v0d --supports--> v02\n- v06 --refutes--> v13\n- v04 --supports--> v06\n- v0b --refutes--> v07\n- v03 --supports--> v04\n- v0c --refutes--> v16\n- v13 --supports--> v0f\n- v0e --supports--> v00\n- v03 --refutes--> v08\n- v05 --supports--> v0c\n- v0c --supports--> v01\n- v07 --supports--> v15\n- v00 --supports--> v12\n- v16 --refutes--> v09\n- v0d --supports--> v0e\n- v16 --supports--> v10\n- v0c --supports--> v0a\n- v12 --supports--> v0b\n- v13 --refutes--> v0d\n- v07 --refutes--> v14\n- v15 --supports--> v05\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e0634960653dd44ecb88e8858e43d2e30f278353954bddfd24e2e7e138f3216c","schema":"livegraph.boolean-propagation.v1","spec_hash":"76f1170eecc7518d8a1736632baed78da2a6ad3490c01e8e04c43ce9f067a0c4","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-b82d17f0ab3d59d2a349","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --supports--> v08\n- v14 --refutes--> v11\n- v03 --refutes--> v01\n- v11 --supports--> v0f\n- v12 --refutes--> v0c\n- v03 --supports--> v06\n- v0d --supports--> v0b\n- v0e --supports--> v14\n- v0b --refutes--> v16\n- v08 --supports--> v04\n- v02 --refutes--> v03\n- v06 --refutes--> v12\n- v08 --supports--> v09\n- v15 --refutes--> v0d\n- v13 --refutes--> v07\n- v16 --refutes--> v00\n- v0c --supports--> v0e\n- v16 --refutes--> v02\n- v0e --supports--> v05\n- v13 --refutes--> v15\n- v10 --refutes--> v13\n- v14 --supports--> v0a\n\n[VERIFIED ROOT]\nv10 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d1fbdf33b19a13597d6af3c0937359e6bc5b2e60c05c126cb68ab3843c35cd06","schema":"livegraph.boolean-propagation.v1","spec_hash":"b82d17f0ab3d59d2a349b8e47daec43daf1ba1033c9c295dc4e448ef358d2d83","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-3f2b1820c21a50c2d2ed","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v0e\n- v11 --refutes--> v16\n- v12 --supports--> v07\n- v07 --refutes--> v0b\n- v0d --refutes--> v11\n- v09 --supports--> v03\n- v03 --supports--> v01\n- v14 --refutes--> v0f\n- v0a --supports--> v0d\n- v13 --supports--> v14\n- v0b --refutes--> v06\n- v0d --refutes--> v04\n- v01 --supports--> v0a\n- v14 --refutes--> v02\n- v07 --refutes--> v15\n- v06 --refutes--> v05\n- v06 --supports--> v09\n- v16 --supports--> v13\n- v06 --supports--> v0c\n- v12 --refutes--> v08\n- v00 --refutes--> v12\n- v02 --refutes--> v10\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0d1de4d20a0205e070f37d007a68131326a4847ee15bf1f27666b5fbf24e770d","schema":"livegraph.boolean-propagation.v1","spec_hash":"3f2b1820c21a50c2d2ed16bc88a07e5ac3858a0e42bf5b32a78f442bcd57d7dd","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-d55577e8cf557b17157f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v07\n- v10 --refutes--> v13\n- v0b --supports--> v01\n- v16 --supports--> v11\n- v12 --supports--> v0d\n- v02 --refutes--> v0c\n- v03 --supports--> v0f\n- v13 --refutes--> v0e\n- v0c --supports--> v12\n- v13 --supports--> v0a\n- v09 --supports--> v02\n- v0c --refutes--> v08\n- v03 --refutes--> v14\n- v04 --refutes--> v0b\n- v0f --supports--> v05\n- v11 --refutes--> v03\n- v0d --refutes--> v16\n- v15 --refutes--> v04\n- v11 --supports--> v06\n- v0f --refutes--> v15\n- v01 --refutes--> v10\n- v13 --refutes--> v00\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0bcb7bb4afd9259d71fb8b80a50418aa4cce3f37ac2729ff4292ffa9f0b2f1d5","schema":"livegraph.boolean-propagation.v1","spec_hash":"d55577e8cf557b17157fd4f1d5b637c6d410fdfb9e36dc14e28baf512e729550","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-2c0f93b480d7529c46fe","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v07\n- v0d --refutes--> v10\n- v00 --supports--> v15\n- v14 --supports--> v13\n- v06 --refutes--> v08\n- v07 --supports--> v03\n- v00 --supports--> v0d\n- v16 --supports--> v01\n- v09 --refutes--> v02\n- v12 --supports--> v14\n- v06 --supports--> v04\n- v00 --refutes--> v0a\n- v02 --supports--> v11\n- v03 --supports--> v0e\n- v13 --refutes--> v0f\n- v10 --supports--> v05\n- v14 --supports--> v0b\n- v0f --refutes--> v00\n- v11 --refutes--> v16\n- v10 --supports--> v09\n- v10 --refutes--> v0c\n- v16 --refutes--> v06\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9b814ce6293240bec3580803550b050ac10b98ba5e7c063422b743973a0c3468","schema":"livegraph.boolean-propagation.v1","spec_hash":"2c0f93b480d7529c46fe2c975be7309b9c8ad1dde577cf2e6c1b391549a8e98e","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-6a918f12a6aca0c04b5e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v08\n- v0f --supports--> v01\n- v16 --refutes--> v0e\n- v07 --supports--> v0a\n- v15 --refutes--> v10\n- v0d --supports--> v16\n- v12 --refutes--> v00\n- v14 --refutes--> v12\n- v10 --refutes--> v02\n- v08 --refutes--> v05\n- v07 --supports--> v09\n- v09 --refutes--> v11\n- v09 --refutes--> v15\n- v00 --refutes--> v0f\n- v16 --supports--> v13\n- v0c --refutes--> v14\n- v02 --refutes--> v06\n- v00 --refutes--> v03\n- v15 --supports--> v0b\n- v06 --refutes--> v0d\n- v07 --supports--> v04\n- v0e --supports--> v0c\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8f14223c400b331b8ae6db7735cde92dae1ed021190ec4bb209ccd451fe8fb0d","schema":"livegraph.boolean-propagation.v1","spec_hash":"6a918f12a6aca0c04b5eb03bf58915c90ea0b1cbe26cb289b48187709c4f3a31","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-2365c1f69ea0f527f496","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --refutes--> v02\n- v02 --refutes--> v07\n- v03 --supports--> v08\n- v0d --refutes--> v13\n- v04 --supports--> v09\n- v11 --refutes--> v0e\n- v08 --refutes--> v0a\n- v08 --refutes--> v16\n- v05 --supports--> v11\n- v05 --refutes--> v06\n- v07 --supports--> v00\n- v0c --supports--> v12\n- v12 --refutes--> v14\n- v0e --refutes--> v10\n- v09 --refutes--> v03\n- v0c --refutes--> v0b\n- v0e --refutes--> v01\n- v0f --refutes--> v0d\n- v08 --supports--> v05\n- v0e --supports--> v0c\n- v00 --supports--> v04\n- v12 --supports--> v0f\n\n[VERIFIED ROOT]\nv15 = TRUE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2366ef8beeffc1a7da161a0528c9b841401691a38e3fa4450ac66d3bdfdb2747","schema":"livegraph.boolean-propagation.v1","spec_hash":"2365c1f69ea0f527f4964ab03d6e4557e64302befa040046d77e66a45cad4858","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v15\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-cca00f52243153529efc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --supports--> v0e\n- v10 --supports--> v0f\n- v00 --refutes--> v16\n- v01 --supports--> v09\n- v02 --supports--> v13\n- v0d --refutes--> v07\n- v10 --refutes--> v14\n- v0b --supports--> v03\n- v0b --refutes--> v08\n- v05 --refutes--> v15\n- v02 --supports--> v06\n- v14 --refutes--> v01\n- v15 --supports--> v04\n- v11 --refutes--> v0a\n- v09 --supports--> v12\n- v04 --supports--> v10\n- v12 --refutes--> v0b\n- v02 --refutes--> v00\n- v12 --supports--> v0c\n- v0a --refutes--> v0d\n- v03 --supports--> v11\n- v00 --supports--> v05\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"759c2c14cdda767ef6e89650dcb0bb48c7a008a35f04d0a894a4e2829ca575c3","schema":"livegraph.boolean-propagation.v1","spec_hash":"cca00f52243153529efcb87b81dce8bff784031b9a25ebc385d85e8e4f165530","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-7fe452f7a9bfdc773e7b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v0f\n- v01 --refutes--> v08\n- v09 --supports--> v16\n- v0c --supports--> v07\n- v14 --refutes--> v04\n- v10 --supports--> v12\n- v0f --supports--> v03\n- v13 --supports--> v01\n- v09 --supports--> v14\n- v11 --supports--> v0d\n- v09 --supports--> v15\n- v01 --refutes--> v0e\n- v08 --refutes--> v02\n- v05 --refutes--> v09\n- v05 --refutes--> v0b\n- v14 --supports--> v06\n- v11 --refutes--> v00\n- v02 --supports--> v0c\n- v0a --refutes--> v10\n- v14 --refutes--> v11\n- v03 --refutes--> v0a\n- v07 --refutes--> v05\n\n[VERIFIED ROOT]\nv13 = FALSE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"996d9ffb884258dc42b961a152b202f6e0c7eccc448519f6aadb59766bbd2684","schema":"livegraph.boolean-propagation.v1","spec_hash":"7fe452f7a9bfdc773e7b2bacf6392d4c020e06509a2e19a6cdc499726392b0e0","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":15,\"root\":\"v13\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-ef6adc52bdbb5c107573","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v11\n- v05 --supports--> v08\n- v14 --supports--> v00\n- v0b --supports--> v04\n- v0d --supports--> v14\n- v0d --refutes--> v16\n- v10 --supports--> v03\n- v0f --refutes--> v05\n- v07 --supports--> v10\n- v01 --refutes--> v02\n- v05 --supports--> v13\n- v12 --refutes--> v0f\n- v01 --supports--> v0c\n- v0c --supports--> v0a\n- v11 --supports--> v0b\n- v10 --supports--> v0d\n- v07 --supports--> v06\n- v14 --supports--> v15\n- v0a --refutes--> v07\n- v10 --refutes--> v0e\n- v09 --refutes--> v01\n- v04 --refutes--> v12\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"86c0a44e872e9c2766f374a87dee62cc81b32c11397f52c6d29b52bd766b1440","schema":"livegraph.boolean-propagation.v1","spec_hash":"ef6adc52bdbb5c1075731bf19eaf60cd9b43f0e8183bde8daba95043db476e82","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":15,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-d53026b299c54921f3c7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v01\n- v02 --refutes--> v12\n- v13 --supports--> v04\n- v0c --refutes--> v05\n- v04 --supports--> v0d\n- v14 --supports--> v09\n- v04 --refutes--> v03\n- v01 --supports--> v08\n- v0e --supports--> v0b\n- v11 --refutes--> v13\n- v12 --supports--> v0e\n- v08 --refutes--> v15\n- v11 --refutes--> v10\n- v0e --refutes--> v0c\n- v04 --refutes--> v06\n- v06 --supports--> v07\n- v04 --refutes--> v0f\n- v09 --supports--> v00\n- v08 --supports--> v11\n- v16 --refutes--> v0a\n- v0a --refutes--> v02\n- v0c --supports--> v14\n\n[VERIFIED ROOT]\nv16 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"82144e1a14baabd74b7b848bbd9fd6b70f0716c9c50cff2cddd8026e0840bcbd","schema":"livegraph.boolean-propagation.v1","spec_hash":"d53026b299c54921f3c74065b2a05d8d57557374456ee3100500286bf588bb83","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v16\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-2e485d5e29d863b80018","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v16 --supports--> v07\n- v09 --refutes--> v13\n- v03 --refutes--> v0a\n- v06 --supports--> v00\n- v12 --supports--> v01\n- v00 --supports--> v11\n- v0b --refutes--> v0c\n- v15 --refutes--> v10\n- v12 --refutes--> v0d\n- v0f --supports--> v12\n- v00 --refutes--> v08\n- v16 --supports--> v06\n- v0c --supports--> v04\n- v00 --refutes--> v0e\n- v13 --supports--> v15\n- v01 --supports--> v0b\n- v12 --refutes--> v14\n- v0b --supports--> v02\n- v0e --supports--> v03\n- v15 --supports--> v05\n- v0a --supports--> v09\n- v04 --refutes--> v16\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5bbd5140652eb7d7b9b6d70dbaa6e89c185990a4984dd03015ea828c6d63867c","schema":"livegraph.boolean-propagation.v1","spec_hash":"2e485d5e29d863b80018e57b8e2ac1d53e93048cf18a67c5e1c319220291da7f","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-209588f9f0794adbdc92","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v0c\n- v13 --refutes--> v00\n- v13 --refutes--> v0a\n- v08 --refutes--> v0b\n- v08 --refutes--> v0f\n- v04 --refutes--> v02\n- v09 --supports--> v01\n- v0d --supports--> v04\n- v01 --supports--> v03\n- v12 --supports--> v13\n- v02 --supports--> v06\n- v15 --refutes--> v10\n- v13 --refutes--> v05\n- v06 --refutes--> v09\n- v15 --refutes--> v07\n- v11 --supports--> v15\n- v08 --refutes--> v11\n- v0d --refutes--> v14\n- v0e --supports--> v12\n- v0c --supports--> v08\n- v07 --supports--> v0d\n- v07 --supports--> v16\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ba834ae1d29905fa24f2407d5ee068af9de4562364cf01375fd87c18498ed32d","schema":"livegraph.boolean-propagation.v1","spec_hash":"209588f9f0794adbdc92c236572218495bae2aed3ce11018c817a0009b8788f7","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-56c1019d9ea99555d4b9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --supports--> v04\n- v00 --supports--> v0c\n- v06 --supports--> v10\n- v0a --refutes--> v16\n- v0e --refutes--> v05\n- v0b --refutes--> v01\n- v07 --supports--> v11\n- v0d --refutes--> v15\n- v0b --refutes--> v07\n- v11 --refutes--> v06\n- v10 --refutes--> v0d\n- v0d --refutes--> v14\n- v16 --supports--> v0b\n- v02 --supports--> v0a\n- v0d --supports--> v12\n- v16 --supports--> v03\n- v04 --refutes--> v00\n- v00 --supports--> v13\n- v05 --supports--> v09\n- v13 --refutes--> v0e\n- v14 --supports--> v0f\n- v13 --supports--> v08\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a4de131057ba77599574b2a9677f07bfbf14df105e91b0b563f1b8141b527b3d","schema":"livegraph.boolean-propagation.v1","spec_hash":"56c1019d9ea99555d4b97d498be5e16bfbdeb26f7ea09b975c6d4a237d48da27","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-4f53b39f3cbd7f7eef12","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v00\n- v06 --refutes--> v11\n- v04 --refutes--> v0c\n- v0e --refutes--> v13\n- v00 --refutes--> v12\n- v02 --supports--> v15\n- v02 --refutes--> v10\n- v03 --supports--> v04\n- v00 --refutes--> v14\n- v00 --refutes--> v0d\n- v08 --supports--> v0a\n- v16 --supports--> v05\n- v04 --supports--> v08\n- v09 --supports--> v02\n- v13 --supports--> v16\n- v04 --supports--> v07\n- v0d --supports--> v09\n- v05 --refutes--> v03\n- v15 --refutes--> v0f\n- v01 --refutes--> v0e\n- v15 --refutes--> v01\n- v08 --supports--> v06\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0cab5ff5ee04650c4104ec4a37ed7592ced90ec5a139a60ee48f7a6ab7251e0b","schema":"livegraph.boolean-propagation.v1","spec_hash":"4f53b39f3cbd7f7eef128b95ec5ab3b58ebf72126ce30e3e9a65caff8f5d9542","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-a7e612c6c71d127e1891","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v0c\n- v09 --supports--> v06\n- v10 --supports--> v11\n- v01 --refutes--> v10\n- v16 --refutes--> v03\n- v12 --refutes--> v15\n- v06 --refutes--> v07\n- v14 --refutes--> v0e\n- v15 --refutes--> v16\n- v13 --refutes--> v0f\n- v10 --supports--> v14\n- v16 --refutes--> v00\n- v05 --supports--> v04\n- v07 --refutes--> v01\n- v09 --supports--> v0b\n- v14 --refutes--> v0d\n- v0e --supports--> v13\n- v0f --refutes--> v08\n- v04 --supports--> v09\n- v0f --refutes--> v12\n- v0e --refutes--> v02\n- v00 --supports--> v0a\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"484230c17f6a38ae2b81415b49ad789deb0916dbe70e9f7944a69dc0531a377d","schema":"livegraph.boolean-propagation.v1","spec_hash":"a7e612c6c71d127e1891922f823fb992a66f1a95bb77c5a67c68253813ac179a","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-bb646f042460a1e09afe","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --refutes--> v0c\n- v07 --refutes--> v0d\n- v13 --supports--> v11\n- v08 --supports--> v0a\n- v00 --refutes--> v05\n- v0c --supports--> v0f\n- v13 --supports--> v03\n- v00 --supports--> v06\n- v0d --refutes--> v16\n- v10 --supports--> v0e\n- v0d --supports--> v02\n- v06 --refutes--> v13\n- v0c --supports--> v15\n- v15 --supports--> v01\n- v0b --refutes--> v08\n- v16 --supports--> v10\n- v01 --supports--> v12\n- v01 --supports--> v09\n- v0e --supports--> v0b\n- v08 --supports--> v04\n- v01 --refutes--> v07\n- v0a --refutes--> v00\n\n[VERIFIED ROOT]\nv14 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"067a4d9e1bd9806f062682ccd8cb5da998384e7d61f7910e0f2542069110b72f","schema":"livegraph.boolean-propagation.v1","spec_hash":"bb646f042460a1e09afeadfcbf374d4874b3c6f7f3bcfbca54876a5be5260eea","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v14\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-39c5e964b21f0cfe09ee","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v0b\n- v0b --supports--> v0d\n- v0d --refutes--> v14\n- v10 --supports--> v03\n- v07 --refutes--> v01\n- v03 --refutes--> v07\n- v10 --supports--> v05\n- v10 --supports--> v12\n- v0d --supports--> v0a\n- v07 --supports--> v16\n- v00 --supports--> v11\n- v16 --refutes--> v02\n- v10 --supports--> v06\n- v0e --supports--> v00\n- v11 --refutes--> v09\n- v14 --supports--> v10\n- v04 --refutes--> v13\n- v0c --refutes--> v0e\n- v15 --refutes--> v04\n- v13 --refutes--> v0c\n- v0b --supports--> v0f\n- v11 --supports--> v08\n\n[VERIFIED ROOT]\nv15 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9c0c9898bd4039192b1750bb4fa07d0fd38db4bb688556446da119f101875993","schema":"livegraph.boolean-propagation.v1","spec_hash":"39c5e964b21f0cfe09ee8ef150d2d23256dd7d8d8c90a110ae348ddf62c062e9","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v15\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-0e205bce2bb38668e4ee","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v16\n- v15 --supports--> v0d\n- v05 --refutes--> v08\n- v11 --refutes--> v0a\n- v0f --supports--> v12\n- v0f --supports--> v02\n- v10 --refutes--> v0e\n- v07 --supports--> v0b\n- v0b --refutes--> v09\n- v00 --supports--> v04\n- v15 --refutes--> v13\n- v00 --supports--> v15\n- v09 --refutes--> v0c\n- v14 --supports--> v05\n- v0c --supports--> v03\n- v0f --supports--> v11\n- v0e --refutes--> v01\n- v0d --supports--> v0f\n- v03 --refutes--> v14\n- v01 --supports--> v00\n- v05 --supports--> v10\n- v0c --supports--> v06\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0c3df9a7466ed9f3fb47b365efe781a1392ba12c6f782b3712f8ac446a5ecc90","schema":"livegraph.boolean-propagation.v1","spec_hash":"0e205bce2bb38668e4eeb84cb02a6e115706ee1b7781e4295afc532b4c3ab69c","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-0bfa5a982771d0340c42","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v02\n- v03 --refutes--> v04\n- v15 --refutes--> v0f\n- v0b --supports--> v09\n- v10 --supports--> v0d\n- v05 --supports--> v16\n- v0b --supports--> v0e\n- v08 --refutes--> v01\n- v12 --supports--> v03\n- v00 --supports--> v12\n- v16 --refutes--> v11\n- v06 --refutes--> v10\n- v08 --refutes--> v0c\n- v15 --refutes--> v0a\n- v04 --refutes--> v0b\n- v0a --refutes--> v00\n- v10 --refutes--> v15\n- v0a --refutes--> v07\n- v16 --supports--> v14\n- v09 --refutes--> v08\n- v02 --supports--> v06\n- v16 --supports--> v13\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"48b9f684eacd8282b108c7b0369e65444f13518216dfca8aeefd91810f367ea7","schema":"livegraph.boolean-propagation.v1","spec_hash":"0bfa5a982771d0340c4270c5c5a62a051ba81d78874c804ef75d5f1e1f11911a","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-479c1266125d2ef6d1c3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --refutes--> v0c\n- v0b --supports--> v0a\n- v11 --supports--> v13\n- v0d --refutes--> v0e\n- v10 --refutes--> v02\n- v0c --refutes--> v0d\n- v0a --supports--> v04\n- v04 --refutes--> v03\n- v06 --supports--> v09\n- v0a --supports--> v12\n- v01 --supports--> v10\n- v08 --refutes--> v01\n- v05 --supports--> v06\n- v08 --supports--> v00\n- v05 --refutes--> v16\n- v04 --refutes--> v14\n- v03 --supports--> v05\n- v0f --refutes--> v11\n- v04 --refutes--> v15\n- v09 --refutes--> v0f\n- v0d --supports--> v08\n- v0c --supports--> v07\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"de82842cb9b3e109308051241cec1d88432f3ac5a37199b96a165e10cf665ba1","schema":"livegraph.boolean-propagation.v1","spec_hash":"479c1266125d2ef6d1c379daf92240652e2640de3be4d8a8e0a97996f731ee24","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":15,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-95c9025d8d0602b6b6f4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v0c\n- v0c --refutes--> v09\n- v03 --supports--> v01\n- v12 --supports--> v00\n- v05 --supports--> v08\n- v09 --refutes--> v11\n- v01 --supports--> v12\n- v03 --refutes--> v04\n- v11 --supports--> v13\n- v14 --refutes--> v05\n- v12 --refutes--> v0e\n- v0b --supports--> v16\n- v07 --refutes--> v14\n- v05 --supports--> v15\n- v02 --refutes--> v10\n- v14 --supports--> v0d\n- v12 --supports--> v07\n- v11 --refutes--> v0f\n- v08 --refutes--> v0b\n- v10 --refutes--> v03\n- v08 --refutes--> v06\n- v13 --refutes--> v02\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv16\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9182279a10c08a3edff2d5f56cab4f804895ed00a39d4566367a1f9d8e26c341","schema":"livegraph.boolean-propagation.v1","spec_hash":"95c9025d8d0602b6b6f4634c5fcdd432ffbee7a2889c6c37b2d1c0047dd83e5c","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":15,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v16\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-665161b3f2ea97b615bf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v00\n- v0b --supports--> v13\n- v13 --supports--> v0a\n- v0e --supports--> v15\n- v14 --supports--> v11\n- v08 --supports--> v0c\n- v08 --refutes--> v16\n- v10 --supports--> v05\n- v00 --refutes--> v14\n- v10 --supports--> v0d\n- v09 --refutes--> v08\n- v15 --refutes--> v02\n- v04 --supports--> v09\n- v05 --supports--> v07\n- v01 --supports--> v10\n- v00 --refutes--> v0f\n- v00 --supports--> v12\n- v03 --refutes--> v0e\n- v02 --refutes--> v04\n- v0c --supports--> v01\n- v10 --supports--> v06\n- v13 --refutes--> v03\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3a8487106a2affe551116c4a112d70ea54e49ee54dbc7079e1fc040d4547b07e","schema":"livegraph.boolean-propagation.v1","spec_hash":"665161b3f2ea97b615bf39a8441f3582e61d4aef949b83982a6bf2c041a3affc","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-d09bc2d3b46abb59fec4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --refutes--> v16\n- v15 --refutes--> v09\n- v03 --refutes--> v0c\n- v0d --supports--> v10\n- v0a --refutes--> v05\n- v02 --supports--> v14\n- v15 --supports--> v00\n- v13 --refutes--> v06\n- v10 --refutes--> v12\n- v02 --refutes--> v0b\n- v13 --refutes--> v0e\n- v09 --refutes--> v02\n- v05 --refutes--> v15\n- v0a --supports--> v01\n- v07 --refutes--> v11\n- v10 --refutes--> v07\n- v15 --supports--> v08\n- v06 --refutes--> v04\n- v0c --refutes--> v0f\n- v0b --refutes--> v0d\n- v04 --supports--> v0a\n- v11 --supports--> v03\n\n[VERIFIED ROOT]\nv13 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d9f04666d07b948c36e2a22a2c40fc4c96aeb2598a1b44ac88b4a9909f8ecd15","schema":"livegraph.boolean-propagation.v1","spec_hash":"d09bc2d3b46abb59fec4381b9666877d1f1d2d96530db2e83939904dad9ce351","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v13\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-a3a8d62c107baae97494","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v0a\n- v12 --supports--> v04\n- v0d --refutes--> v13\n- v0f --refutes--> v06\n- v08 --refutes--> v09\n- v03 --supports--> v15\n- v10 --refutes--> v14\n- v03 --supports--> v0d\n- v0a --supports--> v03\n- v14 --supports--> v02\n- v13 --supports--> v0c\n- v0f --refutes--> v11\n- v0a --supports--> v0e\n- v02 --supports--> v08\n- v07 --supports--> v10\n- v01 --refutes--> v00\n- v16 --supports--> v0f\n- v04 --refutes--> v07\n- v06 --supports--> v01\n- v02 --refutes--> v05\n- v08 --refutes--> v16\n- v06 --supports--> v0b\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1c783c5d8a4ce84d6b252029b8ede15211f66a1fd32d940108592d3ca0da9c41","schema":"livegraph.boolean-propagation.v1","spec_hash":"a3a8d62c107baae97494a67b694c1e8e4a454ff54eeab5bc56a223f529a7016e","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-548a3290e22ccec49cdf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v0b\n- v0d --supports--> v04\n- v15 --refutes--> v06\n- v04 --refutes--> v09\n- v07 --refutes--> v0f\n- v11 --refutes--> v0c\n- v04 --refutes--> v00\n- v13 --supports--> v0a\n- v06 --refutes--> v07\n- v02 --refutes--> v08\n- v07 --refutes--> v02\n- v04 --supports--> v01\n- v07 --supports--> v03\n- v0c --supports--> v14\n- v04 --supports--> v16\n- v07 --supports--> v0e\n- v14 --refutes--> v0d\n- v12 --supports--> v11\n- v08 --supports--> v10\n- v09 --refutes--> v13\n- v0a --refutes--> v05\n- v10 --supports--> v12\n\n[VERIFIED ROOT]\nv15 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b9e210e9b633a5714b09cbeb25649372af858f5682b4481960ff96a97012b455","schema":"livegraph.boolean-propagation.v1","spec_hash":"548a3290e22ccec49cdf7f811729183d128530fd647d9397ac2a374a405b033d","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v15\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-1b542d036582bd8f0cbc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --supports--> v09\n- v16 --refutes--> v0c\n- v0c --refutes--> v15\n- v02 --refutes--> v00\n- v0d --supports--> v16\n- v0a --refutes--> v10\n- v0c --refutes--> v0b\n- v0d --supports--> v0f\n- v0c --supports--> v04\n- v07 --refutes--> v02\n- v14 --refutes--> v11\n- v0e --refutes--> v03\n- v14 --supports--> v08\n- v09 --refutes--> v07\n- v10 --supports--> v14\n- v12 --refutes--> v13\n- v15 --refutes--> v06\n- v07 --refutes--> v05\n- v0e --supports--> v0a\n- v15 --supports--> v01\n- v00 --supports--> v0d\n- v08 --refutes--> v12\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a21310b0ce23e3e55cf4e096bde2569a26dfef170917db1c6a57d679d50f3144","schema":"livegraph.boolean-propagation.v1","spec_hash":"1b542d036582bd8f0cbcef7ed5a86f6d9194a1319f72cfd9830d71e24617e085","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-3bfad463d152497dd09d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --refutes--> v12\n- v01 --refutes--> v0e\n- v0a --refutes--> v08\n- v00 --refutes--> v15\n- v01 --refutes--> v03\n- v02 --refutes--> v0a\n- v07 --refutes--> v0d\n- v10 --refutes--> v13\n- v14 --supports--> v01\n- v12 --refutes--> v10\n- v10 --supports--> v06\n- v0a --supports--> v0b\n- v16 --refutes--> v00\n- v0c --supports--> v07\n- v14 --refutes--> v09\n- v0f --refutes--> v04\n- v0a --supports--> v14\n- v0e --supports--> v0c\n- v0d --supports--> v0f\n- v04 --supports--> v16\n- v0a --supports--> v05\n- v0a --refutes--> v11\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d125de11e1fb2c16e147c5a2777fcce19c09d5982b7d465543198e4739fbbd76","schema":"livegraph.boolean-propagation.v1","spec_hash":"3bfad463d152497dd09ddd99087a2915e96379de1d8945a554e60969f014983c","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-2f1e0c304fc8251a3b57","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --refutes--> v14\n- v10 --refutes--> v0a\n- v0a --supports--> v06\n- v08 --refutes--> v0e\n- v10 --refutes--> v0f\n- v0c --supports--> v10\n- v16 --supports--> v12\n- v00 --refutes--> v0c\n- v0b --refutes--> v07\n- v01 --supports--> v16\n- v04 --refutes--> v03\n- v13 --refutes--> v15\n- v12 --refutes--> v13\n- v14 --refutes--> v11\n- v07 --supports--> v02\n- v07 --refutes--> v08\n- v03 --supports--> v01\n- v0a --supports--> v05\n- v08 --refutes--> v04\n- v0a --supports--> v09\n- v09 --refutes--> v0b\n- v0b --supports--> v0d\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c198490e222dd3e4610cf7930bd229f6317a1c2a4e1626ec7f37b045766b24a3","schema":"livegraph.boolean-propagation.v1","spec_hash":"2f1e0c304fc8251a3b5741128d3aba7ffe281a45e26e0aff8d1db1d416d6164d","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-6c3c8050ce86fda9c711","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v13\n- v14 --refutes--> v02\n- v01 --refutes--> v12\n- v01 --refutes--> v03\n- v10 --refutes--> v07\n- v15 --refutes--> v0f\n- v02 --supports--> v04\n- v0c --supports--> v0d\n- v07 --refutes--> v0e\n- v06 --refutes--> v10\n- v0c --refutes--> v0a\n- v0e --supports--> v08\n- v0e --refutes--> v01\n- v11 --supports--> v00\n- v09 --refutes--> v14\n- v04 --refutes--> v06\n- v0f --supports--> v0c\n- v0d --refutes--> v11\n- v16 --supports--> v15\n- v01 --supports--> v0b\n- v11 --refutes--> v09\n- v11 --refutes--> v05\n\n[VERIFIED ROOT]\nv16 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"96122037f43baa3d383c861d52bdc03916ac97c97a72493d3a96c5fa4d00221d","schema":"livegraph.boolean-propagation.v1","spec_hash":"6c3c8050ce86fda9c711a0d42fbf089824313f6e7e3ca404d38ef2420b9e49c3","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v16\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-ba136f615a4af2139ad4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v0a\n- v00 --refutes--> v07\n- v15 --refutes--> v06\n- v03 --supports--> v08\n- v0d --supports--> v01\n- v08 --supports--> v02\n- v12 --refutes--> v0b\n- v07 --refutes--> v14\n- v03 --refutes--> v10\n- v07 --refutes--> v09\n- v01 --refutes--> v0f\n- v16 --supports--> v05\n- v0f --refutes--> v00\n- v02 --refutes--> v0d\n- v15 --supports--> v16\n- v12 --refutes--> v0e\n- v0e --supports--> v15\n- v0a --refutes--> v03\n- v03 --supports--> v11\n- v15 --refutes--> v13\n- v07 --refutes--> v0c\n- v0c --refutes--> v04\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5b07589aa242005b9628c74d918a4448b73916ba424616c1a7f1fb037c627a50","schema":"livegraph.boolean-propagation.v1","spec_hash":"ba136f615a4af2139ad452b0083de7ba60f79d16712f125144b864bc2d4c60c2","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-ee3dd3c18668fef31f06","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --refutes--> v02\n- v00 --supports--> v14\n- v12 --refutes--> v10\n- v08 --supports--> v0d\n- v08 --supports--> v0c\n- v03 --refutes--> v13\n- v0e --supports--> v0b\n- v0b --refutes--> v08\n- v10 --refutes--> v03\n- v06 --refutes--> v07\n- v09 --refutes--> v04\n- v03 --refutes--> v0a\n- v0b --supports--> v15\n- v00 --supports--> v16\n- v05 --refutes--> v06\n- v0f --refutes--> v05\n- v16 --supports--> v09\n- v12 --supports--> v11\n- v13 --refutes--> v00\n- v04 --refutes--> v0f\n- v0f --supports--> v01\n- v07 --refutes--> v0e\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0ffaa3e1e7dd852cf9dc3389d6a053002c8b4395d019d6435e9e62d07d4b7d93","schema":"livegraph.boolean-propagation.v1","spec_hash":"ee3dd3c18668fef31f0624804a7fbfaeee18de6c58fc17428a5b7ed34519d443","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-239d7e729de9ec7ad3ca","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v16 --refutes--> v00\n- v11 --supports--> v15\n- v06 --refutes--> v01\n- v0d --refutes--> v08\n- v04 --supports--> v07\n- v09 --supports--> v0a\n- v08 --supports--> v0c\n- v12 --supports--> v13\n- v13 --supports--> v02\n- v0f --refutes--> v14\n- v00 --refutes--> v12\n- v09 --refutes--> v0d\n- v0e --refutes--> v04\n- v13 --refutes--> v06\n- v15 --supports--> v03\n- v12 --refutes--> v10\n- v0c --supports--> v0f\n- v0f --refutes--> v0e\n- v01 --refutes--> v09\n- v0d --supports--> v05\n- v15 --refutes--> v16\n- v13 --refutes--> v0b\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6398c5efe89095de4c250e224b8655accf2ef1d668d27dfb6e94b1f78e01372c","schema":"livegraph.boolean-propagation.v1","spec_hash":"239d7e729de9ec7ad3ca50554417c7e18a41ddf24f1bf05a636be8a221950634","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-57c1b268b7de34452487","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --supports--> v06\n- v12 --refutes--> v14\n- v06 --refutes--> v13\n- v14 --refutes--> v0d\n- v0e --refutes--> v16\n- v13 --supports--> v0e\n- v07 --refutes--> v08\n- v16 --supports--> v12\n- v08 --refutes--> v00\n- v08 --refutes--> v10\n- v12 --refutes--> v03\n- v0d --supports--> v0c\n- v0c --supports--> v02\n- v12 --refutes--> v09\n- v11 --supports--> v0b\n- v0c --supports--> v0f\n- v05 --supports--> v11\n- v07 --supports--> v0a\n- v15 --supports--> v01\n- v10 --refutes--> v05\n- v02 --supports--> v07\n- v16 --refutes--> v04\n\n[VERIFIED ROOT]\nv15 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d301b031d0a6ca1f88b1ea6f3efccda76f4e6df3596647a9d05e72c8cac55612","schema":"livegraph.boolean-propagation.v1","spec_hash":"57c1b268b7de34452487649d804559758c0ec494ca4b8bfac1de8f54c2679a8e","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v15\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-40a06eccd0e360d66b1c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v02\n- v01 --refutes--> v05\n- v02 --supports--> v03\n- v0a --refutes--> v06\n- v16 --supports--> v01\n- v06 --supports--> v10\n- v07 --refutes--> v0c\n- v02 --supports--> v0e\n- v0e --supports--> v12\n- v10 --supports--> v0d\n- v04 --refutes--> v15\n- v0e --supports--> v08\n- v00 --refutes--> v11\n- v0c --supports--> v14\n- v04 --refutes--> v07\n- v07 --refutes--> v13\n- v0c --refutes--> v09\n- v09 --supports--> v0a\n- v04 --refutes--> v0b\n- v0e --refutes--> v0f\n- v0d --supports--> v00\n- v08 --refutes--> v16\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"84366433d8429f8fe375392361f8d9650c3440b42b413cb7817a643e9e9bb640","schema":"livegraph.boolean-propagation.v1","spec_hash":"40a06eccd0e360d66b1cae1439338653f5c01bd7a07a97b611e0e584343ecafd","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-741adc6176d67dde53db","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v01\n- v0e --refutes--> v15\n- v01 --supports--> v0b\n- v08 --supports--> v16\n- v0c --refutes--> v03\n- v02 --supports--> v0a\n- v04 --supports--> v0d\n- v10 --supports--> v07\n- v0e --supports--> v11\n- v14 --supports--> v00\n- v0a --supports--> v0c\n- v01 --supports--> v0e\n- v14 --supports--> v09\n- v01 --refutes--> v05\n- v03 --supports--> v04\n- v0e --supports--> v13\n- v16 --supports--> v10\n- v10 --refutes--> v12\n- v0d --refutes--> v14\n- v07 --refutes--> v06\n- v01 --refutes--> v0f\n- v09 --supports--> v08\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e324ba1202a98c9dc3f605bc5be9fc5cf0079876ed83130b93b663ffecbbac24","schema":"livegraph.boolean-propagation.v1","spec_hash":"741adc6176d67dde53dbda44cd95bd81789f29cc7cdf97da8ab64fb634a2c1fc","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-e2f79adb3a4a109bef0b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v0b\n- v0b --refutes--> v0d\n- v16 --refutes--> v08\n- v05 --supports--> v09\n- v0d --refutes--> v07\n- v0c --refutes--> v03\n- v02 --refutes--> v05\n- v00 --supports--> v11\n- v05 --supports--> v0c\n- v0c --refutes--> v06\n- v15 --refutes--> v01\n- v01 --supports--> v0f\n- v0f --supports--> v16\n- v02 --supports--> v0a\n- v14 --supports--> v12\n- v06 --refutes--> v0e\n- v16 --refutes--> v02\n- v16 --refutes--> v04\n- v06 --supports--> v00\n- v12 --supports--> v10\n- v07 --supports--> v13\n- v11 --refutes--> v14\n\n[VERIFIED ROOT]\nv15 = TRUE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"340837b2fff72e4af1a7b204f9c302635f6e1650d3adff1994816925a7ae484c","schema":"livegraph.boolean-propagation.v1","spec_hash":"e2f79adb3a4a109bef0bedd4146ab2348ae952d748a470e8c6e6e1db5d7c37fb","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v15\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-95f6b5256f6e94af3517","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v0d\n- v0b --supports--> v15\n- v0a --supports--> v0e\n- v11 --refutes--> v00\n- v03 --refutes--> v05\n- v16 --supports--> v11\n- v08 --supports--> v0b\n- v07 --supports--> v08\n- v13 --supports--> v09\n- v15 --refutes--> v04\n- v0c --supports--> v07\n- v13 --refutes--> v14\n- v0f --refutes--> v16\n- v00 --refutes--> v03\n- v16 --supports--> v01\n- v06 --supports--> v0f\n- v03 --refutes--> v0c\n- v0f --supports--> v02\n- v13 --refutes--> v0a\n- v08 --refutes--> v10\n- v04 --refutes--> v12\n- v0e --supports--> v06\n\n[VERIFIED ROOT]\nv13 = TRUE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"769a4f1284c190395e4657285f71df2d2f8e39b950a862e5c4ebee1c64da95c9","schema":"livegraph.boolean-propagation.v1","spec_hash":"95f6b5256f6e94af3517efa384f3374a26a942ff5d655df5b3e73705af83f640","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-3e480b4a9f70982f0b72","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v0b\n- v09 --refutes--> v08\n- v0c --supports--> v09\n- v10 --supports--> v05\n- v03 --refutes--> v10\n- v0b --refutes--> v0a\n- v0a --refutes--> v12\n- v13 --refutes--> v00\n- v05 --refutes--> v07\n- v04 --refutes--> v0d\n- v02 --refutes--> v15\n- v12 --refutes--> v16\n- v0d --supports--> v11\n- v04 --refutes--> v0f\n- v0d --supports--> v03\n- v14 --refutes--> v02\n- v05 --refutes--> v0e\n- v14 --supports--> v01\n- v02 --refutes--> v0c\n- v16 --supports--> v14\n- v09 --refutes--> v04\n- v0b --refutes--> v06\n\n[VERIFIED ROOT]\nv13 = FALSE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a90cc81f338d067a5966e34609e100ebcd25a3ef9eaee270d30f5f54d506d5a2","schema":"livegraph.boolean-propagation.v1","spec_hash":"3e480b4a9f70982f0b72b52e8f3fa6007c7ec6ddf91fe0e6a494e3f40897ebcf","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v13\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-8830cbe76bd31b0a80c8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v15\n- v16 --refutes--> v14\n- v16 --supports--> v0f\n- v12 --refutes--> v0e\n- v10 --supports--> v07\n- v09 --refutes--> v08\n- v09 --supports--> v13\n- v15 --supports--> v11\n- v02 --refutes--> v0a\n- v04 --supports--> v03\n- v10 --refutes--> v16\n- v0c --supports--> v06\n- v11 --refutes--> v02\n- v0f --refutes--> v0c\n- v0b --refutes--> v09\n- v12 --supports--> v04\n- v15 --supports--> v00\n- v0f --refutes--> v01\n- v0a --supports--> v05\n- v03 --refutes--> v0d\n- v0a --supports--> v0b\n- v0d --refutes--> v10\n\n[VERIFIED ROOT]\nv12 = FALSE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9bb4c52bdd2fa9694238c4bff9176486859c6e8cb5d4ac4de72bc14f917f09c1","schema":"livegraph.boolean-propagation.v1","spec_hash":"8830cbe76bd31b0a80c80b8a34d2bc00501ab87552d68d4758101928174b5b6a","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v12\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-804b1aec4d0515c03438","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v16 --supports--> v13\n- v14 --refutes--> v0d\n- v11 --supports--> v07\n- v04 --supports--> v12\n- v02 --refutes--> v11\n- v15 --supports--> v00\n- v12 --refutes--> v14\n- v11 --refutes--> v0b\n- v0d --refutes--> v03\n- v0c --refutes--> v01\n- v06 --refutes--> v0f\n- v02 --supports--> v0e\n- v0f --supports--> v02\n- v0b --refutes--> v0a\n- v03 --supports--> v05\n- v04 --refutes--> v09\n- v16 --supports--> v10\n- v03 --refutes--> v06\n- v01 --refutes--> v16\n- v0b --refutes--> v08\n- v15 --supports--> v0c\n- v08 --supports--> v15\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c2a12a85b6917f593bf7abbcdae173ffa484757a5cd73d7284bede0e195a4b05","schema":"livegraph.boolean-propagation.v1","spec_hash":"804b1aec4d0515c0343856ade457b0ca4d56c528bbaeca89e6eeb7b80757c796","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-f3087ceb759e4f59db7a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v16\n- v04 --refutes--> v0f\n- v09 --refutes--> v05\n- v0a --refutes--> v11\n- v14 --refutes--> v13\n- v00 --refutes--> v15\n- v15 --supports--> v02\n- v03 --supports--> v0d\n- v0a --supports--> v06\n- v0d --supports--> v08\n- v12 --refutes--> v07\n- v0d --supports--> v12\n- v0c --refutes--> v01\n- v15 --supports--> v03\n- v0f --refutes--> v0c\n- v0b --supports--> v10\n- v0c --supports--> v0e\n- v13 --refutes--> v0a\n- v10 --refutes--> v09\n- v12 --refutes--> v04\n- v09 --supports--> v14\n- v01 --supports--> v0b\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv16\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"719657380cc52a429141e436cbb5a424f249bf8f2cf482d4aba95bd32dc98eef","schema":"livegraph.boolean-propagation.v1","spec_hash":"f3087ceb759e4f59db7a799b2c6cb30d243e5d4d43c4543246b3eaa207f34539","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v16\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-68c8d8d1bf7a9db8e5ec","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v16\n- v04 --refutes--> v0d\n- v01 --supports--> v03\n- v0b --supports--> v02\n- v12 --refutes--> v0e\n- v0b --refutes--> v11\n- v01 --supports--> v0a\n- v02 --supports--> v00\n- v01 --supports--> v0c\n- v03 --refutes--> v0f\n- v06 --refutes--> v04\n- v03 --supports--> v08\n- v16 --supports--> v09\n- v0e --supports--> v0b\n- v01 --refutes--> v15\n- v16 --supports--> v13\n- v0f --supports--> v05\n- v0d --supports--> v12\n- v10 --supports--> v06\n- v07 --refutes--> v10\n- v02 --supports--> v01\n- v09 --supports--> v14\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv14\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"21847b307ded5dcb8ff6f13ae836a54c65bb1183f75ce63e752f947c17772a1a","schema":"livegraph.boolean-propagation.v1","spec_hash":"68c8d8d1bf7a9db8e5ec020512b955450603ec5e4bee29511fe3596d623949b2","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-71658cc7f2ff179359be","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v07\n- v13 --supports--> v06\n- v07 --supports--> v13\n- v00 --supports--> v16\n- v0e --supports--> v02\n- v04 --refutes--> v0f\n- v0b --supports--> v0c\n- v04 --refutes--> v01\n- v05 --supports--> v0b\n- v14 --supports--> v00\n- v0b --refutes--> v10\n- v08 --refutes--> v14\n- v05 --refutes--> v0d\n- v0f --supports--> v09\n- v04 --supports--> v11\n- v0a --refutes--> v05\n- v13 --refutes--> v15\n- v16 --supports--> v0e\n- v02 --refutes--> v04\n- v13 --supports--> v12\n- v05 --supports--> v03\n- v0c --refutes--> v08\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e8008d3520edf390225e0d9eb3f83315969c6295632b1568c0564dd391b7ba11","schema":"livegraph.boolean-propagation.v1","spec_hash":"71658cc7f2ff179359be852d12c916273496ec0c4643e3c8eccc2c4e4b29f80f","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-3280345662929febc8bc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v00\n- v11 --supports--> v0c\n- v07 --refutes--> v06\n- v11 --refutes--> v02\n- v11 --refutes--> v0e\n- v0f --refutes--> v12\n- v0a --supports--> v05\n- v12 --refutes--> v0b\n- v12 --supports--> v16\n- v0d --refutes--> v04\n- v03 --refutes--> v07\n- v07 --supports--> v01\n- v01 --supports--> v11\n- v15 --refutes--> v0d\n- v00 --refutes--> v03\n- v02 --supports--> v0f\n- v04 --supports--> v09\n- v14 --supports--> v0a\n- v07 --supports--> v13\n- v08 --refutes--> v15\n- v12 --refutes--> v14\n- v09 --supports--> v10\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2ff982f3c9b0bb6475b169eaa530edca44d303557b04f70eda420e22a510ac8e","schema":"livegraph.boolean-propagation.v1","spec_hash":"3280345662929febc8bc0e056418a2656c090408576dd966276195ce8304e6fd","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-9dfa06126ced27013b19","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v06\n- v05 --supports--> v12\n- v0e --refutes--> v0d\n- v00 --refutes--> v09\n- v04 --refutes--> v01\n- v11 --supports--> v13\n- v0e --supports--> v08\n- v11 --supports--> v02\n- v0b --supports--> v05\n- v0a --supports--> v00\n- v01 --supports--> v0e\n- v0f --supports--> v03\n- v12 --refutes--> v0f\n- v09 --supports--> v07\n- v03 --refutes--> v04\n- v09 --supports--> v11\n- v15 --supports--> v0a\n- v0a --refutes--> v10\n- v01 --refutes--> v14\n- v00 --refutes--> v0c\n- v12 --supports--> v16\n- v02 --supports--> v0b\n\n[VERIFIED ROOT]\nv15 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0b45f7b1f4b36df1cac99cdd2f88cd6304bf1af57eef7fdc51fb4439705200a1","schema":"livegraph.boolean-propagation.v1","spec_hash":"9dfa06126ced27013b19440ee6caee265325410b87f673d7d56bb78dc95510a5","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v15\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-bcc5c7ad63fc61a57ac0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v00\n- v12 --refutes--> v10\n- v00 --refutes--> v08\n- v03 --supports--> v04\n- v08 --supports--> v05\n- v16 --refutes--> v02\n- v16 --supports--> v14\n- v0f --refutes--> v0b\n- v0a --supports--> v12\n- v13 --supports--> v03\n- v02 --refutes--> v0d\n- v0a --refutes--> v09\n- v07 --supports--> v0f\n- v0f --supports--> v11\n- v05 --supports--> v13\n- v16 --refutes--> v15\n- v12 --supports--> v16\n- v03 --supports--> v0a\n- v11 --refutes--> v0c\n- v01 --supports--> v0e\n- v01 --supports--> v07\n- v06 --supports--> v01\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8cac6082c1f0f6f03dbf3a8f952a2b34da70b3bcb08951e635ac90c68156639c","schema":"livegraph.boolean-propagation.v1","spec_hash":"bcc5c7ad63fc61a57ac08159f79882deeccb045bf00a7d2c3a3f849b7290b712","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-4b2f34ae54d93a156ce4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v10\n- v0c --supports--> v0f\n- v0e --refutes--> v0b\n- v04 --refutes--> v00\n- v07 --supports--> v0a\n- v09 --supports--> v05\n- v06 --refutes--> v04\n- v12 --refutes--> v08\n- v12 --supports--> v09\n- v14 --refutes--> v0e\n- v11 --refutes--> v14\n- v04 --refutes--> v07\n- v15 --supports--> v0c\n- v0b --supports--> v12\n- v03 --supports--> v02\n- v0f --supports--> v0d\n- v14 --refutes--> v01\n- v07 --refutes--> v15\n- v02 --refutes--> v06\n- v14 --supports--> v16\n- v0d --refutes--> v11\n- v09 --supports--> v13\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"98ac5cc0ba62dddd9f23c10125b7cbeddb86b827a63fefd1cea45bc8722d98e9","schema":"livegraph.boolean-propagation.v1","spec_hash":"4b2f34ae54d93a156ce49069ddec9dbe52d036388195faa4616ef74442c1908e","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-236167831fd39dba6a74","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v0d\n- v0f --supports--> v03\n- v0b --supports--> v15\n- v13 --refutes--> v02\n- v04 --refutes--> v11\n- v02 --refutes--> v16\n- v0a --refutes--> v14\n- v0d --refutes--> v01\n- v07 --refutes--> v0a\n- v07 --supports--> v00\n- v16 --supports--> v0b\n- v0d --supports--> v08\n- v0b --refutes--> v0c\n- v05 --refutes--> v12\n- v0c --supports--> v06\n- v14 --refutes--> v04\n- v0f --supports--> v10\n- v14 --supports--> v09\n- v01 --supports--> v07\n- v12 --supports--> v0f\n- v0b --supports--> v0e\n- v11 --supports--> v13\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"18f9632522745553a4ba4e4197492e49e0470b82e26428b8f97315d9c1d2345a","schema":"livegraph.boolean-propagation.v1","spec_hash":"236167831fd39dba6a740d80a11ce6a1d130e1d42de38436abaafc85e9c957d6","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-a31b34a954f50cc838f8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v05\n- v10 --supports--> v08\n- v12 --refutes--> v0b\n- v16 --supports--> v06\n- v16 --supports--> v01\n- v14 --refutes--> v0c\n- v05 --supports--> v16\n- v11 --supports--> v0f\n- v0b --refutes--> v14\n- v13 --supports--> v07\n- v07 --supports--> v0a\n- v0f --refutes--> v00\n- v0e --refutes--> v10\n- v0a --refutes--> v11\n- v14 --supports--> v0e\n- v0b --refutes--> v09\n- v08 --supports--> v13\n- v16 --supports--> v03\n- v08 --supports--> v15\n- v0e --refutes--> v02\n- v04 --refutes--> v12\n- v04 --supports--> v0d\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"00a5324b12430046bcea89022e0f50cd814551ef4116e76bcb5686b5f45b35dd","schema":"livegraph.boolean-propagation.v1","spec_hash":"a31b34a954f50cc838f8c08fa3c88d636977e933f4f5cd529e5655c797fc2ede","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-c90865d49bf2aa3ae52f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v14\n- v01 --supports--> v11\n- v0f --refutes--> v0c\n- v0d --supports--> v07\n- v0c --refutes--> v0e\n- v14 --refutes--> v06\n- v15 --supports--> v13\n- v10 --refutes--> v08\n- v05 --supports--> v01\n- v10 --supports--> v04\n- v11 --refutes--> v02\n- v0d --refutes--> v0b\n- v06 --supports--> v00\n- v0a --refutes--> v0f\n- v13 --supports--> v10\n- v0b --supports--> v15\n- v14 --supports--> v03\n- v05 --refutes--> v16\n- v06 --supports--> v05\n- v09 --refutes--> v12\n- v11 --supports--> v09\n- v04 --supports--> v0a\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6fb94452b7140843b796c84228af7f56e4e217a5f2b0fa75fc06039afa33fae4","schema":"livegraph.boolean-propagation.v1","spec_hash":"c90865d49bf2aa3ae52f831690abb1244844d78b29dba8e9b235ac3c1ac1cab7","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-b02522fc490be0085315","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v0c\n- v10 --refutes--> v00\n- v07 --supports--> v0a\n- v15 --refutes--> v12\n- v03 --refutes--> v10\n- v00 --supports--> v13\n- v0c --supports--> v15\n- v06 --supports--> v07\n- v0f --supports--> v01\n- v0a --supports--> v11\n- v05 --refutes--> v06\n- v00 --supports--> v02\n- v11 --refutes--> v0f\n- v02 --refutes--> v16\n- v06 --refutes--> v14\n- v0e --refutes--> v05\n- v04 --refutes--> v09\n- v06 --refutes--> v08\n- v04 --refutes--> v0e\n- v15 --refutes--> v03\n- v02 --refutes--> v0b\n- v03 --supports--> v0d\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"094e5c35455f6a674a613166135a13211496408fa3bc7f60c8a750433fd30044","schema":"livegraph.boolean-propagation.v1","spec_hash":"b02522fc490be0085315244fe07c24400cf6cb93339daea820050aaab20983bc","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-9e74c7e1fd60a93a1d60","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v16\n- v0b --supports--> v0d\n- v08 --refutes--> v02\n- v0b --supports--> v0a\n- v02 --supports--> v13\n- v0b --refutes--> v08\n- v07 --supports--> v0e\n- v04 --refutes--> v11\n- v00 --supports--> v09\n- v12 --refutes--> v10\n- v09 --refutes--> v0b\n- v0e --supports--> v01\n- v14 --supports--> v03\n- v01 --supports--> v0c\n- v13 --refutes--> v14\n- v02 --refutes--> v15\n- v12 --supports--> v0f\n- v0f --refutes--> v00\n- v03 --supports--> v07\n- v04 --supports--> v06\n- v0f --refutes--> v05\n- v0c --supports--> v04\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2d77e2b24751fc472cee38fc0086b61362853d2ac01da978c3953cc8594ec333","schema":"livegraph.boolean-propagation.v1","spec_hash":"9e74c7e1fd60a93a1d604d320ad69e8d511d9057fe90b457872f8377b63abc3b","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":15,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-2c314251d346770eae35","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --refutes--> v09\n- v02 --supports--> v10\n- v02 --supports--> v00\n- v01 --refutes--> v16\n- v08 --refutes--> v01\n- v14 --supports--> v0a\n- v02 --refutes--> v13\n- v04 --supports--> v12\n- v15 --supports--> v03\n- v14 --refutes--> v0b\n- v0f --refutes--> v05\n- v12 --supports--> v02\n- v0d --refutes--> v14\n- v13 --supports--> v15\n- v11 --supports--> v04\n- v01 --supports--> v11\n- v0f --refutes--> v06\n- v14 --supports--> v0c\n- v09 --refutes--> v0d\n- v06 --supports--> v07\n- v0b --supports--> v0e\n- v0e --supports--> v0f\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"63ec996f187151080b615cd84b01f282603bf19bd238017dc75d32472974dcce","schema":"livegraph.boolean-propagation.v1","spec_hash":"2c314251d346770eae35e909d8b1c5c5ee48141c2eb2f51ca3808112576de86e","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-75261b725678b32f3a61","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --supports--> v08\n- v04 --supports--> v16\n- v04 --refutes--> v10\n- v0d --supports--> v0e\n- v02 --supports--> v0c\n- v13 --refutes--> v0b\n- v07 --supports--> v0f\n- v03 --refutes--> v13\n- v08 --supports--> v11\n- v05 --refutes--> v15\n- v12 --supports--> v02\n- v0f --refutes--> v0a\n- v03 --supports--> v00\n- v15 --supports--> v01\n- v01 --supports--> v0d\n- v16 --refutes--> v09\n- v13 --refutes--> v05\n- v0f --refutes--> v14\n- v08 --refutes--> v03\n- v16 --refutes--> v06\n- v0c --refutes--> v07\n- v16 --supports--> v12\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bc22c37f0a47a05d57e73f9b56ae00115b2d67a6958cc73cf76292e57cf412c4","schema":"livegraph.boolean-propagation.v1","spec_hash":"75261b725678b32f3a613f2a5f38b1585f19d3e458ae96cd7659c23bffd356ca","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-a7f8083d57e4dd8f641c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v04\n- v0d --supports--> v0b\n- v0b --supports--> v09\n- v0a --supports--> v07\n- v09 --refutes--> v16\n- v03 --supports--> v0f\n- v05 --supports--> v06\n- v0a --refutes--> v10\n- v0c --refutes--> v02\n- v02 --supports--> v12\n- v13 --supports--> v14\n- v14 --refutes--> v08\n- v0d --refutes--> v11\n- v09 --supports--> v01\n- v0f --supports--> v05\n- v08 --supports--> v0d\n- v01 --supports--> v03\n- v05 --supports--> v0c\n- v05 --refutes--> v00\n- v12 --refutes--> v15\n- v10 --supports--> v13\n- v02 --refutes--> v0e\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv15\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"737abe9e2c9476bd5d4fcdfec76c7dc0902601516179bf7c0eedc6cceeb0d7f5","schema":"livegraph.boolean-propagation.v1","spec_hash":"a7f8083d57e4dd8f641c8d09923afe32b0780a2e894846d4b8b94c93628ebea6","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":15,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-11ce018c8d4f98fa8933","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v09\n- v16 --refutes--> v0a\n- v09 --refutes--> v00\n- v0e --supports--> v15\n- v11 --supports--> v01\n- v16 --refutes--> v0e\n- v01 --supports--> v05\n- v12 --supports--> v16\n- v0f --refutes--> v0c\n- v0f --refutes--> v12\n- v05 --refutes--> v03\n- v0e --refutes--> v06\n- v12 --refutes--> v13\n- v08 --supports--> v0d\n- v04 --refutes--> v02\n- v11 --refutes--> v0b\n- v10 --supports--> v04\n- v09 --refutes--> v0f\n- v15 --refutes--> v10\n- v02 --refutes--> v08\n- v0d --supports--> v07\n- v07 --refutes--> v14\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv14\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"83bd552de3271ab7c54806c5aced074eb3579df968cd25acd4b543053cd31a08","schema":"livegraph.boolean-propagation.v1","spec_hash":"11ce018c8d4f98fa89330befef088fd7b5d0e886ce3a6a2a5be96a9e0ac925f0","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-8f840ac67753540286de","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v16 --supports--> v10\n- v12 --supports--> v13\n- v07 --refutes--> v06\n- v14 --refutes--> v05\n- v0f --refutes--> v15\n- v0e --refutes--> v16\n- v09 --supports--> v03\n- v07 --supports--> v02\n- v12 --supports--> v0a\n- v10 --supports--> v00\n- v14 --refutes--> v0d\n- v0a --refutes--> v11\n- v00 --supports--> v07\n- v15 --supports--> v12\n- v16 --supports--> v08\n- v00 --supports--> v04\n- v0c --supports--> v0e\n- v0b --refutes--> v09\n- v0a --supports--> v01\n- v02 --refutes--> v0f\n- v14 --refutes--> v0b\n- v03 --supports--> v0c\n\n[VERIFIED ROOT]\nv14 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fab55957b23049860bde8cc59468acf37f2a621c86ad306796a319a71ed05858","schema":"livegraph.boolean-propagation.v1","spec_hash":"8f840ac67753540286ded8c63df7b7c0c9b0c6e58693b2dffd975c0e992855a7","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v14\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-9b270276c7b071ba467e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v06\n- v08 --supports--> v0b\n- v04 --supports--> v16\n- v0b --supports--> v01\n- v06 --refutes--> v07\n- v12 --supports--> v14\n- v10 --refutes--> v12\n- v0f --supports--> v15\n- v06 --refutes--> v05\n- v02 --refutes--> v04\n- v09 --supports--> v08\n- v12 --refutes--> v0a\n- v0b --supports--> v11\n- v0d --refutes--> v0f\n- v07 --refutes--> v09\n- v16 --refutes--> v0d\n- v15 --supports--> v0c\n- v04 --refutes--> v03\n- v0d --supports--> v0e\n- v13 --refutes--> v10\n- v15 --refutes--> v13\n- v0b --refutes--> v02\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv14\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e08b95cc7f08c431dc73f3977045c008e34998f540fb4a8ff68f25cf346a901d","schema":"livegraph.boolean-propagation.v1","spec_hash":"9b270276c7b071ba467ec08ce1b1390a20e32ddcc464b8002c7dd07bdf44faa3","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-04026548ea6357e7a466","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v12\n- v00 --refutes--> v06\n- v02 --supports--> v08\n- v01 --refutes--> v00\n- v0c --supports--> v02\n- v02 --supports--> v03\n- v09 --refutes--> v13\n- v05 --supports--> v09\n- v0e --refutes--> v11\n- v09 --supports--> v0e\n- v11 --supports--> v10\n- v06 --supports--> v07\n- v06 --supports--> v05\n- v0b --supports--> v0d\n- v0d --refutes--> v04\n- v01 --supports--> v0f\n- v0e --supports--> v15\n- v0c --refutes--> v16\n- v10 --refutes--> v0c\n- v10 --supports--> v0a\n- v04 --supports--> v14\n- v12 --supports--> v0b\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv14\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3cc4fe69d92cc552c84549efdbbd88a90df1d9c1ed152820ebf0bc947b17ed93","schema":"livegraph.boolean-propagation.v1","spec_hash":"04026548ea6357e7a4660f407896c9603069b59405c08b7dd200ab016d84e8ee","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":15,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-972bc9bcaa2074899d57","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v16 --supports--> v02\n- v08 --supports--> v05\n- v04 --refutes--> v14\n- v05 --supports--> v0e\n- v16 --supports--> v0f\n- v07 --refutes--> v00\n- v0b --refutes--> v04\n- v0e --refutes--> v0a\n- v14 --supports--> v0d\n- v03 --refutes--> v16\n- v0e --supports--> v12\n- v0d --supports--> v08\n- v02 --refutes--> v13\n- v00 --supports--> v15\n- v0b --refutes--> v11\n- v13 --supports--> v09\n- v13 --refutes--> v0b\n- v08 --refutes--> v10\n- v12 --supports--> v07\n- v15 --supports--> v06\n- v15 --supports--> v01\n- v16 --supports--> v0c\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"23c0f2c7d5aa22fcc69447ad118d2e24ff7089ee00a340a92538f5efa7209494","schema":"livegraph.boolean-propagation.v1","spec_hash":"972bc9bcaa2074899d57bbd660d0a616a9da8a38a75cebd25e7a93bee830da2c","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-45d36b7b2d0794d1f2e2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v04\n- v01 --supports--> v09\n- v04 --refutes--> v11\n- v0f --refutes--> v01\n- v0d --refutes--> v0a\n- v0e --supports--> v02\n- v0b --supports--> v13\n- v0d --supports--> v0e\n- v09 --supports--> v10\n- v12 --refutes--> v14\n- v04 --supports--> v06\n- v02 --refutes--> v05\n- v03 --refutes--> v0b\n- v15 --refutes--> v08\n- v0b --supports--> v00\n- v00 --refutes--> v07\n- v09 --supports--> v03\n- v12 --refutes--> v0d\n- v11 --refutes--> v15\n- v08 --supports--> v0c\n- v02 --supports--> v0f\n- v00 --refutes--> v16\n\n[VERIFIED ROOT]\nv12 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4e66ba39f7a25423f13bb95e1354fea22ca8947e6f007da7cf46acaba039d054","schema":"livegraph.boolean-propagation.v1","spec_hash":"45d36b7b2d0794d1f2e203fc82089dcd0b1c635e1feedf5e7df44516d4d7756d","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v12\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-401d08d495a9248f8ec1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v04\n- v0f --supports--> v01\n- v02 --supports--> v12\n- v13 --refutes--> v0b\n- v0a --supports--> v02\n- v0e --refutes--> v11\n- v09 --refutes--> v08\n- v08 --supports--> v16\n- v0a --supports--> v06\n- v0b --refutes--> v07\n- v11 --refutes--> v13\n- v01 --supports--> v00\n- v09 --refutes--> v0d\n- v16 --refutes--> v14\n- v05 --supports--> v0f\n- v12 --supports--> v05\n- v0b --supports--> v0c\n- v16 --refutes--> v15\n- v03 --refutes--> v09\n- v16 --refutes--> v10\n- v01 --supports--> v03\n- v14 --supports--> v0e\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"856e006cf6afe190158e12eb547b686a2172fcfe8593b71847c227ddd2a6a681","schema":"livegraph.boolean-propagation.v1","spec_hash":"401d08d495a9248f8ec1e8d7ca10173ce0ac272d7e34c28cad677423f123b71c","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-b7e6be1e465ec5b33766","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v07\n- v13 --refutes--> v0a\n- v01 --refutes--> v10\n- v04 --supports--> v15\n- v14 --supports--> v0c\n- v06 --supports--> v02\n- v15 --refutes--> v08\n- v10 --refutes--> v03\n- v05 --refutes--> v11\n- v0a --refutes--> v01\n- v11 --refutes--> v0b\n- v16 --supports--> v09\n- v16 --refutes--> v0e\n- v16 --supports--> v0f\n- v02 --supports--> v13\n- v15 --supports--> v12\n- v0b --refutes--> v0d\n- v07 --supports--> v16\n- v15 --refutes--> v14\n- v14 --supports--> v05\n- v10 --refutes--> v04\n- v15 --refutes--> v00\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6d717acf9379e5a046cab64e818b5c7f942ee944f84f5a3f8e409377c85d0d11","schema":"livegraph.boolean-propagation.v1","spec_hash":"b7e6be1e465ec5b33766f742c1b518fa6198cdcbfb71ca6ce6b4a696c3d1736c","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-c075018ccfe3e02c07c4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v16\n- v04 --supports--> v0e\n- v0a --supports--> v07\n- v02 --supports--> v01\n- v0c --refutes--> v02\n- v0f --supports--> v11\n- v08 --supports--> v04\n- v00 --refutes--> v06\n- v15 --supports--> v09\n- v00 --refutes--> v08\n- v0d --refutes--> v15\n- v07 --supports--> v12\n- v04 --refutes--> v0c\n- v04 --supports--> v13\n- v09 --refutes--> v03\n- v15 --refutes--> v10\n- v01 --refutes--> v0a\n- v11 --supports--> v00\n- v14 --supports--> v05\n- v0d --refutes--> v0b\n- v16 --supports--> v14\n- v14 --refutes--> v0d\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9446735b2f0b22a4b68ff9a68b5f053a6ddc8f391b2184b6edde48efd637d70f","schema":"livegraph.boolean-propagation.v1","spec_hash":"c075018ccfe3e02c07c4543e890fbe544a7764dea0160471ed15ab02f9b02290","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-bb11c4ecc25b5c0919e1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v0c\n- v01 --supports--> v07\n- v03 --refutes--> v16\n- v15 --refutes--> v05\n- v04 --refutes--> v00\n- v05 --supports--> v13\n- v0d --refutes--> v03\n- v01 --supports--> v11\n- v07 --refutes--> v0e\n- v0d --refutes--> v08\n- v0d --supports--> v0a\n- v13 --refutes--> v12\n- v14 --refutes--> v04\n- v16 --supports--> v15\n- v12 --refutes--> v14\n- v09 --supports--> v01\n- v10 --supports--> v02\n- v10 --supports--> v0f\n- v01 --supports--> v0b\n- v02 --refutes--> v09\n- v12 --refutes--> v06\n- v07 --supports--> v0d\n\n[VERIFIED ROOT]\nv10 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0275da2d88d01a3714f91cd93dd497f8390e22372480cad3e16f6b5e8ae99490","schema":"livegraph.boolean-propagation.v1","spec_hash":"bb11c4ecc25b5c0919e1558b1bdb914a5f90de1524ae89f057e52cb60afec785","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-51431dc4ccac85f6516a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v07\n- v0f --supports--> v14\n- v0e --supports--> v00\n- v0e --supports--> v03\n- v0f --supports--> v0b\n- v11 --refutes--> v0c\n- v16 --refutes--> v0d\n- v01 --refutes--> v0f\n- v07 --refutes--> v05\n- v00 --supports--> v09\n- v13 --supports--> v0a\n- v13 --refutes--> v16\n- v10 --refutes--> v15\n- v07 --refutes--> v10\n- v08 --refutes--> v01\n- v14 --supports--> v0e\n- v15 --refutes--> v12\n- v15 --refutes--> v08\n- v14 --supports--> v06\n- v02 --supports--> v13\n- v0d --refutes--> v11\n- v0e --supports--> v04\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"16c42e056f851e70e55c78c558e7b05f254c121a7a716f2531d66fc277baca9c","schema":"livegraph.boolean-propagation.v1","spec_hash":"51431dc4ccac85f6516a6b26e1e13fe448d1746c8200868fe98635a6b5c8f219","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-07e10d450fd602367d67","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v0a\n- v05 --refutes--> v0e\n- v15 --refutes--> v08\n- v01 --supports--> v03\n- v09 --supports--> v0d\n- v04 --refutes--> v07\n- v04 --refutes--> v16\n- v0d --refutes--> v04\n- v05 --supports--> v00\n- v0a --refutes--> v14\n- v0c --refutes--> v11\n- v10 --supports--> v0c\n- v08 --refutes--> v12\n- v11 --refutes--> v0b\n- v0b --refutes--> v05\n- v05 --refutes--> v02\n- v16 --refutes--> v13\n- v13 --refutes--> v10\n- v00 --supports--> v01\n- v0a --refutes--> v09\n- v08 --refutes--> v0f\n- v11 --supports--> v06\n\n[VERIFIED ROOT]\nv15 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"20a2cf7b6ef851741acd0a5f4b98501f647232087e58f7eabc212445b55fe29a","schema":"livegraph.boolean-propagation.v1","spec_hash":"07e10d450fd602367d675858cb6c037ec3e7885cd36bde6279d5ae48b04a22f0","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v15\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-45a99f1cc57bce087a6f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v00\n- v02 --supports--> v0c\n- v08 --supports--> v0a\n- v12 --refutes--> v07\n- v0a --refutes--> v0f\n- v16 --supports--> v0d\n- v10 --refutes--> v14\n- v09 --refutes--> v16\n- v03 --refutes--> v11\n- v13 --refutes--> v09\n- v00 --refutes--> v01\n- v01 --supports--> v05\n- v10 --supports--> v08\n- v11 --refutes--> v12\n- v01 --supports--> v10\n- v0a --refutes--> v03\n- v08 --refutes--> v04\n- v13 --supports--> v15\n- v10 --refutes--> v06\n- v0c --supports--> v13\n- v07 --supports--> v0b\n- v09 --refutes--> v0e\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e55167229243d4f6ce577c7504f629248fe172921782649d93e42da95119cc80","schema":"livegraph.boolean-propagation.v1","spec_hash":"45a99f1cc57bce087a6fcd13e0ac0980ecad2444d60cde27d9f0f3005162b05f","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-fbc2471386d8b06d04ad","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v09\n- v0e --refutes--> v12\n- v11 --refutes--> v13\n- v0d --supports--> v16\n- v04 --supports--> v06\n- v09 --supports--> v04\n- v10 --refutes--> v02\n- v0a --supports--> v14\n- v04 --supports--> v05\n- v0b --refutes--> v15\n- v14 --refutes--> v08\n- v0e --supports--> v00\n- v03 --refutes--> v0f\n- v0e --refutes--> v11\n- v13 --refutes--> v0d\n- v0f --supports--> v10\n- v15 --supports--> v0a\n- v06 --refutes--> v0b\n- v09 --refutes--> v07\n- v13 --refutes--> v01\n- v10 --refutes--> v0c\n- v0c --supports--> v0e\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e5ff856a3daa71a290f788b6a017d8cf7a10462941aaa7f6a4a68d4d61f36fc6","schema":"livegraph.boolean-propagation.v1","spec_hash":"fbc2471386d8b06d04adccb5ccf035ea4adaa56b5dc56fc1803acbb574b5d70c","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-7035eb10d1c7e3b89d6d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v16 --supports--> v0c\n- v10 --supports--> v02\n- v04 --supports--> v01\n- v13 --supports--> v06\n- v02 --supports--> v05\n- v0c --supports--> v12\n- v00 --refutes--> v08\n- v0d --refutes--> v11\n- v02 --supports--> v09\n- v06 --supports--> v10\n- v11 --refutes--> v03\n- v00 --supports--> v0d\n- v05 --refutes--> v0a\n- v15 --refutes--> v14\n- v0c --supports--> v13\n- v0a --supports--> v15\n- v06 --refutes--> v0f\n- v15 --refutes--> v04\n- v0d --supports--> v07\n- v01 --refutes--> v0e\n- v03 --supports--> v16\n- v10 --supports--> v0b\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f39aea80e5755409c5b37a841d55867a573ac8a1a8f37817ea82704ddf784a59","schema":"livegraph.boolean-propagation.v1","spec_hash":"7035eb10d1c7e3b89d6da62d4e260c6c5cca2d4e0854e1fb8b8e25d0e50e7fd2","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-ddc067217d058a49dab7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v0a\n- v05 --refutes--> v10\n- v09 --supports--> v07\n- v0d --supports--> v05\n- v14 --supports--> v0b\n- v10 --refutes--> v00\n- v01 --supports--> v09\n- v03 --supports--> v0c\n- v14 --supports--> v04\n- v00 --refutes--> v08\n- v0c --refutes--> v14\n- v06 --supports--> v01\n- v00 --supports--> v0e\n- v0b --refutes--> v0d\n- v09 --refutes--> v0f\n- v12 --refutes--> v02\n- v00 --refutes--> v12\n- v09 --supports--> v16\n- v06 --supports--> v11\n- v0a --refutes--> v13\n- v15 --refutes--> v06\n- v13 --supports--> v03\n\n[VERIFIED ROOT]\nv15 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"18fd023f634ec7b17e73c9b071b3496452f3267e39791ce07109e5e606c98749","schema":"livegraph.boolean-propagation.v1","spec_hash":"ddc067217d058a49dab7bf8dcf051be8869e4942e915b961db7d50dad1b36839","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v15\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-fb609b672b17c009bdf8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v11\n- v10 --supports--> v0b\n- v0f --supports--> v04\n- v02 --refutes--> v0d\n- v10 --supports--> v02\n- v14 --refutes--> v0c\n- v01 --supports--> v13\n- v08 --supports--> v00\n- v07 --refutes--> v06\n- v06 --supports--> v15\n- v12 --supports--> v10\n- v14 --refutes--> v16\n- v06 --refutes--> v0f\n- v0e --supports--> v12\n- v12 --refutes--> v0a\n- v00 --refutes--> v14\n- v04 --refutes--> v05\n- v08 --refutes--> v03\n- v14 --refutes--> v09\n- v0d --refutes--> v07\n- v05 --refutes--> v08\n- v0c --supports--> v01\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"86fd7f740f95c7a533e2e8ee754fa48b8088b6bce750be02e33e0a71920863de","schema":"livegraph.boolean-propagation.v1","spec_hash":"fb609b672b17c009bdf87639bac4148dd4d6f73f4d00ac87724f5abb295b71ba","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":15,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-69bb5e3d42719896efb4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v00\n- v09 --refutes--> v0a\n- v0b --supports--> v07\n- v05 --refutes--> v13\n- v0d --supports--> v08\n- v01 --supports--> v12\n- v11 --refutes--> v0b\n- v0d --refutes--> v03\n- v01 --supports--> v05\n- v07 --refutes--> v14\n- v01 --supports--> v15\n- v11 --refutes--> v04\n- v16 --refutes--> v0e\n- v10 --supports--> v0d\n- v13 --refutes--> v02\n- v03 --refutes--> v0c\n- v0c --refutes--> v01\n- v0d --refutes--> v0f\n- v0d --refutes--> v06\n- v0e --supports--> v11\n- v09 --supports--> v10\n- v00 --supports--> v16\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv14\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c6ae4bcb8f7e012577e4203a14ba3346aa27a16860b25bdcd9798b42587b6b76","schema":"livegraph.boolean-propagation.v1","spec_hash":"69bb5e3d42719896efb496f40613d8ead39e1fa695550071a3ea0f86a1e85b55","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-b849ef6299054a5cbacd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v0f\n- v12 --refutes--> v0c\n- v0e --supports--> v13\n- v0c --refutes--> v08\n- v12 --refutes--> v10\n- v16 --supports--> v12\n- v02 --supports--> v16\n- v11 --supports--> v0e\n- v0b --refutes--> v15\n- v0b --supports--> v0d\n- v0f --supports--> v06\n- v08 --refutes--> v11\n- v03 --supports--> v09\n- v14 --refutes--> v07\n- v05 --refutes--> v03\n- v05 --refutes--> v0a\n- v14 --refutes--> v04\n- v06 --supports--> v05\n- v09 --refutes--> v0b\n- v0c --supports--> v01\n- v0c --supports--> v00\n- v0b --supports--> v02\n\n[VERIFIED ROOT]\nv14 = TRUE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a8d18880a0d18b9cbf861a892a67ad65df29ab34e51bf3624813274b5f0aa145","schema":"livegraph.boolean-propagation.v1","spec_hash":"b849ef6299054a5cbacde226d6bd8a5c7dcfa24de299e831909d52a38cfe9ac2","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v14\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-2ed3b4438d8522970b92","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --supports--> v10\n- v15 --supports--> v01\n- v04 --supports--> v07\n- v02 --supports--> v06\n- v10 --supports--> v0d\n- v0a --refutes--> v14\n- v0e --refutes--> v0a\n- v09 --refutes--> v00\n- v13 --supports--> v03\n- v07 --supports--> v0f\n- v10 --refutes--> v13\n- v0c --supports--> v16\n- v0f --refutes--> v15\n- v09 --refutes--> v0c\n- v06 --supports--> v0e\n- v08 --refutes--> v05\n- v05 --supports--> v09\n- v03 --refutes--> v02\n- v0e --refutes--> v11\n- v0c --refutes--> v04\n- v0e --refutes--> v0b\n- v02 --supports--> v12\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv14\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0a5b9b514c8e06f013cfefa145a38eb25d25b4d7a1787fae13d41d04e9159128","schema":"livegraph.boolean-propagation.v1","spec_hash":"2ed3b4438d8522970b920976fc954f131ce06d520ae058dab9832f1a47487af9","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-7774f294e648d3eccfff","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v01\n- v03 --refutes--> v00\n- v03 --refutes--> v0a\n- v00 --supports--> v04\n- v08 --refutes--> v02\n- v11 --supports--> v03\n- v00 --refutes--> v0e\n- v0d --refutes--> v08\n- v16 --refutes--> v0f\n- v15 --supports--> v10\n- v04 --refutes--> v14\n- v04 --refutes--> v06\n- v01 --refutes--> v09\n- v16 --supports--> v0d\n- v02 --refutes--> v11\n- v0b --supports--> v12\n- v0d --supports--> v07\n- v04 --refutes--> v0c\n- v16 --refutes--> v13\n- v06 --refutes--> v05\n- v10 --supports--> v16\n- v05 --supports--> v0b\n\n[VERIFIED ROOT]\nv15 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"25d56163827fc269aa4312c178b00a0ce4b74f8dce9d32682d802daf54c24fea","schema":"livegraph.boolean-propagation.v1","spec_hash":"7774f294e648d3eccfff5973893aba064e927e320bc0e0cd6135c30de48e1872","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v15\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-31dedfbe1d9df3561ab0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v11\n- v0d --refutes--> v16\n- v0e --supports--> v03\n- v03 --refutes--> v15\n- v0d --refutes--> v01\n- v03 --refutes--> v06\n- v13 --supports--> v0b\n- v09 --supports--> v07\n- v07 --supports--> v10\n- v07 --supports--> v04\n- v14 --refutes--> v13\n- v05 --supports--> v0a\n- v0f --refutes--> v0c\n- v0d --refutes--> v05\n- v10 --supports--> v14\n- v13 --refutes--> v0e\n- v11 --supports--> v00\n- v11 --supports--> v02\n- v0c --refutes--> v0d\n- v0a --refutes--> v09\n- v10 --refutes--> v12\n- v15 --refutes--> v08\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f4c7156fc42c002d0a3d5ba10d71311dfcddd0b2df649d19e0f847176b66ada6","schema":"livegraph.boolean-propagation.v1","spec_hash":"31dedfbe1d9df3561ab02c8e5e71e7e7e778e980a3ea7fa80159258f4a81f54e","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-73df90fe21b6d2138b63","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v0b\n- v14 --supports--> v0c\n- v0e --supports--> v12\n- v11 --refutes--> v02\n- v14 --refutes--> v11\n- v0d --refutes--> v04\n- v16 --refutes--> v03\n- v04 --supports--> v0e\n- v00 --supports--> v07\n- v13 --supports--> v0a\n- v0f --supports--> v09\n- v0d --refutes--> v15\n- v0a --supports--> v14\n- v10 --refutes--> v05\n- v09 --supports--> v0d\n- v01 --supports--> v16\n- v13 --supports--> v08\n- v0a --supports--> v06\n- v11 --refutes--> v00\n- v05 --refutes--> v0f\n- v12 --supports--> v01\n- v16 --supports--> v13\n\n[VERIFIED ROOT]\nv10 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7b85c1c08a4ab20f5a7f8572dea0788a5e277aa208df459cd36a82e3dc3a5275","schema":"livegraph.boolean-propagation.v1","spec_hash":"73df90fe21b6d2138b630a84684f792bf3da6c16c60795c421bb45a13de5610a","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-7b04a6c7019c8eb09b14","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v16 --refutes--> v13\n- v08 --supports--> v0a\n- v04 --supports--> v0d\n- v12 --supports--> v0c\n- v0e --refutes--> v14\n- v02 --supports--> v11\n- v08 --supports--> v07\n- v01 --supports--> v02\n- v15 --refutes--> v01\n- v0a --refutes--> v0e\n- v08 --refutes--> v0b\n- v15 --refutes--> v00\n- v11 --refutes--> v06\n- v09 --refutes--> v15\n- v02 --supports--> v10\n- v16 --refutes--> v12\n- v11 --supports--> v0f\n- v0c --refutes--> v08\n- v08 --supports--> v03\n- v14 --refutes--> v05\n- v06 --refutes--> v16\n- v0d --refutes--> v09\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a3332a0da80f16aed653e58f96d035b2c036372591245d043c8f24cf19cef966","schema":"livegraph.boolean-propagation.v1","spec_hash":"7b04a6c7019c8eb09b149356cfa1c0eb1ebfe9857bc49f19d4e7db0caf68cebe","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-8f340563d9ad3ef85558","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --refutes--> v0d\n- v14 --refutes--> v00\n- v03 --refutes--> v0e\n- v0e --refutes--> v02\n- v0f --refutes--> v13\n- v0e --refutes--> v08\n- v00 --refutes--> v03\n- v01 --supports--> v0a\n- v0a --refutes--> v0b\n- v0b --supports--> v14\n- v12 --supports--> v0c\n- v07 --supports--> v12\n- v0a --supports--> v05\n- v0b --refutes--> v11\n- v12 --supports--> v09\n- v04 --supports--> v0f\n- v0e --refutes--> v16\n- v10 --refutes--> v04\n- v0c --supports--> v10\n- v02 --refutes--> v07\n- v01 --refutes--> v15\n- v07 --supports--> v06\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d6e61a7e59dd0be697d8a0b9900745484ec116423c00e69448601daffc8bdf46","schema":"livegraph.boolean-propagation.v1","spec_hash":"8f340563d9ad3ef855586e18f0feeb72c62ba17698793599932541f85e1736ec","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":15,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-5b2435ed3b61feaac80e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --refutes--> v00\n- v05 --supports--> v0b\n- v09 --refutes--> v03\n- v0d --supports--> v0f\n- v02 --supports--> v0d\n- v01 --refutes--> v0e\n- v13 --supports--> v04\n- v03 --refutes--> v10\n- v09 --supports--> v11\n- v10 --refutes--> v05\n- v00 --refutes--> v0c\n- v07 --supports--> v13\n- v01 --refutes--> v09\n- v0c --supports--> v15\n- v0b --supports--> v02\n- v0c --refutes--> v01\n- v0d --refutes--> v12\n- v16 --supports--> v06\n- v09 --supports--> v0a\n- v04 --refutes--> v16\n- v01 --refutes--> v08\n- v0d --supports--> v07\n\n[VERIFIED ROOT]\nv14 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"91ba6a2e0d38d093e69188d6d214a761694a694fb4d38cc0b4b8b2b172af36b7","schema":"livegraph.boolean-propagation.v1","spec_hash":"5b2435ed3b61feaac80e0b98f5fa481bb0d45ec0087367ec466aa087d293ce34","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v14\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-6c650f283fad6b4ef398","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --refutes--> v13\n- v16 --refutes--> v11\n- v11 --refutes--> v09\n- v0b --supports--> v06\n- v06 --supports--> v03\n- v08 --supports--> v0f\n- v03 --supports--> v14\n- v10 --refutes--> v02\n- v16 --supports--> v04\n- v15 --refutes--> v01\n- v07 --refutes--> v0b\n- v00 --supports--> v05\n- v0d --refutes--> v07\n- v0c --supports--> v15\n- v15 --refutes--> v08\n- v0d --refutes--> v0a\n- v11 --supports--> v00\n- v00 --refutes--> v10\n- v10 --supports--> v12\n- v10 --refutes--> v0d\n- v0f --refutes--> v16\n- v13 --supports--> v0e\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"04e63f8b737e6e7c874178a12c0a3eda548051b9868eb8a80f96af8a16614fe3","schema":"livegraph.boolean-propagation.v1","spec_hash":"6c650f283fad6b4ef398a9fcb8e86fe23c0a5e290355c67647ba52279ee951fc","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-a406aeca5a71cee3c471","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v07\n- v01 --refutes--> v06\n- v0c --refutes--> v13\n- v03 --refutes--> v09\n- v13 --refutes--> v11\n- v15 --supports--> v05\n- v05 --supports--> v01\n- v01 --supports--> v04\n- v10 --refutes--> v0e\n- v01 --refutes--> v12\n- v10 --refutes--> v14\n- v11 --refutes--> v10\n- v00 --refutes--> v08\n- v0b --supports--> v0c\n- v09 --supports--> v0f\n- v0d --refutes--> v15\n- v09 --refutes--> v16\n- v02 --refutes--> v00\n- v00 --refutes--> v0d\n- v09 --refutes--> v0b\n- v0d --refutes--> v0a\n- v07 --supports--> v02\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d63d0f61a092b2200a27d82a2fec0c5b7fddeb157b1b1207f90b85be31063b11","schema":"livegraph.boolean-propagation.v1","spec_hash":"a406aeca5a71cee3c471ed8677d608e94c2a4ffe8a440769ceaa53a92f625c51","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-6d02cf74454d85218356","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v05\n- v11 --supports--> v15\n- v16 --refutes--> v11\n- v08 --supports--> v10\n- v0e --refutes--> v12\n- v11 --supports--> v0e\n- v04 --supports--> v08\n- v13 --supports--> v03\n- v0a --refutes--> v01\n- v05 --refutes--> v14\n- v14 --refutes--> v13\n- v11 --refutes--> v07\n- v0f --supports--> v16\n- v08 --refutes--> v09\n- v0e --supports--> v00\n- v0d --refutes--> v04\n- v0e --supports--> v0c\n- v03 --refutes--> v0f\n- v06 --supports--> v0d\n- v0e --refutes--> v0a\n- v06 --supports--> v0b\n- v02 --refutes--> v06\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8196cf4a9e9030b81a656b2dd4f8626d845e25125b91bcf59896fff3bedb33e5","schema":"livegraph.boolean-propagation.v1","spec_hash":"6d02cf74454d8521835688f46197ca5fe79518a63cf7491059ef0bb490a2ff41","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-47fe0efbfd62ba1471ea","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v13\n- v16 --refutes--> v10\n- v11 --refutes--> v00\n- v15 --supports--> v12\n- v00 --supports--> v05\n- v14 --supports--> v16\n- v11 --supports--> v0e\n- v13 --supports--> v04\n- v03 --supports--> v07\n- v00 --supports--> v0b\n- v12 --supports--> v11\n- v06 --refutes--> v03\n- v05 --refutes--> v0f\n- v08 --supports--> v0d\n- v0d --supports--> v06\n- v16 --refutes--> v15\n- v05 --refutes--> v08\n- v04 --refutes--> v14\n- v06 --refutes--> v02\n- v16 --supports--> v0c\n- v04 --refutes--> v01\n- v0a --supports--> v09\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"754f73ebb5db74331555e775ca151479ccec6c34b23960eb55137f3390522427","schema":"livegraph.boolean-propagation.v1","spec_hash":"47fe0efbfd62ba1471ea399bd33365d080df2548a1510667c62c8b0ad2c18260","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-f06fe03d54c43a53c96a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v0e\n- v06 --refutes--> v15\n- v14 --supports--> v07\n- v09 --refutes--> v11\n- v11 --supports--> v0c\n- v16 --refutes--> v12\n- v16 --supports--> v0d\n- v05 --refutes--> v10\n- v0f --refutes--> v03\n- v0e --supports--> v14\n- v08 --refutes--> v16\n- v03 --refutes--> v0a\n- v11 --supports--> v02\n- v10 --refutes--> v09\n- v0e --supports--> v00\n- v0c --supports--> v06\n- v0b --refutes--> v05\n- v09 --refutes--> v01\n- v0e --supports--> v04\n- v16 --supports--> v0f\n- v13 --refutes--> v08\n- v14 --refutes--> v0b\n\n[VERIFIED ROOT]\nv13 = FALSE\n\n[TARGET]\nv15\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5d167583d04e24c4bf1925f76b9ba21f00d467f9eaac01b463a61d01095f9236","schema":"livegraph.boolean-propagation.v1","spec_hash":"f06fe03d54c43a53c96a105cd7fe9f12dea70cfe421ed52eb85662f261818f54","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v13\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-6733ec98936e6b5755e5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v10\n- v13 --refutes--> v04\n- v11 --supports--> v00\n- v16 --refutes--> v09\n- v01 --refutes--> v07\n- v0c --refutes--> v11\n- v05 --refutes--> v0c\n- v05 --supports--> v0e\n- v03 --supports--> v06\n- v11 --supports--> v0f\n- v03 --supports--> v08\n- v0f --supports--> v15\n- v07 --refutes--> v0a\n- v12 --supports--> v0b\n- v0c --refutes--> v14\n- v09 --supports--> v02\n- v02 --refutes--> v13\n- v15 --refutes--> v03\n- v0b --supports--> v01\n- v04 --refutes--> v12\n- v03 --supports--> v16\n- v04 --supports--> v0d\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fa6d0035d212f15f9c275a10e3a5b65ea45a7e9ecab615bce158b53675d7fa3f","schema":"livegraph.boolean-propagation.v1","spec_hash":"6733ec98936e6b5755e548635ee65bb03671f78bd9967b9b675a6f6553a282ac","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-5979074f365e7b130164","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --supports--> v02\n- v0a --supports--> v07\n- v01 --refutes--> v12\n- v02 --refutes--> v11\n- v16 --refutes--> v15\n- v09 --supports--> v13\n- v11 --supports--> v10\n- v10 --supports--> v08\n- v0d --supports--> v0e\n- v10 --refutes--> v09\n- v0f --refutes--> v05\n- v0d --supports--> v0b\n- v02 --refutes--> v0c\n- v0b --refutes--> v0a\n- v15 --supports--> v0d\n- v0f --supports--> v04\n- v0f --refutes--> v14\n- v03 --refutes--> v16\n- v00 --supports--> v01\n- v0d --supports--> v06\n- v0f --supports--> v03\n- v13 --refutes--> v0f\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fddca98c771af50805691a0477afb841a1d7d35aa3450a9526e3fa5f6291a66b","schema":"livegraph.boolean-propagation.v1","spec_hash":"5979074f365e7b130164feebb4b59ccac118d9b7167779c6d3652f4d71d9ecdb","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-a690e257e09f5bb76df5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v12\n- v12 --supports--> v04\n- v04 --supports--> v0c\n- v11 --refutes--> v10\n- v0c --refutes--> v0b\n- v10 --supports--> v07\n- v0d --supports--> v06\n- v10 --supports--> v13\n- v0a --supports--> v01\n- v0a --supports--> v16\n- v03 --supports--> v0f\n- v0f --supports--> v15\n- v0e --supports--> v05\n- v0b --refutes--> v11\n- v05 --supports--> v0a\n- v10 --refutes--> v14\n- v16 --refutes--> v00\n- v0b --refutes--> v02\n- v15 --supports--> v0e\n- v05 --refutes--> v08\n- v16 --refutes--> v0d\n- v0d --refutes--> v09\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"345a43ea48a936912cc6fb7f3e5ce61319e64fb153867d6257269832ff1688bf","schema":"livegraph.boolean-propagation.v1","spec_hash":"a690e257e09f5bb76df56210a93328fff7c38238b5e7f6ac4c97b1901458e344","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-5515ea036f9f9df425f4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v08\n- v15 --refutes--> v09\n- v08 --supports--> v0e\n- v03 --supports--> v0a\n- v04 --refutes--> v14\n- v16 --refutes--> v12\n- v02 --refutes--> v06\n- v07 --refutes--> v13\n- v0d --refutes--> v0b\n- v00 --supports--> v0c\n- v08 --supports--> v0f\n- v0b --supports--> v15\n- v16 --supports--> v11\n- v0e --refutes--> v04\n- v0c --supports--> v0d\n- v03 --refutes--> v05\n- v05 --supports--> v02\n- v11 --supports--> v03\n- v06 --refutes--> v00\n- v08 --refutes--> v01\n- v05 --refutes--> v10\n- v15 --supports--> v07\n\n[VERIFIED ROOT]\nv16 = FALSE\n\n[TARGET]\nv14\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"56db1571d86565305024e32e5762a204efcc5d43f03b6f021ea7a6010af4c6e1","schema":"livegraph.boolean-propagation.v1","spec_hash":"5515ea036f9f9df425f480f3628afdcc6d94fdd3cbb5f9f65eff39c80472b534","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v16\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-68ad11c83479c35a0687","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v16 --supports--> v0e\n- v07 --refutes--> v11\n- v02 --supports--> v07\n- v15 --supports--> v00\n- v07 --supports--> v0d\n- v05 --refutes--> v0f\n- v04 --supports--> v14\n- v0e --supports--> v10\n- v13 --supports--> v02\n- v04 --refutes--> v0c\n- v00 --refutes--> v04\n- v12 --supports--> v01\n- v0e --supports--> v08\n- v0d --supports--> v15\n- v08 --supports--> v03\n- v09 --supports--> v05\n- v14 --refutes--> v06\n- v0f --supports--> v16\n- v00 --supports--> v0a\n- v05 --supports--> v0b\n- v08 --refutes--> v13\n- v12 --refutes--> v09\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4174c6ac2dae0e4dd97b6e70ba0e2ff13c796521f0f46bfa60cc8b2e1210982e","schema":"livegraph.boolean-propagation.v1","spec_hash":"68ad11c83479c35a068706424c44fcdbe9dd89d5988cb99045f4ae324ef075de","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-e80aa1d45fcc3681d945","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --supports--> v0d\n- v0b --supports--> v01\n- v02 --supports--> v13\n- v04 --supports--> v11\n- v14 --supports--> v05\n- v08 --refutes--> v0f\n- v11 --refutes--> v0e\n- v13 --supports--> v15\n- v12 --supports--> v03\n- v14 --supports--> v07\n- v01 --supports--> v00\n- v07 --supports--> v06\n- v13 --supports--> v09\n- v00 --refutes--> v08\n- v0d --supports--> v04\n- v0a --supports--> v0b\n- v16 --refutes--> v14\n- v0f --supports--> v0c\n- v11 --refutes--> v16\n- v0f --supports--> v12\n- v07 --supports--> v02\n- v14 --refutes--> v10\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv15\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2304c24895339cb237a886da33a42d40b931c9b29bd58ff0eb840e4e36d5e688","schema":"livegraph.boolean-propagation.v1","spec_hash":"e80aa1d45fcc3681d94519439ce432b05670c4b3a5924dec5ee64f7b72a09516","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-ee9a805a94db078fed43","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v18 --refutes--> v14\n- v13 --supports--> v10\n- v11 --refutes--> v03\n- v17 --supports--> v07\n- v03 --supports--> v17\n- v14 --refutes--> v06\n- v15 --refutes--> v00\n- v05 --supports--> v11\n- v06 --supports--> v0e\n- v07 --supports--> v12\n- v18 --refutes--> v02\n- v0b --supports--> v18\n- v05 --supports--> v04\n- v0a --supports--> v15\n- v10 --supports--> v05\n- v13 --refutes--> v09\n- v05 --supports--> v16\n- v06 --supports--> v0d\n- v0c --supports--> v0b\n- v03 --supports--> v0f\n- v05 --supports--> v01\n- v00 --supports--> v0c\n- v12 --supports--> v0a\n- v0b --supports--> v08\n\n[VERIFIED ROOT]\nv13 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3555709eeaf00cb4ab4fc00def8d124253e2f9fc926315195915e37a99fcece1","schema":"livegraph.boolean-propagation.v1","spec_hash":"ee9a805a94db078fed439a3112b5d86e12a1d6d6ec79d5f6a6b6cf6b35ef88e0","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v13\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-e97c068e75c5e541b108","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --supports--> v03\n- v07 --supports--> v01\n- v0d --refutes--> v0c\n- v0d --refutes--> v0a\n- v03 --refutes--> v18\n- v05 --refutes--> v0f\n- v08 --supports--> v12\n- v00 --refutes--> v17\n- v13 --supports--> v15\n- v15 --refutes--> v0e\n- v11 --supports--> v09\n- v0e --refutes--> v05\n- v16 --supports--> v06\n- v0f --supports--> v16\n- v15 --supports--> v0b\n- v17 --supports--> v07\n- v06 --supports--> v11\n- v13 --refutes--> v02\n- v14 --refutes--> v10\n- v07 --refutes--> v08\n- v11 --supports--> v14\n- v13 --refutes--> v04\n- v12 --refutes--> v13\n- v03 --refutes--> v0d\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a79250a9eb4d2b1c79eb776d2ef54ffe95b07b2600378c9fe57fb26b2f629cc4","schema":"livegraph.boolean-propagation.v1","spec_hash":"e97c068e75c5e541b108b0ba3cff4562dc7d4c0effbe6f3cff586ee6b9a427a5","spec_json":"{\"edges\":[{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-96fb469d9294994b6648","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --refutes--> v03\n- v0d --refutes--> v0e\n- v11 --supports--> v0c\n- v14 --refutes--> v0d\n- v07 --supports--> v15\n- v18 --refutes--> v17\n- v0a --refutes--> v09\n- v0e --refutes--> v0f\n- v05 --refutes--> v16\n- v09 --refutes--> v14\n- v14 --supports--> v00\n- v10 --refutes--> v18\n- v02 --supports--> v0a\n- v01 --refutes--> v02\n- v10 --refutes--> v06\n- v0f --refutes--> v11\n- v11 --refutes--> v07\n- v09 --refutes--> v0b\n- v17 --refutes--> v12\n- v15 --supports--> v10\n- v12 --refutes--> v08\n- v0a --refutes--> v13\n- v09 --supports--> v04\n- v05 --supports--> v01\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f16af0ba78481a12295e10c76b1db288a585ee4b8040822752aba47f908a0e7f","schema":"livegraph.boolean-propagation.v1","spec_hash":"96fb469d9294994b664806847a408d7ebb54fe248312b2180725dae81f5a921e","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-4a81f0cd86e392e14238","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v02\n- v0c --refutes--> v0b\n- v18 --refutes--> v05\n- v06 --refutes--> v0c\n- v05 --refutes--> v06\n- v0f --supports--> v07\n- v0c --refutes--> v13\n- v11 --supports--> v10\n- v08 --refutes--> v0a\n- v09 --supports--> v11\n- v17 --refutes--> v0d\n- v0f --supports--> v15\n- v02 --supports--> v0e\n- v04 --supports--> v01\n- v0f --supports--> v14\n- v02 --refutes--> v09\n- v10 --refutes--> v12\n- v01 --refutes--> v0f\n- v10 --refutes--> v18\n- v0d --supports--> v08\n- v14 --refutes--> v03\n- v14 --refutes--> v00\n- v04 --supports--> v16\n- v13 --supports--> v04\n\n[VERIFIED ROOT]\nv17 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d4b8b625e7241c3085d4cd2cedf51ee84830ae6073ce5d40d136713274bf4d4a","schema":"livegraph.boolean-propagation.v1","spec_hash":"4a81f0cd86e392e14238e3fb516b16cbfe63fcdeccb1c290bb5bf04fab953027","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v17\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-1c9a5dc253fb03437827","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --supports--> v06\n- v10 --supports--> v0c\n- v05 --refutes--> v0f\n- v08 --refutes--> v0e\n- v03 --refutes--> v14\n- v0d --supports--> v01\n- v17 --supports--> v16\n- v00 --supports--> v11\n- v13 --refutes--> v15\n- v0d --supports--> v04\n- v05 --supports--> v08\n- v0e --supports--> v18\n- v0a --supports--> v10\n- v01 --refutes--> v12\n- v0a --supports--> v02\n- v0e --supports--> v0a\n- v06 --refutes--> v0d\n- v16 --refutes--> v13\n- v11 --refutes--> v03\n- v12 --refutes--> v17\n- v14 --refutes--> v07\n- v17 --supports--> v09\n- v10 --supports--> v0b\n- v13 --refutes--> v00\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0f716e009dd022cd612cb91c2a52b282a906cc4a93f63fecd34346e99498b9cb","schema":"livegraph.boolean-propagation.v1","spec_hash":"1c9a5dc253fb03437827e11e7654f1455f8de25a2de3d3881a714a52f2c14fcc","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-a29cca3853f9b6461902","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v01\n- v12 --supports--> v0a\n- v15 --supports--> v06\n- v11 --refutes--> v04\n- v0b --refutes--> v07\n- v18 --supports--> v14\n- v06 --refutes--> v03\n- v14 --supports--> v0b\n- v14 --refutes--> v10\n- v03 --refutes--> v0d\n- v0e --supports--> v09\n- v06 --refutes--> v05\n- v13 --refutes--> v16\n- v17 --refutes--> v08\n- v0b --supports--> v11\n- v04 --refutes--> v15\n- v0a --supports--> v00\n- v03 --refutes--> v17\n- v0c --supports--> v13\n- v16 --supports--> v12\n- v15 --supports--> v02\n- v16 --supports--> v0f\n- v09 --refutes--> v0c\n- v0a --refutes--> v18\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6c6b9e69837cd2163ef10badb140800fea14a205625719452e668c29f326ae74","schema":"livegraph.boolean-propagation.v1","spec_hash":"a29cca3853f9b6461902883405ccfd5a126761697b0cf2c45d1b313083af6d6c","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-fccd14934bcec3bec638","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --supports--> v18\n- v17 --refutes--> v10\n- v00 --refutes--> v08\n- v17 --refutes--> v00\n- v02 --supports--> v04\n- v07 --refutes--> v09\n- v13 --supports--> v07\n- v0b --supports--> v13\n- v07 --refutes--> v03\n- v08 --supports--> v05\n- v08 --refutes--> v15\n- v0f --supports--> v11\n- v0d --refutes--> v12\n- v14 --supports--> v0c\n- v07 --refutes--> v17\n- v0e --supports--> v01\n- v05 --refutes--> v0a\n- v01 --refutes--> v02\n- v0f --supports--> v06\n- v04 --supports--> v0b\n- v05 --refutes--> v0f\n- v14 --refutes--> v0e\n- v11 --supports--> v0d\n- v04 --refutes--> v16\n\n[VERIFIED ROOT]\nv14 = TRUE\n\n[TARGET]\nv18\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7bc9c7b9563c42316e2276e2139f2ded53ba8046cc7822fa7ab31774963465ac","schema":"livegraph.boolean-propagation.v1","spec_hash":"fccd14934bcec3bec63884753106514a03913ad4a0ed38dff443b7524c4cfec5","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v14\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v18\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-7dcbb58f53e596f29a8d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v18\n- v14 --refutes--> v0b\n- v11 --supports--> v04\n- v12 --refutes--> v06\n- v0d --supports--> v02\n- v13 --refutes--> v01\n- v04 --supports--> v0f\n- v02 --supports--> v17\n- v07 --supports--> v08\n- v17 --supports--> v11\n- v13 --supports--> v14\n- v13 --refutes--> v0c\n- v12 --refutes--> v05\n- v07 --supports--> v10\n- v14 --supports--> v16\n- v03 --refutes--> v0d\n- v0e --supports--> v00\n- v0f --supports--> v07\n- v15 --refutes--> v0e\n- v00 --supports--> v12\n- v0f --refutes--> v09\n- v08 --refutes--> v0a\n- v06 --supports--> v03\n- v08 --supports--> v13\n\n[VERIFIED ROOT]\nv15 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0caae30fd97756a44548fabf61002b16cb21deaab79e55366ac743e703358faf","schema":"livegraph.boolean-propagation.v1","spec_hash":"7dcbb58f53e596f29a8d920a805ab240ee14106687724135d9aee1f1481e8e5b","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v15\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-25a8396975f77e9b7b98","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --supports--> v12\n- v0c --supports--> v04\n- v05 --supports--> v0b\n- v04 --supports--> v0f\n- v04 --supports--> v05\n- v02 --supports--> v0a\n- v0b --supports--> v13\n- v10 --supports--> v18\n- v01 --supports--> v07\n- v0a --refutes--> v0c\n- v09 --refutes--> v15\n- v00 --refutes--> v09\n- v15 --refutes--> v0e\n- v12 --refutes--> v00\n- v0b --supports--> v17\n- v12 --refutes--> v0d\n- v07 --supports--> v02\n- v03 --refutes--> v01\n- v0e --refutes--> v10\n- v10 --refutes--> v06\n- v07 --refutes--> v16\n- v07 --refutes--> v11\n- v15 --refutes--> v14\n- v05 --supports--> v08\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bdb11471e3f1ce9bd53dbd4e8fe68c6a81f5e810f949d12452a56354693f2bb0","schema":"livegraph.boolean-propagation.v1","spec_hash":"25a8396975f77e9b7b989373476004cd0c3c0b30eb8b89951b813ba0f67277a9","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":16,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-04a7fa83d88ea2f0de09","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v0b\n- v16 --refutes--> v12\n- v0c --refutes--> v14\n- v10 --supports--> v02\n- v04 --supports--> v0a\n- v16 --refutes--> v0d\n- v17 --refutes--> v10\n- v0e --refutes--> v0c\n- v12 --supports--> v00\n- v04 --refutes--> v0e\n- v0b --supports--> v09\n- v00 --supports--> v06\n- v0e --supports--> v03\n- v01 --supports--> v08\n- v0b --supports--> v05\n- v10 --refutes--> v18\n- v0b --refutes--> v17\n- v15 --supports--> v16\n- v07 --supports--> v01\n- v14 --refutes--> v0f\n- v04 --supports--> v13\n- v08 --supports--> v11\n- v06 --refutes--> v04\n- v10 --refutes--> v15\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b4e3d31409e8ea0f6668c0da436bcc98aa4f82039a380cc09a387b02d3643b12","schema":"livegraph.boolean-propagation.v1","spec_hash":"04a7fa83d88ea2f0de099606d02b81a5bf864ca51de454742d70df5f606066c2","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-b9727b3448d905f934f0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --refutes--> v0d\n- v0f --refutes--> v0c\n- v02 --refutes--> v16\n- v17 --refutes--> v03\n- v02 --supports--> v0b\n- v0e --supports--> v13\n- v0b --supports--> v0a\n- v0d --refutes--> v07\n- v18 --refutes--> v15\n- v18 --supports--> v04\n- v13 --supports--> v17\n- v12 --supports--> v0e\n- v00 --supports--> v18\n- v04 --supports--> v14\n- v13 --supports--> v08\n- v0e --refutes--> v01\n- v11 --supports--> v0f\n- v04 --refutes--> v06\n- v13 --refutes--> v05\n- v0b --supports--> v10\n- v03 --supports--> v09\n- v0f --refutes--> v02\n- v0a --refutes--> v12\n- v09 --refutes--> v00\n\n[VERIFIED ROOT]\nv11 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2944691263b0060f3836fab2a6e2516f4c32f70ac32e02ae3c8e0ee18ab53665","schema":"livegraph.boolean-propagation.v1","spec_hash":"b9727b3448d905f934f06ef0cdf7e05520f55319f8e54156831da55c27ea8c85","spec_json":"{\"edges\":[{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-f5f983b989d4b467cb5a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v08\n- v04 --refutes--> v17\n- v0a --refutes--> v18\n- v0c --supports--> v05\n- v18 --supports--> v14\n- v07 --supports--> v04\n- v01 --refutes--> v13\n- v0c --supports--> v0f\n- v02 --refutes--> v16\n- v03 --supports--> v10\n- v0b --supports--> v11\n- v04 --refutes--> v02\n- v14 --supports--> v06\n- v0c --refutes--> v00\n- v03 --refutes--> v01\n- v16 --supports--> v09\n- v06 --supports--> v0c\n- v11 --refutes--> v07\n- v09 --refutes--> v03\n- v16 --supports--> v0e\n- v0c --supports--> v0d\n- v0a --supports--> v15\n- v0d --supports--> v12\n- v13 --refutes--> v0a\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a2a5129ea8b0c37cfbd5b5c8ae8654e2d8424c7048ee787828ee81a2e6ee9e95","schema":"livegraph.boolean-propagation.v1","spec_hash":"f5f983b989d4b467cb5a9b790ae40c5731d3682e1ca213777c40450fd1510300","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-70ca379defb6186d2d3e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v0d\n- v0d --refutes--> v12\n- v14 --refutes--> v0a\n- v11 --refutes--> v01\n- v05 --refutes--> v10\n- v0e --supports--> v15\n- v15 --supports--> v18\n- v0c --refutes--> v06\n- v0d --supports--> v13\n- v11 --supports--> v0f\n- v11 --refutes--> v03\n- v0a --refutes--> v0b\n- v18 --refutes--> v08\n- v0a --refutes--> v04\n- v05 --refutes--> v09\n- v04 --supports--> v00\n- v0c --refutes--> v05\n- v05 --supports--> v16\n- v0f --refutes--> v14\n- v16 --supports--> v02\n- v12 --refutes--> v07\n- v00 --supports--> v0e\n- v08 --refutes--> v0c\n- v08 --supports--> v17\n\n[VERIFIED ROOT]\nv11 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c8bd4ca7ce1e94fd888262c5629a045d329ae525fb19b7a0a22ef9e9881c2662","schema":"livegraph.boolean-propagation.v1","spec_hash":"70ca379defb6186d2d3e6e21418a63ab51185045b3832e8b6567b9da4fedff17","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-3a47aec9331e7678d042","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v0b\n- v14 --supports--> v09\n- v05 --supports--> v04\n- v02 --supports--> v13\n- v03 --refutes--> v0f\n- v04 --supports--> v0d\n- v17 --supports--> v0c\n- v07 --refutes--> v05\n- v18 --refutes--> v02\n- v05 --supports--> v15\n- v04 --supports--> v18\n- v0f --supports--> v08\n- v04 --supports--> v10\n- v09 --refutes--> v01\n- v05 --supports--> v12\n- v08 --refutes--> v16\n- v00 --refutes--> v14\n- v04 --supports--> v0a\n- v11 --refutes--> v03\n- v16 --refutes--> v07\n- v09 --refutes--> v11\n- v0c --refutes--> v06\n- v16 --supports--> v0e\n- v02 --supports--> v17\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dbfa1db26a0bf905788772e8d3b74852c98667aa94d15ae7138321c06bd1f0c4","schema":"livegraph.boolean-propagation.v1","spec_hash":"3a47aec9331e7678d042d0735fa76ae4f90e8658664d06131e97804c69feab0f","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-b014168885dc3319ba6d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v07\n- v00 --refutes--> v0d\n- v18 --refutes--> v05\n- v0e --supports--> v00\n- v14 --refutes--> v09\n- v0f --supports--> v0e\n- v01 --supports--> v16\n- v05 --refutes--> v0a\n- v10 --refutes--> v12\n- v09 --supports--> v17\n- v08 --supports--> v14\n- v01 --supports--> v0b\n- v09 --supports--> v10\n- v06 --refutes--> v18\n- v16 --supports--> v02\n- v18 --refutes--> v11\n- v10 --supports--> v01\n- v09 --supports--> v04\n- v00 --refutes--> v06\n- v0a --refutes--> v0c\n- v03 --refutes--> v13\n- v13 --supports--> v08\n- v02 --supports--> v0f\n- v18 --supports--> v15\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8b62dcd1dc75f1b0bf87a974f328b77f57791b7934057558d886e0a6fe8f47fa","schema":"livegraph.boolean-propagation.v1","spec_hash":"b014168885dc3319ba6d0731be735c8715d4646bc9dcc194613b7991701b75d3","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-f3d20a83fcb050ac2143","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v09\n- v17 --refutes--> v18\n- v0c --refutes--> v05\n- v04 --refutes--> v0f\n- v0e --supports--> v04\n- v00 --refutes--> v10\n- v15 --supports--> v13\n- v11 --supports--> v07\n- v11 --supports--> v17\n- v16 --refutes--> v15\n- v0b --supports--> v0c\n- v04 --refutes--> v06\n- v12 --supports--> v0b\n- v15 --supports--> v08\n- v01 --refutes--> v0e\n- v0c --refutes--> v03\n- v0a --refutes--> v11\n- v17 --supports--> v01\n- v09 --supports--> v02\n- v15 --refutes--> v12\n- v03 --refutes--> v00\n- v00 --refutes--> v0a\n- v02 --refutes--> v16\n- v15 --refutes--> v14\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"12f9fee13c3a99799807546f314db859631b22aa6465733520accbca93a1e240","schema":"livegraph.boolean-propagation.v1","spec_hash":"f3d20a83fcb050ac2143c57b7c0510bd467d7141ddfb6af104036a69e86f6218","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-84faf8422036ddde0606","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v05\n- v15 --refutes--> v09\n- v12 --refutes--> v0d\n- v11 --supports--> v18\n- v0d --supports--> v0b\n- v0c --refutes--> v00\n- v15 --refutes--> v16\n- v08 --supports--> v03\n- v15 --refutes--> v01\n- v01 --supports--> v0c\n- v01 --refutes--> v17\n- v0f --supports--> v07\n- v14 --refutes--> v11\n- v13 --supports--> v10\n- v11 --refutes--> v08\n- v0e --supports--> v06\n- v05 --supports--> v14\n- v0e --refutes--> v02\n- v0c --refutes--> v13\n- v06 --refutes--> v12\n- v0a --supports--> v0e\n- v0c --supports--> v04\n- v07 --refutes--> v15\n- v08 --supports--> v0f\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a2aba881f266e53a417d73d0d242d578e2e18ead0e3f51a3a1a8bf886a91a30c","schema":"livegraph.boolean-propagation.v1","spec_hash":"84faf8422036ddde060688bee2a471069bf2b948f274f1ab4da3d6cf9505b386","spec_json":"{\"edges\":[{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":16,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-081acf1210a70651c915","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v17\n- v02 --supports--> v09\n- v0f --supports--> v0b\n- v06 --supports--> v05\n- v03 --supports--> v07\n- v13 --supports--> v0a\n- v06 --supports--> v0e\n- v08 --supports--> v04\n- v07 --supports--> v18\n- v01 --refutes--> v16\n- v14 --supports--> v12\n- v07 --supports--> v08\n- v17 --supports--> v10\n- v12 --supports--> v11\n- v0d --refutes--> v02\n- v12 --supports--> v0d\n- v0a --refutes--> v14\n- v01 --supports--> v00\n- v0f --refutes--> v15\n- v16 --supports--> v0f\n- v0f --supports--> v13\n- v0c --supports--> v01\n- v02 --refutes--> v06\n- v05 --refutes--> v03\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e275e88320286e3f0addf44702c367f191cce81ce71d06b9b2ab4f82a29bb56e","schema":"livegraph.boolean-propagation.v1","spec_hash":"081acf1210a70651c91543d43c37d93ff18039ac91dff382632741d23c2cf22b","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-c3a673b23974a4d255b7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v01\n- v03 --supports--> v06\n- v13 --supports--> v11\n- v0b --supports--> v05\n- v08 --refutes--> v0b\n- v13 --supports--> v12\n- v07 --supports--> v18\n- v04 --refutes--> v02\n- v0c --refutes--> v04\n- v0a --supports--> v0e\n- v07 --supports--> v09\n- v0b --supports--> v13\n- v15 --refutes--> v0a\n- v09 --supports--> v0d\n- v17 --supports--> v08\n- v0f --refutes--> v14\n- v0d --supports--> v00\n- v16 --supports--> v0f\n- v0a --refutes--> v0c\n- v09 --supports--> v10\n- v00 --refutes--> v15\n- v04 --supports--> v03\n- v0f --refutes--> v17\n- v12 --refutes--> v07\n\n[VERIFIED ROOT]\nv16 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6afbe03e244361d020a842f70c7291d6ea9b3d7de2b0d96653f8c031b2c3b094","schema":"livegraph.boolean-propagation.v1","spec_hash":"c3a673b23974a4d255b7f8ba4349ef13086f84bf0c4a8d20ab517e55b3350fdb","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v16\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-30af690a6f4e54badcab","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v10\n- v08 --refutes--> v18\n- v15 --refutes--> v0d\n- v07 --refutes--> v16\n- v0f --supports--> v13\n- v0d --supports--> v01\n- v05 --supports--> v09\n- v09 --refutes--> v06\n- v0d --supports--> v05\n- v14 --refutes--> v04\n- v03 --refutes--> v0c\n- v09 --refutes--> v12\n- v00 --refutes--> v07\n- v08 --supports--> v0b\n- v13 --supports--> v00\n- v02 --refutes--> v17\n- v16 --supports--> v03\n- v0f --supports--> v0e\n- v11 --refutes--> v0f\n- v18 --refutes--> v0a\n- v14 --refutes--> v15\n- v03 --supports--> v14\n- v17 --refutes--> v11\n- v18 --refutes--> v02\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dc1bdd50dced171714d66fe4ce37f80fe3d26194a78bed0578edbbf1a7eb8631","schema":"livegraph.boolean-propagation.v1","spec_hash":"30af690a6f4e54badcab0ef18719f780fbd84c488c16ef9e7c90055d96a6362a","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-7559a19e898340cb807e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v00\n- v15 --refutes--> v0d\n- v11 --supports--> v17\n- v13 --refutes--> v08\n- v08 --refutes--> v09\n- v0b --supports--> v13\n- v0f --refutes--> v0e\n- v11 --refutes--> v18\n- v06 --refutes--> v01\n- v17 --refutes--> v04\n- v16 --refutes--> v0a\n- v16 --supports--> v14\n- v17 --supports--> v16\n- v12 --supports--> v0b\n- v0c --supports--> v06\n- v0d --supports--> v11\n- v16 --refutes--> v0f\n- v13 --supports--> v07\n- v11 --refutes--> v03\n- v01 --refutes--> v10\n- v0d --supports--> v05\n- v09 --refutes--> v02\n- v10 --supports--> v15\n- v0e --supports--> v12\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5bca6e2281cd9596f1e1dac6a177feec72340b7ce75d9ff6e615e0b5771a0b89","schema":"livegraph.boolean-propagation.v1","spec_hash":"7559a19e898340cb807e88e39c9a45473aa68fbd4b6cd88d7fa65757c5705b15","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-a2094d7747dc6f3d805e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --refutes--> v09\n- v00 --supports--> v17\n- v15 --refutes--> v18\n- v13 --supports--> v0f\n- v07 --refutes--> v11\n- v09 --refutes--> v05\n- v00 --refutes--> v14\n- v0b --refutes--> v06\n- v06 --supports--> v02\n- v05 --supports--> v12\n- v0b --refutes--> v08\n- v12 --refutes--> v07\n- v14 --refutes--> v0a\n- v03 --refutes--> v15\n- v13 --refutes--> v10\n- v02 --supports--> v03\n- v11 --refutes--> v0e\n- v0f --supports--> v0c\n- v00 --supports--> v16\n- v0c --refutes--> v0b\n- v03 --supports--> v0d\n- v07 --supports--> v04\n- v00 --refutes--> v01\n- v0e --supports--> v13\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv18\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"940fe126b281de25e0154e6cc613a9947f1c329330578059ece2cec72d7de23d","schema":"livegraph.boolean-propagation.v1","spec_hash":"a2094d7747dc6f3d805ea17f426600d853fb6c8c804701e25d6d57ce941f1276","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":16,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v18\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-65e6b924ffa29538a7b7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v11\n- v12 --refutes--> v14\n- v06 --supports--> v15\n- v05 --refutes--> v0a\n- v0b --supports--> v0d\n- v0f --refutes--> v13\n- v10 --refutes--> v06\n- v0c --refutes--> v09\n- v0c --refutes--> v18\n- v11 --supports--> v01\n- v15 --refutes--> v0f\n- v08 --supports--> v10\n- v00 --supports--> v07\n- v0d --refutes--> v04\n- v13 --refutes--> v0e\n- v07 --refutes--> v12\n- v11 --refutes--> v02\n- v05 --refutes--> v0c\n- v13 --refutes--> v05\n- v02 --supports--> v16\n- v02 --supports--> v03\n- v02 --refutes--> v17\n- v03 --refutes--> v08\n- v12 --supports--> v0b\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv18\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d8345ac952598bf22270c18037130057768f0d4a34b1d6697a1d3e7d214a44d5","schema":"livegraph.boolean-propagation.v1","spec_hash":"65e6b924ffa29538a7b7e6e717b4d89fdff75252a8615ee8e84e813dd9b569d5","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":16,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v18\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-e36f0be03a6a1b1c92cb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --refutes--> v03\n- v08 --refutes--> v05\n- v11 --supports--> v08\n- v03 --refutes--> v14\n- v17 --refutes--> v0f\n- v18 --supports--> v11\n- v14 --refutes--> v10\n- v12 --supports--> v07\n- v0a --refutes--> v12\n- v10 --supports--> v0d\n- v0e --supports--> v0b\n- v07 --refutes--> v15\n- v14 --supports--> v13\n- v0e --refutes--> v0a\n- v11 --refutes--> v02\n- v0e --refutes--> v06\n- v0d --supports--> v17\n- v05 --refutes--> v0e\n- v03 --refutes--> v16\n- v00 --refutes--> v18\n- v0f --supports--> v09\n- v18 --refutes--> v04\n- v05 --refutes--> v01\n- v17 --supports--> v0c\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c16a84d716e2242b4b206c1dcf9bab8f8edd8c85979070db374e9165c01e0ccf","schema":"livegraph.boolean-propagation.v1","spec_hash":"e36f0be03a6a1b1c92cb16cf34aa6ba68f2a1c7d2e1134f59dd730c61ad56df1","spec_json":"{\"edges\":[{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-e163d52219c793d2938e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v01\n- v0d --supports--> v0a\n- v16 --refutes--> v02\n- v12 --supports--> v16\n- v05 --refutes--> v0f\n- v11 --supports--> v04\n- v09 --supports--> v14\n- v0c --refutes--> v08\n- v10 --refutes--> v07\n- v14 --refutes--> v18\n- v11 --refutes--> v12\n- v18 --supports--> v0e\n- v03 --refutes--> v17\n- v05 --refutes--> v15\n- v16 --supports--> v13\n- v0e --supports--> v0d\n- v06 --refutes--> v10\n- v0d --supports--> v05\n- v0d --supports--> v00\n- v0b --refutes--> v11\n- v17 --supports--> v09\n- v0f --supports--> v0c\n- v10 --refutes--> v0b\n- v16 --refutes--> v03\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"936a5352bf90f720573b293465e0ef3f12efffaa96fffc8610b8e66dffcc5902","schema":"livegraph.boolean-propagation.v1","spec_hash":"e163d52219c793d2938ec26064e1b92189acbada637159ce95eb66e3f9beddbb","spec_json":"{\"edges\":[{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-dcad0e71082678aa746d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --supports--> v01\n- v18 --refutes--> v17\n- v18 --supports--> v07\n- v18 --refutes--> v04\n- v15 --refutes--> v08\n- v07 --supports--> v06\n- v0f --supports--> v16\n- v0b --refutes--> v0e\n- v12 --refutes--> v14\n- v0d --refutes--> v18\n- v02 --refutes--> v12\n- v14 --refutes--> v03\n- v03 --refutes--> v0c\n- v01 --refutes--> v0d\n- v0e --supports--> v15\n- v07 --refutes--> v02\n- v18 --supports--> v13\n- v15 --supports--> v0a\n- v05 --refutes--> v0f\n- v0f --supports--> v00\n- v03 --supports--> v09\n- v09 --refutes--> v05\n- v0a --supports--> v11\n- v14 --refutes--> v10\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv16\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"df4d80980d6466500a459bd43269ff8f7cff9175faf2b3550153ac7259a9796a","schema":"livegraph.boolean-propagation.v1","spec_hash":"dcad0e71082678aa746dca1b6c57a2e9dcaa335d06f5ae3e9382aea7b0a01569","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v16\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-4af83258fd95ac419751","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v16 --refutes--> v17\n- v08 --refutes--> v0b\n- v03 --refutes--> v13\n- v13 --supports--> v10\n- v0b --refutes--> v0d\n- v05 --supports--> v12\n- v00 --refutes--> v05\n- v0a --supports--> v09\n- v0a --refutes--> v03\n- v10 --supports--> v06\n- v07 --refutes--> v18\n- v04 --supports--> v08\n- v04 --supports--> v15\n- v0d --refutes--> v0f\n- v17 --supports--> v0a\n- v0d --refutes--> v01\n- v0d --refutes--> v11\n- v01 --refutes--> v00\n- v07 --supports--> v02\n- v18 --refutes--> v04\n- v10 --supports--> v14\n- v0b --supports--> v0c\n- v14 --supports--> v07\n- v07 --refutes--> v0e\n\n[VERIFIED ROOT]\nv16 = FALSE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a41000b60eb1f1381fc16ea7e17452558e5fef745ee1a08f0077a355cf58f825","schema":"livegraph.boolean-propagation.v1","spec_hash":"4af83258fd95ac419751d0446197898f95928736ccd43e3a95ffc28556ddc9c8","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v16\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-91632ac04c81be081dc4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v06\n- v16 --refutes--> v14\n- v10 --supports--> v03\n- v13 --refutes--> v09\n- v13 --refutes--> v0f\n- v05 --supports--> v0e\n- v18 --refutes--> v07\n- v02 --supports--> v17\n- v18 --refutes--> v16\n- v06 --supports--> v02\n- v02 --supports--> v15\n- v06 --refutes--> v0d\n- v14 --supports--> v11\n- v09 --supports--> v08\n- v0c --refutes--> v0b\n- v17 --refutes--> v18\n- v0a --supports--> v0c\n- v0e --refutes--> v13\n- v13 --supports--> v00\n- v03 --refutes--> v05\n- v06 --refutes--> v04\n- v05 --supports--> v01\n- v03 --refutes--> v12\n- v11 --supports--> v10\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"831a37f55efa2b1c3192b7502a7216a8099119277955e17c7ce53ae1aaabd129","schema":"livegraph.boolean-propagation.v1","spec_hash":"91632ac04c81be081dc455f6d858da176d61a339205950226438b8d7a9a77331","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-43f1d9593f4a091b9348","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v10\n- v08 --supports--> v13\n- v0e --refutes--> v18\n- v18 --refutes--> v11\n- v05 --supports--> v0a\n- v01 --refutes--> v16\n- v01 --supports--> v0e\n- v14 --refutes--> v15\n- v14 --supports--> v02\n- v06 --supports--> v14\n- v0d --supports--> v17\n- v02 --refutes--> v12\n- v09 --refutes--> v07\n- v0c --refutes--> v0b\n- v18 --supports--> v05\n- v17 --refutes--> v00\n- v02 --refutes--> v0d\n- v0d --supports--> v0f\n- v00 --refutes--> v0c\n- v01 --supports--> v04\n- v03 --supports--> v09\n- v0a --refutes--> v08\n- v09 --refutes--> v01\n- v13 --supports--> v06\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c08a3c7b515c7688486d81fbef87bad0e6480520b248af7ab0e1e85e49a6e0ba","schema":"livegraph.boolean-propagation.v1","spec_hash":"43f1d9593f4a091b9348e6b673e729b8fc3535cdb6d70cedd13b25ed9cf00db9","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-898217537ff46d66f2ad","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v12\n- v03 --supports--> v0d\n- v03 --refutes--> v18\n- v06 --refutes--> v10\n- v12 --refutes--> v06\n- v09 --refutes--> v16\n- v00 --refutes--> v0a\n- v08 --refutes--> v07\n- v0e --supports--> v05\n- v0c --supports--> v0e\n- v07 --refutes--> v01\n- v0a --supports--> v08\n- v13 --refutes--> v03\n- v06 --supports--> v09\n- v12 --supports--> v14\n- v05 --supports--> v00\n- v07 --supports--> v0b\n- v0d --refutes--> v04\n- v04 --supports--> v15\n- v11 --supports--> v13\n- v0b --supports--> v02\n- v04 --supports--> v0c\n- v03 --refutes--> v0f\n- v06 --refutes--> v17\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv16\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"65b1b470ebb05052c602795e5cde747f934fcde70b0229313b7a116af5c254be","schema":"livegraph.boolean-propagation.v1","spec_hash":"898217537ff46d66f2ada09cf2b3d57287692c3baa585b9d1bb5a4dd9fdd826c","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":16,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v16\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-c30e1e8827f2dcbec847","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v0a\n- v0b --supports--> v11\n- v07 --refutes--> v04\n- v00 --supports--> v16\n- v04 --refutes--> v0f\n- v10 --supports--> v00\n- v03 --refutes--> v07\n- v12 --refutes--> v08\n- v17 --supports--> v0c\n- v12 --supports--> v02\n- v14 --refutes--> v0d\n- v16 --refutes--> v12\n- v0c --supports--> v18\n- v01 --supports--> v14\n- v09 --supports--> v10\n- v0b --refutes--> v06\n- v09 --supports--> v15\n- v09 --refutes--> v0e\n- v12 --refutes--> v01\n- v06 --supports--> v05\n- v05 --supports--> v17\n- v04 --supports--> v0b\n- v17 --refutes--> v13\n- v18 --supports--> v09\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0c005893145230589bbe90ae41e8bb553535c0b4cb31e9080881bf47b2c22249","schema":"livegraph.boolean-propagation.v1","spec_hash":"c30e1e8827f2dcbec847ccfa657f4c0721003b2226a1b0fd2878afb05f72504a","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-cd7b342845b6da1168c0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v08\n- v0a --refutes--> v13\n- v0a --refutes--> v0d\n- v0b --supports--> v15\n- v14 --refutes--> v0a\n- v07 --refutes--> v0c\n- v0c --refutes--> v11\n- v16 --supports--> v14\n- v12 --supports--> v0e\n- v04 --refutes--> v03\n- v07 --supports--> v10\n- v0d --supports--> v12\n- v05 --supports--> v07\n- v0f --supports--> v16\n- v08 --supports--> v05\n- v0c --refutes--> v18\n- v06 --supports--> v04\n- v0c --supports--> v0b\n- v04 --supports--> v0f\n- v00 --supports--> v06\n- v0d --supports--> v09\n- v0a --refutes--> v17\n- v07 --supports--> v01\n- v0e --supports--> v02\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv15\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"356e36760f9cc09447efbe3dff80bf92cd73eda19ab653ffa7727d47018c097f","schema":"livegraph.boolean-propagation.v1","spec_hash":"cd7b342845b6da1168c0c4cceab9a4505d580c6a14b7689f62dfc138a9ac50d2","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":16,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-8e1377839a617747af45","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --supports--> v0b\n- v09 --refutes--> v13\n- v13 --refutes--> v0c\n- v12 --refutes--> v01\n- v07 --refutes--> v06\n- v14 --supports--> v17\n- v18 --refutes--> v16\n- v08 --refutes--> v15\n- v0f --supports--> v0e\n- v0a --refutes--> v11\n- v07 --refutes--> v0d\n- v10 --refutes--> v0f\n- v15 --refutes--> v03\n- v11 --supports--> v18\n- v0b --refutes--> v14\n- v0f --supports--> v12\n- v12 --refutes--> v02\n- v10 --supports--> v04\n- v17 --refutes--> v05\n- v07 --supports--> v09\n- v18 --refutes--> v07\n- v12 --supports--> v00\n- v0c --refutes--> v08\n- v05 --refutes--> v10\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"26f2c781c645d1661b24fa240cf8240b152c08e4507dd3fbc6155a8ed44f3be8","schema":"livegraph.boolean-propagation.v1","spec_hash":"8e1377839a617747af459523244d6f3da40078c85d7be6f3656facbc54bd1b6b","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-ffc07a675f24fd0c3d14","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v0b\n- v0c --supports--> v10\n- v02 --supports--> v18\n- v15 --refutes--> v0a\n- v11 --refutes--> v04\n- v05 --refutes--> v08\n- v13 --supports--> v02\n- v17 --refutes--> v13\n- v12 --supports--> v15\n- v01 --supports--> v11\n- v0c --supports--> v05\n- v16 --refutes--> v0c\n- v01 --supports--> v0e\n- v0b --refutes--> v09\n- v00 --refutes--> v12\n- v18 --refutes--> v07\n- v0b --refutes--> v01\n- v14 --supports--> v00\n- v05 --supports--> v0f\n- v16 --refutes--> v06\n- v0a --refutes--> v17\n- v12 --refutes--> v03\n- v05 --refutes--> v0d\n- v05 --supports--> v14\n\n[VERIFIED ROOT]\nv16 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3fd9943419bf8429b6d55d3934d43dc57b4e06188cf91218fed4a98e3ac8b82c","schema":"livegraph.boolean-propagation.v1","spec_hash":"ffc07a675f24fd0c3d14764c80ddf7548e671a2ac3e5ed4a78ffe9b9d001834d","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v16\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-81d61a737f6cee3d24d0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v18 --refutes--> v0e\n- v0e --refutes--> v0d\n- v09 --supports--> v02\n- v15 --refutes--> v17\n- v09 --supports--> v0c\n- v08 --supports--> v14\n- v03 --refutes--> v08\n- v07 --refutes--> v06\n- v09 --supports--> v15\n- v14 --supports--> v0a\n- v09 --supports--> v01\n- v08 --supports--> v04\n- v11 --refutes--> v16\n- v00 --refutes--> v18\n- v06 --supports--> v13\n- v15 --supports--> v05\n- v0d --supports--> v0b\n- v0a --supports--> v00\n- v05 --supports--> v11\n- v12 --supports--> v09\n- v16 --supports--> v03\n- v05 --refutes--> v10\n- v18 --refutes--> v0f\n- v06 --supports--> v12\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"98c559b84a3e463f53b4f6e91879afc2427e63de9d04dfd775361ff699ec6084","schema":"livegraph.boolean-propagation.v1","spec_hash":"81d61a737f6cee3d24d04b7bafc3b348e0e93d03893464ada8f7498c06a99d14","spec_json":"{\"edges\":[{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-dfab11413fe07b4ddfed","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v0c\n- v16 --refutes--> v14\n- v0b --refutes--> v00\n- v07 --refutes--> v02\n- v16 --refutes--> v11\n- v04 --refutes--> v15\n- v18 --refutes--> v17\n- v06 --supports--> v0b\n- v13 --supports--> v06\n- v07 --refutes--> v12\n- v03 --supports--> v0a\n- v13 --supports--> v01\n- v0e --supports--> v04\n- v0d --supports--> v07\n- v05 --supports--> v10\n- v05 --supports--> v03\n- v0f --refutes--> v0d\n- v17 --supports--> v0e\n- v07 --supports--> v05\n- v0b --refutes--> v18\n- v09 --refutes--> v13\n- v11 --supports--> v08\n- v03 --refutes--> v16\n- v08 --supports--> v09\n\n[VERIFIED ROOT]\nv0f = FALSE\n\n[TARGET]\nv15\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"62e6157524793fd256cefb55c60422354ae65cced450d59b54e1c71fcc06273a","schema":"livegraph.boolean-propagation.v1","spec_hash":"dfab11413fe07b4ddfedf4bae95afd0994ed73777cc6f96f88881d66343876bb","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-1589fa78810423225ba0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v18\n- v16 --refutes--> v0e\n- v01 --refutes--> v14\n- v15 --supports--> v01\n- v00 --refutes--> v0d\n- v02 --refutes--> v0f\n- v0b --supports--> v15\n- v13 --refutes--> v06\n- v0e --supports--> v03\n- v06 --refutes--> v05\n- v0a --refutes--> v02\n- v07 --supports--> v11\n- v14 --refutes--> v04\n- v07 --refutes--> v08\n- v08 --refutes--> v16\n- v09 --refutes--> v13\n- v14 --refutes--> v09\n- v00 --supports--> v0a\n- v0e --refutes--> v0c\n- v03 --refutes--> v0b\n- v13 --supports--> v17\n- v02 --supports--> v12\n- v05 --refutes--> v00\n- v0a --refutes--> v10\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ef26bf00921529721949e73c25d901bebacb4d1a70270ed546b4e247cf79a0fa","schema":"livegraph.boolean-propagation.v1","spec_hash":"1589fa78810423225ba0f3149d3da1a5f6a80796fa990273d57cc5b228d0c997","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":16,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-c4730b0954bef419d7f1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v17\n- v0b --supports--> v12\n- v0a --supports--> v0c\n- v00 --refutes--> v14\n- v07 --refutes--> v0a\n- v05 --supports--> v15\n- v05 --supports--> v0d\n- v0b --supports--> v04\n- v02 --supports--> v0f\n- v08 --refutes--> v05\n- v0c --refutes--> v0b\n- v11 --refutes--> v00\n- v0c --supports--> v01\n- v18 --supports--> v10\n- v00 --supports--> v08\n- v0e --refutes--> v07\n- v0f --supports--> v0e\n- v18 --refutes--> v09\n- v17 --refutes--> v13\n- v08 --refutes--> v03\n- v05 --supports--> v06\n- v04 --refutes--> v18\n- v06 --supports--> v02\n- v17 --refutes--> v16\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv16\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6a486ab7f8c19ff6f2aa1620281e454774254c9aae701a367af58a7973a9e45a","schema":"livegraph.boolean-propagation.v1","spec_hash":"c4730b0954bef419d7f125c35036229c6743a6a1a75d5c24abbc69cdffc9ddd0","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v16\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-6b032e1bfacaadf319bd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v17\n- v10 --refutes--> v08\n- v18 --refutes--> v00\n- v18 --refutes--> v0c\n- v05 --refutes--> v03\n- v08 --refutes--> v18\n- v0c --supports--> v13\n- v0d --supports--> v01\n- v0e --refutes--> v04\n- v12 --refutes--> v02\n- v13 --refutes--> v0a\n- v03 --refutes--> v11\n- v0e --supports--> v10\n- v13 --supports--> v12\n- v09 --refutes--> v0b\n- v02 --supports--> v09\n- v0c --refutes--> v15\n- v01 --refutes--> v14\n- v0b --refutes--> v0d\n- v01 --supports--> v05\n- v07 --supports--> v0e\n- v11 --supports--> v16\n- v01 --refutes--> v06\n- v11 --refutes--> v0f\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dc3742efa751a73f0a08eeacbb4985bca7dbb5f28a514f607b610b3993ba1aef","schema":"livegraph.boolean-propagation.v1","spec_hash":"6b032e1bfacaadf319bdb766542f62461cc71df3f198867f41ee36ac5a191827","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-e1c71e53b42c75b6dacc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v18 --supports--> v0e\n- v06 --refutes--> v03\n- v01 --supports--> v13\n- v02 --supports--> v09\n- v00 --supports--> v15\n- v04 --refutes--> v11\n- v06 --supports--> v0d\n- v12 --supports--> v04\n- v0a --refutes--> v14\n- v09 --refutes--> v07\n- v15 --refutes--> v02\n- v05 --supports--> v0f\n- v07 --refutes--> v10\n- v0a --supports--> v18\n- v09 --refutes--> v17\n- v06 --refutes--> v00\n- v0c --supports--> v01\n- v09 --refutes--> v16\n- v0e --refutes--> v05\n- v05 --refutes--> v12\n- v13 --supports--> v06\n- v10 --refutes--> v0a\n- v01 --refutes--> v0b\n- v09 --refutes--> v08\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b453059ab1758c631f1009b85daad0ebfdbaeed3ea32ae86cb69b777f1eb8300","schema":"livegraph.boolean-propagation.v1","spec_hash":"e1c71e53b42c75b6dacc82f64ea2ee0e4c605d55540d685364790dd3a7afb995","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-4d7f8c5311ee1319762a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v02\n- v0c --refutes--> v08\n- v00 --supports--> v0c\n- v0a --supports--> v06\n- v04 --supports--> v0e\n- v15 --supports--> v0d\n- v16 --refutes--> v12\n- v04 --supports--> v0b\n- v18 --refutes--> v16\n- v0f --supports--> v17\n- v0a --supports--> v01\n- v10 --supports--> v0a\n- v0b --refutes--> v0f\n- v14 --supports--> v03\n- v07 --supports--> v00\n- v11 --supports--> v04\n- v17 --supports--> v09\n- v17 --refutes--> v10\n- v13 --refutes--> v07\n- v08 --refutes--> v15\n- v15 --supports--> v11\n- v14 --refutes--> v13\n- v15 --supports--> v05\n- v16 --refutes--> v14\n\n[VERIFIED ROOT]\nv18 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9d56db5977de0dfe7b888a8516155684ee6ad0f480c274853df85496620c8c69","schema":"livegraph.boolean-propagation.v1","spec_hash":"4d7f8c5311ee1319762aed09145f763476f1b59ed49c775788f47c05e3a5069b","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v18\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-6c409250d4d0e723b76c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --supports--> v0d\n- v0e --supports--> v09\n- v0b --refutes--> v0e\n- v0d --refutes--> v08\n- v05 --refutes--> v04\n- v02 --supports--> v14\n- v14 --refutes--> v11\n- v12 --refutes--> v10\n- v03 --refutes--> v07\n- v0a --refutes--> v0f\n- v11 --supports--> v13\n- v0f --supports--> v01\n- v0e --refutes--> v00\n- v10 --supports--> v06\n- v04 --refutes--> v15\n- v01 --supports--> v12\n- v04 --refutes--> v0a\n- v03 --refutes--> v05\n- v0f --refutes--> v0c\n- v02 --supports--> v17\n- v03 --supports--> v16\n- v10 --supports--> v02\n- v05 --supports--> v18\n- v09 --refutes--> v03\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b76fad86c5ae17d27719833efe1dc7df740870621dcf52028bc9d48ef55109d2","schema":"livegraph.boolean-propagation.v1","spec_hash":"6c409250d4d0e723b76c680da5105616a75c405657f6e5dcea3425a3135aab63","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":16,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-37b64d6a351707866f5a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v07\n- v12 --supports--> v00\n- v0e --refutes--> v15\n- v06 --supports--> v0e\n- v05 --supports--> v14\n- v18 --refutes--> v01\n- v15 --supports--> v04\n- v05 --supports--> v0d\n- v10 --refutes--> v13\n- v12 --supports--> v16\n- v0b --refutes--> v18\n- v04 --refutes--> v09\n- v10 --refutes--> v0f\n- v11 --refutes--> v06\n- v13 --supports--> v05\n- v01 --refutes--> v03\n- v01 --refutes--> v11\n- v07 --supports--> v02\n- v10 --supports--> v0a\n- v0b --supports--> v0c\n- v00 --refutes--> v0b\n- v0b --supports--> v17\n- v0d --supports--> v12\n- v0e --supports--> v08\n\n[VERIFIED ROOT]\nv10 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"80c865876fc2b25bdad1cd73ca712e3da28ed1d1356939a1a1eec2a3a382a8d1","schema":"livegraph.boolean-propagation.v1","spec_hash":"37b64d6a351707866f5a9732b1385a6cb62ec93f9b365b7cb590e01187b3ae72","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-4adbdd3414f128820c7e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v12\n- v09 --supports--> v0b\n- v04 --refutes--> v11\n- v0b --refutes--> v0d\n- v17 --refutes--> v14\n- v12 --supports--> v03\n- v15 --supports--> v17\n- v15 --supports--> v00\n- v0f --refutes--> v02\n- v12 --supports--> v0e\n- v07 --refutes--> v01\n- v02 --refutes--> v08\n- v02 --supports--> v18\n- v0a --supports--> v05\n- v0d --supports--> v10\n- v07 --supports--> v09\n- v14 --supports--> v07\n- v04 --supports--> v16\n- v0a --supports--> v04\n- v0b --supports--> v13\n- v0d --supports--> v06\n- v16 --supports--> v0f\n- v10 --refutes--> v0a\n- v0e --refutes--> v15\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv18\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"74a22c698a8ee03227dcd1d71d86bb5503e794750d5daa0623e429dbc3a09635","schema":"livegraph.boolean-propagation.v1","spec_hash":"4adbdd3414f128820c7e6de21a962896c180473ace2d98dbc390c422b3f8686b","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v18\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-5177ed272955cf68d901","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v14\n- v01 --refutes--> v16\n- v00 --refutes--> v03\n- v03 --supports--> v09\n- v0e --refutes--> v01\n- v04 --refutes--> v0a\n- v04 --supports--> v15\n- v0b --supports--> v11\n- v0e --supports--> v17\n- v18 --refutes--> v0e\n- v02 --supports--> v0f\n- v08 --supports--> v02\n- v02 --supports--> v06\n- v07 --supports--> v12\n- v16 --refutes--> v04\n- v00 --supports--> v13\n- v18 --supports--> v0d\n- v0c --supports--> v10\n- v09 --supports--> v08\n- v14 --supports--> v0c\n- v0c --refutes--> v0b\n- v11 --refutes--> v00\n- v12 --supports--> v18\n- v0e --supports--> v05\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"232ec7281a2e4309c5e523570b5030ad8a8607b42fb3175412b1c8ad01b58a2d","schema":"livegraph.boolean-propagation.v1","spec_hash":"5177ed272955cf68d9012f449807b83f4d9af56ebbc94df012ed9ea95eed340b","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-10f42c15d628b3d26a96","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v11\n- v01 --supports--> v18\n- v09 --refutes--> v0f\n- v0c --refutes--> v0a\n- v05 --refutes--> v01\n- v00 --refutes--> v05\n- v07 --supports--> v0d\n- v04 --refutes--> v13\n- v0f --supports--> v0e\n- v17 --refutes--> v06\n- v18 --refutes--> v04\n- v0f --supports--> v10\n- v15 --refutes--> v00\n- v06 --refutes--> v15\n- v0e --supports--> v07\n- v04 --refutes--> v09\n- v08 --refutes--> v0c\n- v17 --supports--> v12\n- v07 --refutes--> v03\n- v0a --supports--> v17\n- v14 --supports--> v08\n- v0e --supports--> v0b\n- v15 --supports--> v02\n- v0c --refutes--> v16\n\n[VERIFIED ROOT]\nv14 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f4d1502e49ba16f1137c171d703c80fe44a2eda33344ed98ccf28d08227be6f0","schema":"livegraph.boolean-propagation.v1","spec_hash":"10f42c15d628b3d26a96049d6492f0ee6fd33cd43b065d3f0da2e36f4f9a757e","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v14\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-01ef58ddd985c38fcf3c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v13\n- v04 --refutes--> v07\n- v0d --refutes--> v17\n- v13 --refutes--> v15\n- v12 --supports--> v0f\n- v04 --refutes--> v18\n- v11 --supports--> v05\n- v02 --supports--> v08\n- v16 --refutes--> v10\n- v14 --supports--> v06\n- v08 --supports--> v03\n- v08 --supports--> v14\n- v08 --supports--> v0e\n- v12 --supports--> v00\n- v12 --supports--> v11\n- v05 --supports--> v0b\n- v16 --refutes--> v0d\n- v12 --refutes--> v0a\n- v0c --refutes--> v09\n- v17 --supports--> v04\n- v0b --refutes--> v01\n- v14 --refutes--> v16\n- v07 --supports--> v12\n- v09 --supports--> v02\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv15\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"115ff49a5117e6c6a667f409b003f0af26025762b3c682f469b6e277e51db6dc","schema":"livegraph.boolean-propagation.v1","spec_hash":"01ef58ddd985c38fcf3c7ddbd6736cf729513a466bd39c27411daf42da46701f","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-284c567a539b930bcfc8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v0e\n- v01 --refutes--> v0d\n- v03 --supports--> v18\n- v13 --supports--> v0f\n- v17 --refutes--> v02\n- v02 --refutes--> v06\n- v08 --refutes--> v00\n- v16 --refutes--> v15\n- v0d --supports--> v09\n- v02 --supports--> v13\n- v10 --refutes--> v0b\n- v14 --supports--> v0c\n- v05 --supports--> v16\n- v0b --refutes--> v12\n- v01 --refutes--> v04\n- v16 --supports--> v0a\n- v13 --supports--> v01\n- v09 --refutes--> v03\n- v0c --supports--> v17\n- v07 --refutes--> v14\n- v00 --refutes--> v05\n- v14 --supports--> v11\n- v03 --refutes--> v08\n- v16 --supports--> v10\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"899a7ab682b75b18a8bc9b65588a1ea691502948ea0eea55ac8376f46cd78718","schema":"livegraph.boolean-propagation.v1","spec_hash":"284c567a539b930bcfc8566fd117f5c0505ac78a96069c2f4c845639cb2d2231","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-80628a50a427017bdbee","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v02\n- v10 --refutes--> v15\n- v03 --refutes--> v11\n- v0d --refutes--> v10\n- v00 --supports--> v04\n- v0b --refutes--> v0d\n- v0e --supports--> v13\n- v11 --supports--> v14\n- v15 --supports--> v17\n- v09 --refutes--> v07\n- v00 --refutes--> v0e\n- v11 --refutes--> v01\n- v0c --supports--> v05\n- v07 --refutes--> v18\n- v15 --refutes--> v09\n- v06 --supports--> v00\n- v05 --refutes--> v12\n- v0c --supports--> v0a\n- v18 --supports--> v08\n- v18 --supports--> v03\n- v05 --supports--> v16\n- v10 --supports--> v0f\n- v16 --supports--> v06\n- v13 --refutes--> v0b\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7ef0f4bc607968dd8ca564d7a99a0a7bd26ef5bef98b253fe54eef739414e0e8","schema":"livegraph.boolean-propagation.v1","spec_hash":"80628a50a427017bdbeea7fec7067a626972d23257d725866a6a03b206ec34d6","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-7931b22a1338a016ad83","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --refutes--> v06\n- v0b --refutes--> v10\n- v16 --refutes--> v0a\n- v14 --refutes--> v01\n- v0f --refutes--> v12\n- v02 --refutes--> v0f\n- v0a --supports--> v14\n- v0d --supports--> v03\n- v09 --supports--> v08\n- v04 --refutes--> v07\n- v0c --refutes--> v15\n- v08 --supports--> v17\n- v09 --refutes--> v05\n- v01 --refutes--> v02\n- v01 --supports--> v18\n- v0d --supports--> v00\n- v12 --refutes--> v0e\n- v17 --supports--> v04\n- v0b --supports--> v09\n- v0c --refutes--> v0d\n- v12 --supports--> v0b\n- v07 --supports--> v0c\n- v04 --refutes--> v11\n- v13 --refutes--> v16\n\n[VERIFIED ROOT]\nv13 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fbb31e71e9b7e992a2d9e0d29cfa6136daa0a5fd5a3385ca50c67944a3d4a9fa","schema":"livegraph.boolean-propagation.v1","spec_hash":"7931b22a1338a016ad8387069667db7659ccf8450e89453a8299afca04a4370b","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-e9e8794665dbaa48db70","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v08\n- v14 --refutes--> v05\n- v0b --supports--> v12\n- v00 --refutes--> v14\n- v17 --supports--> v0d\n- v14 --supports--> v0b\n- v0d --refutes--> v00\n- v04 --refutes--> v0a\n- v10 --refutes--> v11\n- v14 --supports--> v16\n- v03 --refutes--> v0c\n- v18 --supports--> v01\n- v10 --refutes--> v04\n- v0e --refutes--> v03\n- v0a --supports--> v02\n- v03 --supports--> v15\n- v01 --supports--> v0e\n- v0d --supports--> v07\n- v18 --supports--> v13\n- v0a --supports--> v17\n- v09 --refutes--> v10\n- v06 --refutes--> v18\n- v0f --refutes--> v06\n- v0c --supports--> v09\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"56b35bb2f34d9c8401b5b62555d5838428ba91650b17d0b7002fc4e5c711d1f2","schema":"livegraph.boolean-propagation.v1","spec_hash":"e9e8794665dbaa48db70aba02f82b9a2bbd6bf24a477ded6e69fab0b41e4b95c","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-10a4a881358b92733a89","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v04\n- v03 --supports--> v11\n- v12 --supports--> v05\n- v0b --supports--> v06\n- v07 --supports--> v0f\n- v04 --refutes--> v08\n- v02 --refutes--> v0e\n- v0b --refutes--> v16\n- v0f --supports--> v17\n- v05 --supports--> v0d\n- v11 --supports--> v13\n- v00 --supports--> v15\n- v11 --supports--> v10\n- v01 --supports--> v00\n- v00 --refutes--> v12\n- v0f --refutes--> v01\n- v08 --supports--> v07\n- v03 --refutes--> v09\n- v0f --refutes--> v14\n- v0c --refutes--> v0a\n- v05 --supports--> v03\n- v10 --supports--> v02\n- v16 --refutes--> v0c\n- v16 --refutes--> v18\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"159e376631a75aa5df8bdb359bf34f798c7cb38693ef6269ad056a00200c7b88","schema":"livegraph.boolean-propagation.v1","spec_hash":"10a4a881358b92733a890f13eb52294fa0b6cc621917d2170346fc419bf46c2b","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":16,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-87705c5b07edbaf3779c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v0f\n- v02 --refutes--> v00\n- v03 --supports--> v0c\n- v13 --refutes--> v15\n- v12 --refutes--> v08\n- v03 --supports--> v0a\n- v17 --refutes--> v09\n- v09 --supports--> v03\n- v11 --supports--> v12\n- v09 --refutes--> v16\n- v13 --refutes--> v0b\n- v05 --supports--> v13\n- v05 --refutes--> v07\n- v0d --supports--> v06\n- v0e --refutes--> v0d\n- v00 --supports--> v0e\n- v0b --supports--> v14\n- v18 --supports--> v01\n- v06 --refutes--> v10\n- v0a --refutes--> v05\n- v0b --supports--> v11\n- v0e --refutes--> v04\n- v12 --supports--> v02\n- v01 --refutes--> v17\n\n[VERIFIED ROOT]\nv18 = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"93c6aabe306e9097a09ccca985b6ffc2718a9da8682e17e013f099bec5383362","schema":"livegraph.boolean-propagation.v1","spec_hash":"87705c5b07edbaf3779c365259b88925da5b7953ad6dd2acb2afc3c15104840a","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v18\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-a5627eaf4161d1fac5fd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v16\n- v06 --refutes--> v0f\n- v04 --supports--> v0a\n- v15 --refutes--> v02\n- v0d --supports--> v07\n- v0a --supports--> v0b\n- v0a --supports--> v17\n- v03 --supports--> v06\n- v02 --supports--> v18\n- v08 --refutes--> v15\n- v03 --supports--> v13\n- v01 --supports--> v08\n- v15 --supports--> v0e\n- v07 --refutes--> v09\n- v09 --refutes--> v0c\n- v12 --supports--> v0d\n- v18 --supports--> v05\n- v0f --refutes--> v00\n- v18 --refutes--> v03\n- v06 --supports--> v10\n- v12 --refutes--> v14\n- v00 --supports--> v12\n- v0c --supports--> v04\n- v02 --refutes--> v11\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv16\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"585a438717e9c6f6283003df3f56d20b239f6d250092b145bfbd83418939ea49","schema":"livegraph.boolean-propagation.v1","spec_hash":"a5627eaf4161d1fac5fd6f0fe0624f8b967593501ff45e38b71e43827f5b71f6","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v16\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-7fc8b16bddd1113160ce","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --supports--> v08\n- v06 --supports--> v15\n- v10 --refutes--> v01\n- v00 --supports--> v0f\n- v18 --refutes--> v00\n- v02 --refutes--> v0b\n- v07 --refutes--> v0c\n- v0f --supports--> v10\n- v0c --supports--> v16\n- v10 --refutes--> v17\n- v15 --supports--> v02\n- v16 --refutes--> v12\n- v11 --refutes--> v18\n- v16 --refutes--> v09\n- v14 --supports--> v04\n- v0f --supports--> v0e\n- v11 --supports--> v0a\n- v12 --refutes--> v14\n- v14 --refutes--> v05\n- v10 --supports--> v06\n- v05 --refutes--> v11\n- v11 --refutes--> v03\n- v08 --supports--> v07\n- v00 --refutes--> v0d\n\n[VERIFIED ROOT]\nv13 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"326ba8fb7d45758c9b497a47a4d02c513213fc2e6423848249736f5886248cac","schema":"livegraph.boolean-propagation.v1","spec_hash":"7fc8b16bddd1113160ce3ded0111f6ebb556214c71e82a3c752fb11539f5dc86","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v13\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-5aff0c3124875f9e3dd0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v12\n- v02 --supports--> v08\n- v05 --supports--> v0d\n- v0f --refutes--> v05\n- v11 --supports--> v03\n- v17 --refutes--> v0e\n- v11 --refutes--> v00\n- v10 --supports--> v18\n- v05 --supports--> v16\n- v08 --supports--> v01\n- v0a --refutes--> v04\n- v15 --refutes--> v09\n- v00 --refutes--> v0f\n- v0f --supports--> v0c\n- v0d --supports--> v07\n- v0e --refutes--> v13\n- v13 --refutes--> v10\n- v0a --supports--> v02\n- v06 --supports--> v17\n- v01 --refutes--> v06\n- v09 --supports--> v0a\n- v09 --supports--> v14\n- v17 --supports--> v0b\n- v0d --refutes--> v15\n\n[VERIFIED ROOT]\nv11 = FALSE\n\n[TARGET]\nv18\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d99074d016ae803c8f08d797c9f1fd9b2b697cbfacb14b5e88bca8c410000992","schema":"livegraph.boolean-propagation.v1","spec_hash":"5aff0c3124875f9e3dd0dd53ebde642cd8f809ae792de948858f64b4b710de9a","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v18\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-b7e7d82f6dc5ba847525","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v0e\n- v06 --refutes--> v05\n- v08 --refutes--> v0a\n- v14 --supports--> v16\n- v17 --supports--> v13\n- v10 --refutes--> v01\n- v07 --refutes--> v00\n- v14 --refutes--> v18\n- v01 --refutes--> v15\n- v08 --refutes--> v10\n- v14 --refutes--> v0c\n- v12 --refutes--> v17\n- v09 --refutes--> v0f\n- v04 --supports--> v11\n- v15 --supports--> v02\n- v06 --supports--> v0d\n- v05 --refutes--> v04\n- v13 --supports--> v14\n- v17 --supports--> v0b\n- v15 --supports--> v09\n- v18 --refutes--> v06\n- v03 --supports--> v12\n- v00 --supports--> v08\n- v09 --supports--> v03\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"edd0d0b286ed1bd22fc8bd554eef3cb38c662d11c17af00883ba54a9aabdb644","schema":"livegraph.boolean-propagation.v1","spec_hash":"b7e7d82f6dc5ba847525be3d8f3fad36709c28fe614b7baf0219dd3757cf9429","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-89dab172772cc2d0ef6b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v17\n- v0d --supports--> v0e\n- v09 --refutes--> v01\n- v09 --refutes--> v11\n- v0c --refutes--> v0f\n- v13 --refutes--> v12\n- v0e --supports--> v05\n- v0a --supports--> v08\n- v13 --supports--> v02\n- v0e --refutes--> v16\n- v00 --refutes--> v0a\n- v0f --supports--> v0b\n- v06 --refutes--> v0c\n- v02 --supports--> v09\n- v11 --refutes--> v0d\n- v15 --refutes--> v10\n- v08 --supports--> v06\n- v04 --refutes--> v15\n- v03 --supports--> v04\n- v02 --refutes--> v07\n- v06 --supports--> v14\n- v0b --supports--> v03\n- v0e --refutes--> v00\n- v08 --supports--> v18\n\n[VERIFIED ROOT]\nv13 = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"40a26b455c0a923c60800fda084710670ef13c8879cdd244ae4341cc6517555d","schema":"livegraph.boolean-propagation.v1","spec_hash":"89dab172772cc2d0ef6bc665be3b0135bdadee9278ce2f8fe0909f9d5239e310","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":16,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-81729bcb2ea9fbb5dbb3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v13\n- v00 --refutes--> v16\n- v18 --supports--> v09\n- v0f --refutes--> v03\n- v15 --refutes--> v14\n- v0a --refutes--> v0b\n- v11 --refutes--> v00\n- v14 --supports--> v12\n- v01 --supports--> v17\n- v10 --refutes--> v07\n- v14 --supports--> v0e\n- v00 --supports--> v04\n- v0d --refutes--> v10\n- v0b --refutes--> v11\n- v16 --supports--> v15\n- v10 --supports--> v05\n- v03 --supports--> v08\n- v0d --refutes--> v02\n- v12 --refutes--> v01\n- v03 --refutes--> v0a\n- v09 --supports--> v0f\n- v01 --supports--> v0c\n- v0d --supports--> v06\n- v05 --refutes--> v18\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"10d8bdc6946be1f400851ccd95cb35742c256b2d7d952553c897880fffbc6b15","schema":"livegraph.boolean-propagation.v1","spec_hash":"81729bcb2ea9fbb5dbb3476926f75d62a980455df8fc6146a07fbe5c498af780","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-0e22f7fb8e803629ae19","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v16\n- v18 --supports--> v10\n- v0a --supports--> v06\n- v13 --refutes--> v17\n- v11 --refutes--> v0b\n- v12 --supports--> v08\n- v05 --refutes--> v11\n- v13 --refutes--> v14\n- v08 --refutes--> v01\n- v17 --refutes--> v00\n- v01 --supports--> v05\n- v16 --refutes--> v03\n- v0c --supports--> v02\n- v10 --supports--> v0e\n- v10 --refutes--> v0c\n- v06 --refutes--> v0f\n- v0c --supports--> v13\n- v18 --refutes--> v15\n- v0f --refutes--> v07\n- v11 --supports--> v09\n- v11 --supports--> v0d\n- v0a --refutes--> v04\n- v00 --supports--> v12\n- v16 --refutes--> v18\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"aa29562af52eef332721f9b2331bd8fecdeed131fdedbfb275a536efb1d09f04","schema":"livegraph.boolean-propagation.v1","spec_hash":"0e22f7fb8e803629ae191ff2ba7a0ec37c53c68b44da7285fd87f846e2de5655","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-180211673a254b360dda","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v0e\n- v09 --refutes--> v05\n- v10 --refutes--> v04\n- v09 --supports--> v0a\n- v14 --supports--> v0b\n- v0f --refutes--> v06\n- v02 --supports--> v0c\n- v0d --supports--> v16\n- v0f --supports--> v00\n- v08 --supports--> v17\n- v16 --refutes--> v01\n- v17 --refutes--> v09\n- v18 --refutes--> v14\n- v02 --supports--> v08\n- v01 --refutes--> v02\n- v08 --supports--> v07\n- v04 --supports--> v13\n- v18 --refutes--> v15\n- v18 --refutes--> v12\n- v05 --refutes--> v18\n- v01 --supports--> v03\n- v00 --refutes--> v0d\n- v11 --supports--> v10\n- v13 --refutes--> v0f\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5d0e5fc5428b7bad72d59c6f89344e5b36fafe21dcdb67febe3a3d302f29c456","schema":"livegraph.boolean-propagation.v1","spec_hash":"180211673a254b360ddac55e6fa6db0bf2d64a2953ea0d3912f51cba16195ad8","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-c7be575e917d0d6b8d5a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v10\n- v16 --refutes--> v14\n- v05 --supports--> v01\n- v0d --supports--> v06\n- v0d --refutes--> v00\n- v16 --supports--> v17\n- v0c --refutes--> v0a\n- v0a --refutes--> v15\n- v04 --supports--> v12\n- v02 --supports--> v18\n- v08 --supports--> v05\n- v00 --refutes--> v0e\n- v03 --supports--> v0d\n- v14 --supports--> v02\n- v14 --supports--> v0b\n- v04 --refutes--> v11\n- v0a --refutes--> v03\n- v07 --supports--> v16\n- v11 --supports--> v0f\n- v05 --supports--> v04\n- v0e --refutes--> v07\n- v09 --supports--> v0c\n- v08 --refutes--> v13\n- v0f --supports--> v09\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv18\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"addd109ab7123f5e0c4ef03d070d1274fb2d53427ea51cc84040bf93c5175f48","schema":"livegraph.boolean-propagation.v1","spec_hash":"c7be575e917d0d6b8d5a23e5893f29058649a0a746fdbf95ee094cf65889a6f9","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":16,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v18\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-97983b4191424db0a41d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v17\n- v18 --refutes--> v0a\n- v13 --refutes--> v0e\n- v09 --refutes--> v05\n- v04 --supports--> v00\n- v0b --refutes--> v16\n- v12 --supports--> v15\n- v0f --refutes--> v13\n- v16 --supports--> v06\n- v0f --refutes--> v08\n- v13 --refutes--> v11\n- v16 --refutes--> v04\n- v15 --supports--> v03\n- v10 --refutes--> v01\n- v12 --supports--> v14\n- v15 --supports--> v0b\n- v04 --refutes--> v09\n- v10 --refutes--> v0f\n- v05 --refutes--> v0d\n- v0d --supports--> v10\n- v0c --supports--> v12\n- v0e --refutes--> v18\n- v0a --refutes--> v07\n- v13 --supports--> v02\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv17\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"df183c7eaf8342ead1f8a1d7e4f9d9516353d7deea2b6f6675b12230778f1626","schema":"livegraph.boolean-propagation.v1","spec_hash":"97983b4191424db0a41d5d2490b30e0d68710a3205188695a06e8a097036b8a7","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v17\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-1474f40356a5629eeb56","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v16\n- v06 --supports--> v14\n- v04 --supports--> v0f\n- v02 --supports--> v17\n- v16 --refutes--> v0b\n- v06 --supports--> v0e\n- v04 --supports--> v10\n- v0e --refutes--> v03\n- v08 --supports--> v09\n- v0d --refutes--> v06\n- v12 --supports--> v00\n- v05 --refutes--> v04\n- v09 --refutes--> v02\n- v0b --supports--> v08\n- v12 --refutes--> v0a\n- v0f --refutes--> v11\n- v04 --supports--> v07\n- v0f --refutes--> v18\n- v17 --refutes--> v12\n- v02 --supports--> v13\n- v18 --supports--> v01\n- v18 --refutes--> v15\n- v01 --refutes--> v0d\n- v0e --refutes--> v0c\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"197008a98803ff80c7fe7a2385401ffda0462eff8ea8d25ee3a075148afe1a3f","schema":"livegraph.boolean-propagation.v1","spec_hash":"1474f40356a5629eeb569bc21ff5f0ed10ec6ffe294b7a1dec97ef4efa947a7d","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-bc759930ad182ca85da5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v0c\n- v07 --refutes--> v0b\n- v04 --supports--> v03\n- v16 --refutes--> v14\n- v05 --supports--> v02\n- v04 --refutes--> v09\n- v12 --supports--> v0e\n- v12 --refutes--> v01\n- v07 --supports--> v15\n- v12 --supports--> v00\n- v18 --refutes--> v11\n- v12 --supports--> v0a\n- v17 --refutes--> v18\n- v14 --supports--> v07\n- v03 --supports--> v17\n- v10 --refutes--> v08\n- v14 --refutes--> v0f\n- v02 --supports--> v10\n- v08 --supports--> v16\n- v13 --supports--> v06\n- v11 --refutes--> v13\n- v15 --refutes--> v04\n- v06 --supports--> v12\n- v18 --supports--> v0d\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f0a72e7780bdae86e5b1ea5abdcbab659f614ffa447f0bbf10e2c2898b0ff71d","schema":"livegraph.boolean-propagation.v1","spec_hash":"bc759930ad182ca85da5af6a5de433c52178e8599c5f7fe9903e9ed7e5d2bf3f","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-3e52c25580a993d92222","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v0f\n- v10 --supports--> v09\n- v0e --supports--> v0a\n- v0b --refutes--> v0e\n- v0a --refutes--> v12\n- v04 --supports--> v08\n- v13 --supports--> v00\n- v01 --refutes--> v05\n- v16 --refutes--> v15\n- v03 --supports--> v10\n- v00 --supports--> v03\n- v01 --supports--> v06\n- v02 --supports--> v0b\n- v07 --supports--> v17\n- v09 --supports--> v07\n- v07 --refutes--> v11\n- v02 --supports--> v0d\n- v16 --supports--> v0c\n- v07 --refutes--> v14\n- v08 --supports--> v16\n- v03 --refutes--> v18\n- v15 --supports--> v13\n- v17 --supports--> v02\n- v01 --supports--> v04\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"84e048f561e779e339d2bab6b6a360e56e19655b53131acac258355ba787f1df","schema":"livegraph.boolean-propagation.v1","spec_hash":"3e52c25580a993d92222cdfa60a0b7ccd5cae1b1d5961d379e309fd4c3d85f3f","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-730a5afc37949b96d21e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v17 --refutes--> v09\n- v01 --supports--> v03\n- v02 --refutes--> v05\n- v12 --supports--> v10\n- v0d --supports--> v01\n- v06 --supports--> v00\n- v00 --refutes--> v0b\n- v16 --refutes--> v06\n- v10 --supports--> v17\n- v09 --supports--> v08\n- v03 --refutes--> v16\n- v05 --refutes--> v18\n- v01 --refutes--> v0c\n- v0e --supports--> v02\n- v08 --supports--> v0f\n- v17 --supports--> v14\n- v16 --refutes--> v04\n- v0f --supports--> v0d\n- v03 --supports--> v13\n- v18 --supports--> v12\n- v00 --refutes--> v07\n- v03 --refutes--> v11\n- v17 --supports--> v15\n- v0d --supports--> v0a\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2c801d187f1bed450be28cbe44e730a6eb849b81c374f5e2289443c571e660b9","schema":"livegraph.boolean-propagation.v1","spec_hash":"730a5afc37949b96d21e68a1170493b6df918bb266920b91e3035b21a7acf7ca","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-3fa9686370b79ae76966","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v16 --supports--> v08\n- v17 --refutes--> v14\n- v10 --refutes--> v11\n- v0a --refutes--> v0e\n- v04 --refutes--> v00\n- v05 --refutes--> v12\n- v06 --supports--> v05\n- v0e --refutes--> v0d\n- v0b --supports--> v18\n- v02 --supports--> v17\n- v0b --refutes--> v13\n- v06 --refutes--> v09\n- v03 --supports--> v10\n- v18 --supports--> v16\n- v17 --supports--> v03\n- v12 --refutes--> v0f\n- v08 --supports--> v04\n- v11 --supports--> v0c\n- v0d --refutes--> v02\n- v12 --refutes--> v0a\n- v11 --refutes--> v0b\n- v02 --refutes--> v15\n- v04 --supports--> v01\n- v10 --refutes--> v07\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e389b1c562fa4de795ea0861ce8cb558182ca25e060197cbb3b52b492513e0ca","schema":"livegraph.boolean-propagation.v1","spec_hash":"3fa9686370b79ae769669e6e41ff6af90554012cc92262e283bae5932e219625","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-eddf4b22ed39d7cd5caf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v02\n- v18 --refutes--> v16\n- v07 --supports--> v08\n- v0b --supports--> v04\n- v0e --refutes--> v06\n- v09 --refutes--> v03\n- v02 --supports--> v0e\n- v11 --supports--> v15\n- v12 --refutes--> v13\n- v08 --refutes--> v0b\n- v0a --supports--> v07\n- v04 --supports--> v14\n- v0b --refutes--> v00\n- v0f --refutes--> v18\n- v0e --refutes--> v10\n- v0b --refutes--> v17\n- v0e --refutes--> v01\n- v12 --refutes--> v0a\n- v16 --supports--> v09\n- v0c --supports--> v0d\n- v04 --refutes--> v11\n- v06 --refutes--> v0f\n- v0a --refutes--> v05\n- v03 --refutes--> v12\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv15\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"59f409243b13e2fb1c18ac6a04633bb8883edcc17f4157e3f3ebd7e504110376","schema":"livegraph.boolean-propagation.v1","spec_hash":"eddf4b22ed39d7cd5cafc49d30b1564ba9abbc5c0a726923898000107d6a2ed2","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-457bd3c7c319c7427fed","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v16 --supports--> v04\n- v02 --supports--> v10\n- v18 --refutes--> v0b\n- v0e --supports--> v14\n- v10 --supports--> v07\n- v14 --refutes--> v12\n- v02 --refutes--> v0a\n- v17 --refutes--> v02\n- v18 --refutes--> v0c\n- v06 --supports--> v18\n- v0c --supports--> v11\n- v16 --refutes--> v17\n- v12 --refutes--> v0d\n- v15 --supports--> v16\n- v12 --supports--> v00\n- v07 --refutes--> v03\n- v01 --supports--> v15\n- v0c --refutes--> v13\n- v0e --supports--> v0f\n- v0c --supports--> v09\n- v01 --supports--> v05\n- v08 --supports--> v01\n- v0d --refutes--> v08\n- v03 --refutes--> v06\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f4e101d0446dc0f62a4bd98628d5c49c15c518cba9e2a8e609a9b879a149a2b9","schema":"livegraph.boolean-propagation.v1","spec_hash":"457bd3c7c319c7427fed17802dfa7df18b755670f418ac82ca72d3b3c65dcb3d","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-0b1d7c7d03956d510b6a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v07\n- v14 --refutes--> v11\n- v0d --supports--> v00\n- v17 --refutes--> v18\n- v14 --supports--> v04\n- v05 --refutes--> v0e\n- v07 --refutes--> v09\n- v16 --refutes--> v0a\n- v15 --supports--> v03\n- v0a --refutes--> v08\n- v14 --supports--> v01\n- v0b --refutes--> v0c\n- v05 --refutes--> v12\n- v09 --refutes--> v17\n- v10 --supports--> v05\n- v0d --refutes--> v15\n- v03 --supports--> v0b\n- v0c --supports--> v10\n- v18 --supports--> v0f\n- v08 --refutes--> v14\n- v03 --refutes--> v02\n- v10 --refutes--> v13\n- v0f --refutes--> v16\n- v16 --refutes--> v06\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1fc714f51b2691e183142112f0bb41fc9dc7de34c9c3245d6cc7fe9e642e4e53","schema":"livegraph.boolean-propagation.v1","spec_hash":"0b1d7c7d03956d510b6a749c06c0605605b9f59177685c7dc362aa7d1af389f6","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-4996b6ec92db54378491","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v17\n- v13 --supports--> v0a\n- v0d --refutes--> v0f\n- v16 --supports--> v04\n- v13 --refutes--> v08\n- v15 --refutes--> v0d\n- v12 --supports--> v16\n- v0e --refutes--> v12\n- v06 --refutes--> v0c\n- v0d --refutes--> v0e\n- v0a --refutes--> v10\n- v06 --refutes--> v11\n- v06 --supports--> v0b\n- v05 --supports--> v13\n- v18 --refutes--> v01\n- v15 --supports--> v14\n- v0a --refutes--> v07\n- v12 --supports--> v00\n- v01 --supports--> v15\n- v17 --refutes--> v18\n- v16 --supports--> v02\n- v04 --refutes--> v06\n- v11 --supports--> v03\n- v03 --supports--> v05\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c865f8e19e54d7b0b3546f8f563de2250ca32824c7c5209f67268072c8c55f87","schema":"livegraph.boolean-propagation.v1","spec_hash":"4996b6ec92db5437849153048117c9a19e8d5df3fd674ed01575b0f59af809e1","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-476c664211150bda782a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v02\n- v11 --supports--> v13\n- v00 --refutes--> v14\n- v07 --refutes--> v0a\n- v06 --supports--> v00\n- v18 --supports--> v01\n- v15 --supports--> v12\n- v16 --refutes--> v17\n- v13 --refutes--> v0d\n- v10 --supports--> v16\n- v12 --refutes--> v09\n- v04 --supports--> v0b\n- v02 --supports--> v07\n- v13 --refutes--> v0e\n- v14 --refutes--> v04\n- v17 --supports--> v05\n- v17 --refutes--> v0f\n- v12 --refutes--> v18\n- v13 --supports--> v15\n- v10 --supports--> v03\n- v06 --supports--> v08\n- v11 --supports--> v0c\n- v01 --supports--> v10\n- v05 --refutes--> v06\n\n[VERIFIED ROOT]\nv11 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0ab64823c09411e64d3a9cba967daedfd3cd156e15494c0da06ab8ec22e10ece","schema":"livegraph.boolean-propagation.v1","spec_hash":"476c664211150bda782a661821fb7ab87285c300993c85d3655d4fdd351eacfa","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-7965b1301885c932b0e3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v16\n- v10 --refutes--> v0e\n- v0d --refutes--> v00\n- v0c --supports--> v10\n- v0d --supports--> v12\n- v0d --supports--> v0f\n- v06 --refutes--> v17\n- v07 --supports--> v04\n- v0e --supports--> v01\n- v0b --refutes--> v05\n- v11 --supports--> v14\n- v10 --refutes--> v0a\n- v09 --supports--> v08\n- v06 --supports--> v03\n- v08 --supports--> v15\n- v05 --refutes--> v09\n- v00 --supports--> v0b\n- v03 --refutes--> v07\n- v14 --supports--> v0d\n- v0e --refutes--> v18\n- v10 --supports--> v13\n- v04 --supports--> v11\n- v01 --supports--> v06\n- v05 --supports--> v02\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv15\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ca916898733d97328c2bd05aaf271c45e094af67177a4cd12cb146da2758fd9d","schema":"livegraph.boolean-propagation.v1","spec_hash":"7965b1301885c932b0e34fe0c1bae1c4f68adbfcb0258031787aa4ca516c22b0","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":16,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-53ce17aeaed948f53f1f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v04\n- v13 --refutes--> v0d\n- v17 --refutes--> v08\n- v07 --supports--> v06\n- v09 --refutes--> v03\n- v17 --supports--> v0c\n- v14 --supports--> v05\n- v07 --supports--> v02\n- v13 --refutes--> v01\n- v0b --supports--> v13\n- v02 --supports--> v0f\n- v16 --supports--> v15\n- v14 --refutes--> v00\n- v00 --refutes--> v18\n- v18 --supports--> v12\n- v15 --supports--> v07\n- v09 --supports--> v17\n- v0f --supports--> v0a\n- v02 --supports--> v0e\n- v12 --refutes--> v11\n- v12 --refutes--> v09\n- v04 --supports--> v16\n- v0a --supports--> v0b\n- v13 --refutes--> v10\n\n[VERIFIED ROOT]\nv14 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cf3a82538cdee043ed8392846549dc43aab8a0daaf7237976474794aa34f4d5a","schema":"livegraph.boolean-propagation.v1","spec_hash":"53ce17aeaed948f53f1f326dced14188fd02295afff0128512ea3a152bc2ea71","spec_json":"{\"edges\":[{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v14\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-ab216559296c383932e4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v13\n- v10 --supports--> v0a\n- v10 --supports--> v15\n- v18 --supports--> v11\n- v01 --refutes--> v06\n- v0d --supports--> v04\n- v09 --refutes--> v17\n- v04 --refutes--> v12\n- v08 --refutes--> v0c\n- v06 --supports--> v14\n- v06 --refutes--> v16\n- v02 --supports--> v01\n- v0f --supports--> v0d\n- v06 --refutes--> v07\n- v11 --supports--> v02\n- v16 --supports--> v09\n- v13 --refutes--> v08\n- v0d --refutes--> v0e\n- v13 --supports--> v0b\n- v15 --refutes--> v03\n- v03 --refutes--> v0f\n- v17 --supports--> v10\n- v10 --refutes--> v00\n- v15 --refutes--> v05\n\n[VERIFIED ROOT]\nv18 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dcaa205c9d94a69a73f8b595a5d0bb2e3c9c9454963dfd266cb052882a1bc8b0","schema":"livegraph.boolean-propagation.v1","spec_hash":"ab216559296c383932e4969769fe47fcc6afd464be0e8116113723c6835a2602","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v18\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-11977760660937e078ce","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v16 --supports--> v0d\n- v03 --refutes--> v15\n- v02 --supports--> v00\n- v0b --refutes--> v05\n- v18 --refutes--> v11\n- v03 --supports--> v0c\n- v00 --supports--> v16\n- v0b --refutes--> v01\n- v18 --refutes--> v07\n- v0d --refutes--> v08\n- v11 --refutes--> v09\n- v17 --supports--> v13\n- v01 --supports--> v03\n- v12 --supports--> v0b\n- v0a --refutes--> v04\n- v03 --refutes--> v06\n- v15 --refutes--> v18\n- v0f --supports--> v02\n- v11 --supports--> v17\n- v04 --supports--> v0f\n- v00 --refutes--> v0e\n- v12 --refutes--> v10\n- v14 --supports--> v12\n- v17 --refutes--> v0a\n\n[VERIFIED ROOT]\nv14 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"640edfab612e986ac1bbb8ade8db2baf2113506a3758fe169538964b8d0d8f60","schema":"livegraph.boolean-propagation.v1","spec_hash":"11977760660937e078ce7a11088aed48b5ac49a12bcede07d68163c45bc7e5c6","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v14\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-3bc8bc799f32c966c426","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v09\n- v15 --refutes--> v0d\n- v03 --refutes--> v13\n- v02 --supports--> v12\n- v10 --refutes--> v06\n- v18 --refutes--> v0b\n- v03 --supports--> v0e\n- v0c --supports--> v16\n- v10 --refutes--> v03\n- v09 --refutes--> v17\n- v09 --refutes--> v0f\n- v01 --refutes--> v00\n- v09 --refutes--> v07\n- v0f --supports--> v18\n- v13 --refutes--> v02\n- v15 --supports--> v14\n- v05 --refutes--> v11\n- v01 --refutes--> v08\n- v00 --supports--> v15\n- v16 --supports--> v0a\n- v0a --refutes--> v01\n- v0b --supports--> v05\n- v11 --refutes--> v0c\n- v0c --refutes--> v04\n\n[VERIFIED ROOT]\nv10 = FALSE\n\n[TARGET]\nv14\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"13234db0d30cc6a4d53dcbaf83f1e0c0239770db44371a3f6a39279eff436860","schema":"livegraph.boolean-propagation.v1","spec_hash":"3bc8bc799f32c966c426105e6de8860b4aed47c9929e6977cd2e228bb7fe88d8","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-07d091cc7b5a8579d4fd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v0a\n- v01 --refutes--> v06\n- v04 --supports--> v00\n- v11 --supports--> v0c\n- v17 --refutes--> v01\n- v17 --refutes--> v08\n- v13 --refutes--> v05\n- v15 --refutes--> v18\n- v0d --supports--> v09\n- v04 --refutes--> v10\n- v0d --refutes--> v07\n- v05 --supports--> v15\n- v0a --refutes--> v17\n- v07 --supports--> v0f\n- v04 --supports--> v0d\n- v02 --supports--> v03\n- v07 --supports--> v11\n- v03 --refutes--> v14\n- v0e --refutes--> v16\n- v18 --refutes--> v0b\n- v0c --refutes--> v02\n- v13 --refutes--> v12\n- v0b --refutes--> v04\n- v0e --supports--> v13\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"af24a82c3651867a43c21046b18759b460e200ca7fbb99b1e7afe5ce0a879ade","schema":"livegraph.boolean-propagation.v1","spec_hash":"07d091cc7b5a8579d4fd0e0322e2b4dc49c54813104d4fb5f5a9bf8c01106fdb","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-c87a7e33be055da62b45","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v01\n- v17 --supports--> v13\n- v00 --refutes--> v0b\n- v04 --supports--> v11\n- v0b --supports--> v05\n- v0f --refutes--> v10\n- v0f --supports--> v08\n- v02 --refutes--> v06\n- v0b --supports--> v17\n- v09 --refutes--> v02\n- v01 --refutes--> v0d\n- v0f --supports--> v0a\n- v03 --refutes--> v18\n- v02 --supports--> v00\n- v0c --supports--> v14\n- v17 --supports--> v15\n- v18 --refutes--> v04\n- v08 --refutes--> v07\n- v13 --refutes--> v0c\n- v14 --supports--> v0f\n- v18 --supports--> v16\n- v08 --supports--> v0e\n- v0d --refutes--> v12\n- v12 --refutes--> v09\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e9ea8c4cd4016d0b036124c87c74f84eed5107a64c9e0ccc5149d0af7b746d5e","schema":"livegraph.boolean-propagation.v1","spec_hash":"c87a7e33be055da62b45ec4f08183581ba6ed3f9ca836b00300c8ea516b73a0c","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-0b023165a89e7db2f33e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --supports--> v0e\n- v0a --refutes--> v0b\n- v18 --refutes--> v16\n- v01 --refutes--> v07\n- v0b --supports--> v10\n- v04 --supports--> v02\n- v10 --supports--> v12\n- v10 --refutes--> v0d\n- v03 --refutes--> v06\n- v08 --refutes--> v0f\n- v0b --supports--> v13\n- v00 --supports--> v04\n- v04 --supports--> v0c\n- v0f --supports--> v09\n- v0d --supports--> v03\n- v06 --supports--> v01\n- v0a --supports--> v14\n- v0e --supports--> v08\n- v15 --supports--> v17\n- v00 --supports--> v05\n- v04 --refutes--> v11\n- v16 --supports--> v0a\n- v01 --refutes--> v15\n- v0c --refutes--> v18\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"45c8a14126f68d04c2cf4fe4cb530963f97a98404979d905fd3bcea3e32f583f","schema":"livegraph.boolean-propagation.v1","spec_hash":"0b023165a89e7db2f33e1f922227827f158731d4c7779eb3daf5ff80207e59c8","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-b9e9a51e826ede751b95","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v07\n- v10 --refutes--> v00\n- v17 --supports--> v03\n- v12 --supports--> v11\n- v11 --supports--> v15\n- v06 --refutes--> v0a\n- v0d --supports--> v09\n- v0c --supports--> v06\n- v02 --refutes--> v18\n- v16 --refutes--> v0f\n- v17 --refutes--> v01\n- v0f --refutes--> v02\n- v11 --refutes--> v17\n- v04 --supports--> v16\n- v02 --refutes--> v12\n- v0d --supports--> v05\n- v06 --refutes--> v10\n- v10 --supports--> v0d\n- v03 --supports--> v14\n- v0b --refutes--> v08\n- v08 --supports--> v04\n- v0e --supports--> v0b\n- v14 --supports--> v0c\n- v06 --supports--> v13\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c73a6a5a317709ed8a6fb2d0fc1db17ecaaa115ef0a5f2041bdb171920761e66","schema":"livegraph.boolean-propagation.v1","spec_hash":"b9e9a51e826ede751b95c2689254eff8616991bb163c3bf4884a611c4038b1a2","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-d984b264d91b3d487f43","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v10\n- v04 --refutes--> v11\n- v17 --refutes--> v02\n- v16 --supports--> v05\n- v07 --refutes--> v0b\n- v16 --supports--> v0f\n- v11 --supports--> v0a\n- v10 --refutes--> v14\n- v09 --supports--> v17\n- v00 --supports--> v0d\n- v11 --supports--> v16\n- v01 --refutes--> v06\n- v13 --supports--> v12\n- v01 --supports--> v18\n- v15 --refutes--> v01\n- v13 --supports--> v04\n- v0d --refutes--> v13\n- v07 --supports--> v00\n- v0e --supports--> v07\n- v10 --supports--> v0e\n- v05 --refutes--> v09\n- v05 --refutes--> v0c\n- v00 --supports--> v03\n- v06 --refutes--> v08\n\n[VERIFIED ROOT]\nv15 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"76ef4da89758038192d80b67bbd99792d3da3c88909fda920469d139944e6d18","schema":"livegraph.boolean-propagation.v1","spec_hash":"d984b264d91b3d487f43a947a516f6438e19c8982cd0bd047f7d70be325c14f0","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v15\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-ca220e811517a580a9a5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v06\n- v18 --supports--> v17\n- v0d --refutes--> v12\n- v12 --supports--> v16\n- v12 --refutes--> v09\n- v00 --supports--> v15\n- v00 --supports--> v05\n- v07 --supports--> v11\n- v15 --supports--> v04\n- v0e --supports--> v00\n- v09 --refutes--> v0c\n- v0f --supports--> v0d\n- v17 --refutes--> v07\n- v00 --refutes--> v08\n- v14 --supports--> v0a\n- v13 --supports--> v0f\n- v03 --supports--> v0b\n- v07 --supports--> v02\n- v03 --refutes--> v14\n- v15 --refutes--> v13\n- v02 --refutes--> v03\n- v07 --refutes--> v01\n- v10 --supports--> v0e\n- v0c --supports--> v18\n\n[VERIFIED ROOT]\nv10 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"da64e974cbb0ad056659c46192d1d44bb6cd34faacde8089343e6132281ea0b0","schema":"livegraph.boolean-propagation.v1","spec_hash":"ca220e811517a580a9a56e8b9691c5e745ddb572f91356fda1fb4e785be2b7c2","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-3d4a89c9230aaf2df04f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v04\n- v11 --refutes--> v0a\n- v01 --supports--> v06\n- v0f --refutes--> v00\n- v04 --refutes--> v07\n- v0f --refutes--> v11\n- v18 --supports--> v0d\n- v0d --supports--> v02\n- v0d --refutes--> v0e\n- v0f --supports--> v05\n- v09 --refutes--> v0f\n- v03 --supports--> v15\n- v07 --supports--> v12\n- v07 --supports--> v08\n- v0f --refutes--> v14\n- v13 --refutes--> v03\n- v10 --supports--> v13\n- v0a --supports--> v18\n- v16 --refutes--> v17\n- v12 --supports--> v10\n- v15 --supports--> v0c\n- v15 --refutes--> v16\n- v17 --refutes--> v09\n- v09 --refutes--> v0b\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fbf08d2e567de4cc4bc6cdd2ef20eb89b78c518b7ff5073f02bc2de60ffc5f5e","schema":"livegraph.boolean-propagation.v1","spec_hash":"3d4a89c9230aaf2df04f1ba0227fbf121498e7f9778dd3656044e72368a1ced7","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-7b78e3d18dbafc962734","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v0b\n- v10 --refutes--> v01\n- v0b --supports--> v0e\n- v04 --refutes--> v14\n- v02 --refutes--> v0f\n- v14 --refutes--> v08\n- v08 --supports--> v12\n- v13 --supports--> v18\n- v0b --supports--> v13\n- v02 --supports--> v16\n- v0a --supports--> v15\n- v0d --supports--> v10\n- v0f --supports--> v03\n- v09 --refutes--> v17\n- v04 --supports--> v00\n- v01 --supports--> v09\n- v15 --supports--> v0c\n- v14 --refutes--> v07\n- v18 --supports--> v0a\n- v18 --supports--> v05\n- v0f --refutes--> v04\n- v0f --supports--> v11\n- v17 --refutes--> v02\n- v08 --refutes--> v06\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d49329e2849f0c164e37c24891a03cbe15c2de124a0d3064ae5cb349b344f7e2","schema":"livegraph.boolean-propagation.v1","spec_hash":"7b78e3d18dbafc962734e3bfe312611904de7360be1b28b15041af56d5ef1a30","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-b9fa59d72c8598d866dc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v00\n- v11 --refutes--> v13\n- v04 --supports--> v0a\n- v0c --supports--> v0f\n- v0f --supports--> v05\n- v15 --refutes--> v0b\n- v14 --refutes--> v16\n- v05 --refutes--> v04\n- v11 --refutes--> v0d\n- v16 --refutes--> v08\n- v11 --supports--> v14\n- v18 --refutes--> v0e\n- v10 --refutes--> v06\n- v09 --refutes--> v10\n- v01 --refutes--> v18\n- v12 --refutes--> v02\n- v16 --supports--> v07\n- v05 --refutes--> v17\n- v18 --refutes--> v11\n- v02 --supports--> v03\n- v0a --supports--> v01\n- v02 --supports--> v0c\n- v10 --supports--> v15\n- v16 --refutes--> v09\n\n[VERIFIED ROOT]\nv12 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c81faf9ec1aab27c11e69589204746344f1b63ad4bbe04a61ab56523948dda06","schema":"livegraph.boolean-propagation.v1","spec_hash":"b9fa59d72c8598d866dc40a1f142a19d7f25579fda30262568486dfaeb9fff87","spec_json":"{\"edges\":[{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v12\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-7bbaad8f044bb6637976","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v18\n- v18 --refutes--> v15\n- v08 --refutes--> v0c\n- v09 --refutes--> v00\n- v01 --supports--> v12\n- v01 --refutes--> v0f\n- v17 --refutes--> v02\n- v12 --refutes--> v04\n- v12 --refutes--> v0d\n- v17 --refutes--> v0a\n- v15 --supports--> v0e\n- v09 --refutes--> v05\n- v0d --refutes--> v07\n- v11 --supports--> v14\n- v00 --refutes--> v03\n- v17 --supports--> v10\n- v07 --supports--> v11\n- v0c --refutes--> v13\n- v15 --refutes--> v01\n- v0c --supports--> v09\n- v14 --refutes--> v17\n- v0b --refutes--> v16\n- v12 --refutes--> v06\n- v10 --refutes--> v0b\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv16\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"56ac1ee4e4de23d3131380fc0db7e7dff0b1a225f2c253b0c2d33f783a503ec2","schema":"livegraph.boolean-propagation.v1","spec_hash":"7bbaad8f044bb66379767ebf1a17b4862e85d0765a18a2e9290b60861b57e9d6","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v16\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-8e408cacc4e717bed1f2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v0f\n- v16 --refutes--> v01\n- v12 --refutes--> v17\n- v09 --supports--> v0d\n- v09 --refutes--> v10\n- v11 --refutes--> v18\n- v0b --refutes--> v0a\n- v08 --refutes--> v03\n- v01 --refutes--> v11\n- v0f --supports--> v13\n- v18 --refutes--> v12\n- v14 --supports--> v05\n- v0f --refutes--> v00\n- v12 --refutes--> v14\n- v06 --supports--> v0e\n- v05 --refutes--> v0c\n- v08 --refutes--> v06\n- v0e --refutes--> v0b\n- v0d --refutes--> v08\n- v11 --refutes--> v15\n- v01 --refutes--> v02\n- v13 --refutes--> v16\n- v18 --refutes--> v04\n- v0d --supports--> v07\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8a7c08e3308de209f6fd5347ab3fd3ed3ea8ae00f3aff4f2c08cbf6e48838efe","schema":"livegraph.boolean-propagation.v1","spec_hash":"8e408cacc4e717bed1f28f11e5f7ea134448813bb9aeec18daa4b949c54cc5b1","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-0bea5596ac75f055bb31","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v0e\n- v0d --refutes--> v0f\n- v03 --refutes--> v04\n- v0c --refutes--> v15\n- v13 --refutes--> v10\n- v18 --supports--> v13\n- v07 --refutes--> v00\n- v17 --refutes--> v03\n- v15 --refutes--> v0d\n- v12 --refutes--> v17\n- v05 --refutes--> v0c\n- v02 --supports--> v08\n- v07 --supports--> v05\n- v08 --supports--> v01\n- v12 --supports--> v11\n- v15 --supports--> v16\n- v08 --refutes--> v0a\n- v13 --supports--> v07\n- v0d --refutes--> v06\n- v01 --refutes--> v18\n- v04 --supports--> v02\n- v02 --refutes--> v14\n- v13 --refutes--> v0b\n- v0f --supports--> v09\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4af87c397ce7303c6f33d4959680fad743e631b0b27a0b9ae8d0604cc6f2d8b3","schema":"livegraph.boolean-propagation.v1","spec_hash":"0bea5596ac75f055bb316053a9428313fc22b142d7c464799613992d88e1a3d1","spec_json":"{\"edges\":[{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-f4260acca67c23068fb9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v08\n- v10 --supports--> v00\n- v14 --supports--> v04\n- v08 --supports--> v0e\n- v0a --supports--> v16\n- v0e --refutes--> v03\n- v16 --refutes--> v0c\n- v06 --refutes--> v09\n- v0e --supports--> v0f\n- v03 --refutes--> v14\n- v15 --supports--> v01\n- v03 --supports--> v13\n- v09 --refutes--> v05\n- v17 --refutes--> v06\n- v18 --supports--> v07\n- v16 --supports--> v18\n- v15 --supports--> v0a\n- v02 --supports--> v17\n- v0b --supports--> v02\n- v03 --refutes--> v11\n- v05 --refutes--> v15\n- v02 --supports--> v12\n- v18 --refutes--> v10\n- v14 --refutes--> v0d\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"521312d6ef96a6c4cef03d3d114ca8233a172148928c974fb69bc287c74a8d5c","schema":"livegraph.boolean-propagation.v1","spec_hash":"f4260acca67c23068fb9411ccac5327f4a643a2fe062efde9db32e31c07c74ad","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-6f60c521861d4d0409a6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v16 --refutes--> v13\n- v11 --refutes--> v15\n- v17 --refutes--> v0a\n- v05 --refutes--> v06\n- v03 --supports--> v0d\n- v07 --refutes--> v05\n- v0e --refutes--> v10\n- v06 --supports--> v18\n- v06 --refutes--> v01\n- v13 --supports--> v07\n- v0d --supports--> v11\n- v17 --refutes--> v12\n- v06 --refutes--> v17\n- v12 --refutes--> v0e\n- v09 --refutes--> v08\n- v04 --refutes--> v03\n- v0f --refutes--> v14\n- v15 --supports--> v0b\n- v0f --refutes--> v16\n- v0e --supports--> v04\n- v12 --refutes--> v00\n- v08 --refutes--> v0f\n- v06 --supports--> v0c\n- v0f --refutes--> v02\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c1e1b6b2dd58a52992df598381395633d49df8578de82627ad4a52e289f43aa8","schema":"livegraph.boolean-propagation.v1","spec_hash":"6f60c521861d4d0409a673d560682849807e8edf44e8720a9af653ba2dbad83b","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-da333114f89eb0c7db7e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v0d\n- v14 --supports--> v0a\n- v11 --refutes--> v07\n- v06 --supports--> v14\n- v05 --supports--> v15\n- v17 --refutes--> v01\n- v00 --supports--> v10\n- v00 --supports--> v09\n- v0d --supports--> v05\n- v15 --refutes--> v13\n- v12 --supports--> v0f\n- v11 --refutes--> v0c\n- v03 --supports--> v12\n- v11 --supports--> v17\n- v16 --supports--> v00\n- v17 --supports--> v0e\n- v0f --refutes--> v16\n- v12 --refutes--> v02\n- v0b --supports--> v11\n- v0b --refutes--> v18\n- v14 --refutes--> v08\n- v04 --refutes--> v06\n- v13 --supports--> v04\n- v14 --refutes--> v03\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cf47c40529274ef9925495501c8868716a829d161e06baf9eac51652c745d094","schema":"livegraph.boolean-propagation.v1","spec_hash":"da333114f89eb0c7db7eff1378073b72b9429bc05ff718e1af2b6f6fca8562db","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-ee233c38b636e9955d72","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v0f\n- v12 --supports--> v18\n- v12 --supports--> v09\n- v03 --refutes--> v01\n- v02 --refutes--> v13\n- v05 --refutes--> v04\n- v0f --supports--> v07\n- v14 --refutes--> v0c\n- v09 --refutes--> v08\n- v0a --refutes--> v14\n- v03 --refutes--> v06\n- v05 --refutes--> v11\n- v05 --refutes--> v0a\n- v13 --refutes--> v10\n- v07 --refutes--> v03\n- v16 --refutes--> v12\n- v06 --refutes--> v05\n- v0c --refutes--> v0d\n- v09 --refutes--> v17\n- v0c --refutes--> v16\n- v13 --refutes--> v00\n- v15 --supports--> v02\n- v0c --supports--> v0e\n- v10 --refutes--> v0b\n\n[VERIFIED ROOT]\nv15 = FALSE\n\n[TARGET]\nv17\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"221683541b5058023548cc03e1f219f37e6d036f6dc1583c0dd4e064ecd026d6","schema":"livegraph.boolean-propagation.v1","spec_hash":"ee233c38b636e9955d72b39a60ed7303f726b6e8e93eb3e3828ca166ccef4053","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":16,\"root\":\"v15\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v17\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-2b401bf82df2ff2be490","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v07\n- v18 --refutes--> v08\n- v00 --refutes--> v11\n- v0a --supports--> v04\n- v10 --supports--> v01\n- v01 --refutes--> v0f\n- v17 --refutes--> v12\n- v0b --supports--> v02\n- v05 --supports--> v16\n- v18 --refutes--> v15\n- v16 --supports--> v0c\n- v16 --refutes--> v18\n- v10 --refutes--> v0e\n- v12 --supports--> v14\n- v02 --supports--> v06\n- v15 --supports--> v00\n- v01 --refutes--> v0d\n- v02 --supports--> v03\n- v12 --refutes--> v13\n- v03 --supports--> v17\n- v00 --refutes--> v10\n- v09 --refutes--> v0b\n- v0d --refutes--> v09\n- v04 --supports--> v05\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv14\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"24ff80d4f2c6521a964bb731929d2c48cd818c1fedc252aac570c546386fd68f","schema":"livegraph.boolean-propagation.v1","spec_hash":"2b401bf82df2ff2be490f6fd73e3d7ab5c859b375ccff553c2905915dc815578","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-952ec2c65b6bdadeb4c5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v0c\n- v0c --refutes--> v12\n- v01 --refutes--> v14\n- v12 --refutes--> v00\n- v08 --supports--> v02\n- v07 --refutes--> v10\n- v0f --supports--> v0d\n- v11 --supports--> v07\n- v15 --refutes--> v0a\n- v02 --refutes--> v18\n- v14 --refutes--> v13\n- v11 --supports--> v0e\n- v18 --supports--> v0f\n- v12 --refutes--> v15\n- v09 --refutes--> v06\n- v10 --refutes--> v09\n- v05 --supports--> v01\n- v02 --refutes--> v03\n- v11 --refutes--> v04\n- v18 --supports--> v17\n- v0d --refutes--> v11\n- v12 --refutes--> v0b\n- v13 --supports--> v08\n- v0c --supports--> v16\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ced3bcb7ae7eb3809f517877bba9ca9bf3728f41b6126f7ff419d88b69b19fe0","schema":"livegraph.boolean-propagation.v1","spec_hash":"952ec2c65b6bdadeb4c5ff0749ddb12b1245619855f47de5a30e381591d28ff9","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-9c3312263b935b1ac86f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v15\n- v03 --refutes--> v02\n- v02 --refutes--> v09\n- v00 --refutes--> v11\n- v06 --refutes--> v0b\n- v01 --supports--> v0f\n- v0b --supports--> v0e\n- v0b --refutes--> v08\n- v10 --refutes--> v12\n- v0c --refutes--> v04\n- v0c --supports--> v07\n- v00 --supports--> v17\n- v05 --supports--> v0a\n- v0c --supports--> v14\n- v0a --supports--> v01\n- v18 --refutes--> v0c\n- v16 --refutes--> v18\n- v0f --refutes--> v0d\n- v0e --supports--> v05\n- v17 --refutes--> v16\n- v10 --supports--> v06\n- v06 --supports--> v13\n- v02 --refutes--> v00\n- v14 --refutes--> v10\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"361cda048e3392bec464d1fe1328c06a89471bdefdbab1641970e949e129d069","schema":"livegraph.boolean-propagation.v1","spec_hash":"9c3312263b935b1ac86f1d2212797243cc99b445ebda736056d2cf367c8bce25","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-7f0ce88d08408adf71c0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v11\n- v00 --refutes--> v0e\n- v10 --refutes--> v05\n- v05 --supports--> v15\n- v07 --supports--> v00\n- v18 --refutes--> v07\n- v0a --refutes--> v0f\n- v17 --supports--> v16\n- v17 --supports--> v13\n- v0e --supports--> v04\n- v15 --refutes--> v0b\n- v01 --refutes--> v17\n- v0b --supports--> v0c\n- v03 --supports--> v10\n- v11 --refutes--> v0d\n- v0d --refutes--> v0a\n- v15 --supports--> v12\n- v16 --supports--> v03\n- v18 --supports--> v08\n- v15 --supports--> v09\n- v0c --refutes--> v14\n- v0e --supports--> v01\n- v0b --refutes--> v02\n- v0e --supports--> v06\n\n[VERIFIED ROOT]\nv18 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"84d4e57667a9c98ceb3e778b86d9ecdd0bf620ee9fa100d17b51067f788cbbb1","schema":"livegraph.boolean-propagation.v1","spec_hash":"7f0ce88d08408adf71c01f364ea57923ee01ec7a208d27249e938c46f5b79ebb","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v18\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-7616b0da4fb772fe74c4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --supports--> v0f\n- v14 --refutes--> v16\n- v02 --refutes--> v00\n- v13 --supports--> v17\n- v0f --refutes--> v03\n- v10 --supports--> v15\n- v08 --refutes--> v0e\n- v0c --supports--> v18\n- v16 --refutes--> v07\n- v0c --refutes--> v06\n- v15 --supports--> v13\n- v04 --refutes--> v10\n- v06 --refutes--> v12\n- v03 --supports--> v08\n- v07 --supports--> v01\n- v16 --supports--> v0b\n- v0d --supports--> v05\n- v01 --refutes--> v0c\n- v13 --supports--> v02\n- v02 --refutes--> v14\n- v0f --supports--> v09\n- v08 --supports--> v0a\n- v0c --refutes--> v11\n- v08 --supports--> v0d\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"25c908607371d61957dd76da49148ce3a2b5cb0dbe114859dde9412397843c38","schema":"livegraph.boolean-propagation.v1","spec_hash":"7616b0da4fb772fe74c466338a233606e7457bbb00df803b3299763c8b153cf6","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":16,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-3c58dd16819296002c46","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v04\n- v08 --refutes--> v14\n- v00 --supports--> v02\n- v15 --supports--> v10\n- v11 --refutes--> v05\n- v00 --refutes--> v18\n- v0d --refutes--> v15\n- v04 --refutes--> v06\n- v0a --refutes--> v0b\n- v04 --supports--> v0a\n- v17 --refutes--> v0c\n- v02 --supports--> v17\n- v17 --refutes--> v09\n- v0a --supports--> v12\n- v14 --supports--> v00\n- v13 --supports--> v01\n- v0d --supports--> v0e\n- v05 --refutes--> v13\n- v01 --refutes--> v0f\n- v17 --refutes--> v16\n- v0a --supports--> v03\n- v08 --refutes--> v07\n- v03 --supports--> v11\n- v0f --supports--> v0d\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c723e1aea8209cfd4a8bc423e66db692cc17df3e063586de655190f22d8a7ae4","schema":"livegraph.boolean-propagation.v1","spec_hash":"3c58dd16819296002c46dfba9b085c39a560a2a0fd4f684bd6e22fa92b0943cb","spec_json":"{\"edges\":[{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-974a881acd81ce225e49","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --supports--> v09\n- v0e --refutes--> v10\n- v08 --refutes--> v04\n- v09 --refutes--> v12\n- v0d --supports--> v0f\n- v0b --refutes--> v0c\n- v12 --supports--> v02\n- v16 --supports--> v18\n- v04 --refutes--> v00\n- v08 --refutes--> v03\n- v00 --refutes--> v07\n- v14 --refutes--> v17\n- v10 --refutes--> v0d\n- v06 --supports--> v11\n- v07 --supports--> v15\n- v09 --refutes--> v0a\n- v0b --refutes--> v14\n- v13 --refutes--> v06\n- v06 --refutes--> v16\n- v02 --refutes--> v08\n- v0d --refutes--> v13\n- v16 --refutes--> v0b\n- v07 --refutes--> v01\n- v01 --refutes--> v05\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"df7372305e660d73ae0a4c0618af75873419bb20bbde8747eb15714aefa1ae07","schema":"livegraph.boolean-propagation.v1","spec_hash":"974a881acd81ce225e49b34a0ac387f105e2f57f7d8489c56d78d46c1e19356c","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":16,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-174f8e763bab2b2cb7b6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v0b\n- v04 --refutes--> v0a\n- v10 --supports--> v04\n- v18 --refutes--> v08\n- v17 --supports--> v09\n- v12 --refutes--> v14\n- v03 --supports--> v01\n- v0e --refutes--> v12\n- v15 --supports--> v11\n- v01 --refutes--> v05\n- v17 --refutes--> v02\n- v04 --supports--> v03\n- v12 --refutes--> v18\n- v08 --supports--> v15\n- v18 --supports--> v16\n- v10 --refutes--> v13\n- v15 --refutes--> v17\n- v0b --supports--> v0f\n- v02 --supports--> v10\n- v05 --supports--> v0c\n- v0e --refutes--> v07\n- v01 --refutes--> v0d\n- v00 --supports--> v0e\n- v0f --supports--> v00\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"446a708af6c673895c737a09958afca069bc50139e63ddea790526856b5869ad","schema":"livegraph.boolean-propagation.v1","spec_hash":"174f8e763bab2b2cb7b6d59449e9d91081c9520b90d53f78f22f84c7a60de2ac","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-9e48e61ca351ff6fe56c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v06\n- v02 --supports--> v16\n- v0b --supports--> v05\n- v17 --supports--> v10\n- v04 --supports--> v18\n- v06 --refutes--> v0c\n- v02 --refutes--> v04\n- v0f --supports--> v08\n- v06 --refutes--> v0d\n- v14 --supports--> v13\n- v09 --refutes--> v11\n- v00 --supports--> v09\n- v0c --supports--> v14\n- v13 --supports--> v0a\n- v12 --supports--> v02\n- v03 --refutes--> v15\n- v18 --supports--> v0f\n- v0c --refutes--> v07\n- v08 --refutes--> v01\n- v13 --refutes--> v0b\n- v10 --refutes--> v12\n- v08 --refutes--> v0e\n- v09 --supports--> v03\n- v08 --supports--> v00\n\n[VERIFIED ROOT]\nv17 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3e7f5f5aac7873a556da6c557115edd6ce661da5ab9b68197dcaf552b3241b96","schema":"livegraph.boolean-propagation.v1","spec_hash":"9e48e61ca351ff6fe56cd72921c247ce4a758cce7b2e8d0bd195102154347224","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v17\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-f0ca70a2460977e87fe1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v01\n- v07 --refutes--> v11\n- v01 --refutes--> v10\n- v12 --refutes--> v0c\n- v11 --refutes--> v09\n- v05 --refutes--> v06\n- v0c --supports--> v02\n- v15 --refutes--> v18\n- v0c --supports--> v14\n- v0b --refutes--> v03\n- v0e --refutes--> v00\n- v18 --supports--> v05\n- v05 --supports--> v0a\n- v0b --refutes--> v12\n- v10 --supports--> v07\n- v17 --supports--> v04\n- v14 --refutes--> v0e\n- v07 --supports--> v08\n- v01 --refutes--> v13\n- v00 --supports--> v0d\n- v07 --refutes--> v16\n- v0d --refutes--> v0f\n- v0d --refutes--> v15\n- v17 --refutes--> v0b\n\n[VERIFIED ROOT]\nv17 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3674f89b12d3bd4b0b12cfd65e6f0b5838d6dd1fd9eff5b18e8f21de63098b32","schema":"livegraph.boolean-propagation.v1","spec_hash":"f0ca70a2460977e87fe124a94cc27f7d9e800e3e13abcf40c0def2fcd3fa4f31","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v17\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-ccf29ac684630e1e1884","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v14\n- v02 --refutes--> v16\n- v0d --refutes--> v0b\n- v02 --supports--> v18\n- v11 --refutes--> v0e\n- v13 --refutes--> v0f\n- v17 --refutes--> v0c\n- v0a --refutes--> v09\n- v10 --supports--> v0a\n- v14 --supports--> v03\n- v11 --refutes--> v01\n- v18 --refutes--> v17\n- v02 --supports--> v00\n- v05 --refutes--> v0d\n- v04 --supports--> v10\n- v12 --refutes--> v04\n- v17 --supports--> v05\n- v01 --supports--> v02\n- v07 --supports--> v13\n- v14 --refutes--> v11\n- v0a --refutes--> v07\n- v04 --refutes--> v15\n- v0d --refutes--> v06\n- v0b --supports--> v08\n\n[VERIFIED ROOT]\nv12 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5badd12837374c64ca3b2f659db4fb32ca46b41fc3dd0392f1f22c858932da4e","schema":"livegraph.boolean-propagation.v1","spec_hash":"ccf29ac684630e1e18845a7e0fe2b012a24c56a9c611fe8f5f471fa993b8542a","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v12\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-759b183e61e194c14fe4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v05\n- v0f --refutes--> v10\n- v07 --refutes--> v0c\n- v10 --supports--> v0a\n- v0d --supports--> v0f\n- v11 --refutes--> v06\n- v12 --supports--> v11\n- v05 --supports--> v0b\n- v15 --supports--> v0e\n- v14 --refutes--> v08\n- v05 --supports--> v16\n- v13 --supports--> v0d\n- v0e --supports--> v00\n- v12 --supports--> v17\n- v01 --supports--> v18\n- v05 --supports--> v04\n- v10 --refutes--> v07\n- v06 --refutes--> v13\n- v18 --supports--> v03\n- v0b --supports--> v14\n- v0d --refutes--> v09\n- v14 --supports--> v01\n- v03 --refutes--> v12\n- v06 --refutes--> v02\n\n[VERIFIED ROOT]\nv15 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c2a92fd3ea1947958731c6b1faf35dd2d1f1e715ed0f81e1652a9f347a5443c0","schema":"livegraph.boolean-propagation.v1","spec_hash":"759b183e61e194c14fe405a691660a594b9a9cfa9cdc88d90e85fe64115ed321","spec_json":"{\"edges\":[{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v15\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-e205022e3a770df47d8d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v07\n- v0c --supports--> v0b\n- v0b --supports--> v01\n- v04 --refutes--> v0c\n- v10 --supports--> v18\n- v0f --refutes--> v02\n- v16 --supports--> v12\n- v01 --refutes--> v17\n- v16 --supports--> v04\n- v00 --supports--> v0a\n- v10 --supports--> v08\n- v14 --refutes--> v00\n- v0b --supports--> v05\n- v0e --refutes--> v15\n- v0c --supports--> v09\n- v0c --refutes--> v11\n- v0a --refutes--> v10\n- v0d --refutes--> v0f\n- v03 --refutes--> v0d\n- v02 --refutes--> v16\n- v01 --refutes--> v13\n- v15 --supports--> v03\n- v17 --supports--> v14\n- v14 --supports--> v06\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"da0ac82f165603d59b0ae18f8d4118d02b582ebcf9d4b93313ace5588a0a0777","schema":"livegraph.boolean-propagation.v1","spec_hash":"e205022e3a770df47d8d08271d82a8c4b482c93347dab4162caae20f25fdf47d","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-88825e1e5149e227ac77","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v09\n- v07 --supports--> v13\n- v00 --refutes--> v0d\n- v00 --refutes--> v14\n- v04 --refutes--> v12\n- v12 --supports--> v01\n- v0f --refutes--> v02\n- v08 --supports--> v00\n- v02 --refutes--> v18\n- v16 --supports--> v03\n- v0e --refutes--> v06\n- v04 --refutes--> v0b\n- v0d --supports--> v0e\n- v13 --supports--> v0c\n- v08 --supports--> v11\n- v12 --refutes--> v17\n- v0d --refutes--> v15\n- v16 --supports--> v0a\n- v0d --supports--> v05\n- v17 --refutes--> v08\n- v10 --supports--> v07\n- v0a --supports--> v04\n- v06 --supports--> v0f\n- v0c --refutes--> v16\n\n[VERIFIED ROOT]\nv10 = TRUE\n\n[TARGET]\nv18\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b6c86f7f8d24aa76a61e2b9904e93af35bddbbc60adcdca9353b97c6472a4bfc","schema":"livegraph.boolean-propagation.v1","spec_hash":"88825e1e5149e227ac77dced2b5a7ad7ee2a9597366ae82fd021d867aefb636f","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v18\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-77d78d729489d19bc257","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v16\n- v0d --supports--> v08\n- v18 --refutes--> v17\n- v00 --refutes--> v0b\n- v0c --refutes--> v0a\n- v16 --supports--> v14\n- v15 --supports--> v07\n- v17 --supports--> v15\n- v09 --refutes--> v04\n- v16 --refutes--> v12\n- v0a --refutes--> v13\n- v0a --supports--> v0f\n- v0c --supports--> v03\n- v00 --supports--> v05\n- v01 --refutes--> v02\n- v0b --supports--> v10\n- v09 --refutes--> v06\n- v18 --refutes--> v0e\n- v02 --supports--> v18\n- v04 --refutes--> v0d\n- v0f --supports--> v09\n- v14 --supports--> v0c\n- v0a --refutes--> v11\n- v0b --supports--> v01\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"04b9f55121f449ddb4d8810a85057b19b4a88aa54f809668ef2c08bae2ed01d1","schema":"livegraph.boolean-propagation.v1","spec_hash":"77d78d729489d19bc257457d23a659506c3f6c8b3e5c3cfb8f9e3473822a5dc7","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-af775262d7331f76f139","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v0e\n- v13 --supports--> v11\n- v14 --supports--> v07\n- v00 --supports--> v14\n- v00 --refutes--> v17\n- v12 --supports--> v10\n- v02 --supports--> v18\n- v0b --refutes--> v09\n- v13 --refutes--> v0c\n- v07 --refutes--> v06\n- v12 --refutes--> v16\n- v08 --refutes--> v0d\n- v15 --refutes--> v0f\n- v0a --supports--> v00\n- v02 --supports--> v0b\n- v09 --supports--> v13\n- v06 --supports--> v01\n- v0c --refutes--> v03\n- v0f --refutes--> v12\n- v12 --supports--> v05\n- v06 --refutes--> v04\n- v03 --refutes--> v08\n- v05 --refutes--> v02\n- v08 --supports--> v0a\n\n[VERIFIED ROOT]\nv15 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3c531ac471b192c9df0a5d0ce44c375d4a2922c4dda8136062ee181472cf6918","schema":"livegraph.boolean-propagation.v1","spec_hash":"af775262d7331f76f139e21ad22def4c60f3f1a5ca10740078548ee6d5347aba","spec_json":"{\"edges\":[{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":16,\"root\":\"v15\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-d5493a31169e53058ca4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v0a\n- v06 --refutes--> v15\n- v02 --refutes--> v18\n- v0c --refutes--> v0d\n- v02 --supports--> v05\n- v0b --supports--> v17\n- v14 --refutes--> v02\n- v13 --supports--> v08\n- v0e --refutes--> v12\n- v10 --supports--> v07\n- v08 --supports--> v0b\n- v15 --supports--> v0c\n- v0e --refutes--> v13\n- v07 --supports--> v06\n- v0d --refutes--> v03\n- v04 --supports--> v16\n- v0a --supports--> v10\n- v04 --refutes--> v09\n- v05 --refutes--> v04\n- v10 --refutes--> v11\n- v04 --supports--> v01\n- v10 --refutes--> v0f\n- v16 --refutes--> v0e\n- v14 --refutes--> v00\n\n[VERIFIED ROOT]\nv14 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2732ed0f5d5bb6b16d71cfbc1eea8bb4782237c3e92b4d8f7753e28e794439be","schema":"livegraph.boolean-propagation.v1","spec_hash":"d5493a31169e53058ca4601d21dafba1bef9a25bfdcf8492b3cb148701d22090","spec_json":"{\"edges\":[{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":16,\"root\":\"v14\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-ec4bdf15cf35ae594ff2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v00\n- v18 --refutes--> v0c\n- v15 --refutes--> v06\n- v0e --refutes--> v17\n- v14 --refutes--> v08\n- v02 --supports--> v0a\n- v05 --refutes--> v0e\n- v17 --supports--> v18\n- v0e --refutes--> v0f\n- v0b --supports--> v14\n- v11 --supports--> v02\n- v14 --supports--> v03\n- v08 --refutes--> v13\n- v06 --supports--> v11\n- v18 --supports--> v0d\n- v0d --supports--> v12\n- v0a --refutes--> v0b\n- v0e --supports--> v07\n- v0d --refutes--> v10\n- v13 --refutes--> v09\n- v01 --supports--> v05\n- v18 --supports--> v04\n- v0a --refutes--> v16\n- v09 --supports--> v01\n\n[VERIFIED ROOT]\nv15 = FALSE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"212b2951dc33abd4b58059382e830430b9d309a2fb547a83a7007dac5243ff1a","schema":"livegraph.boolean-propagation.v1","spec_hash":"ec4bdf15cf35ae594ff22ab871e2a6963e8a24a80f1a6893816ef2f3564ba905","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v15\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-820c8e25f2b11cf7334b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v00\n- v03 --refutes--> v02\n- v0e --supports--> v17\n- v14 --refutes--> v0e\n- v0d --supports--> v14\n- v01 --refutes--> v09\n- v11 --supports--> v10\n- v10 --supports--> v13\n- v0c --supports--> v03\n- v14 --supports--> v06\n- v10 --refutes--> v12\n- v01 --supports--> v08\n- v05 --refutes--> v07\n- v09 --supports--> v0c\n- v13 --refutes--> v0d\n- v18 --refutes--> v05\n- v17 --supports--> v15\n- v03 --supports--> v0f\n- v0a --refutes--> v11\n- v0d --supports--> v0b\n- v0c --refutes--> v16\n- v00 --supports--> v0a\n- v05 --refutes--> v04\n- v17 --supports--> v01\n\n[VERIFIED ROOT]\nv18 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1892cba8575977c34f290e6db6aeaa12450e0b222d8a96274708a1ebb4fcded9","schema":"livegraph.boolean-propagation.v1","spec_hash":"820c8e25f2b11cf7334bd6a8fd1218c910366d902296d2a453afd70e2622aec6","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v18\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-0e39b653d70a85645948","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v05\n- v02 --refutes--> v0a\n- v0c --supports--> v08\n- v04 --refutes--> v0c\n- v00 --refutes--> v03\n- v15 --refutes--> v0e\n- v14 --refutes--> v10\n- v15 --refutes--> v13\n- v13 --refutes--> v02\n- v0d --refutes--> v07\n- v15 --supports--> v18\n- v06 --supports--> v0d\n- v05 --refutes--> v17\n- v11 --supports--> v16\n- v04 --refutes--> v09\n- v10 --supports--> v11\n- v00 --refutes--> v06\n- v08 --refutes--> v12\n- v02 --refutes--> v14\n- v00 --supports--> v0b\n- v08 --refutes--> v0f\n- v17 --supports--> v00\n- v14 --refutes--> v01\n- v07 --supports--> v15\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv16\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a2425f0d1e5459334a794f2cc55edd6dd9bddba1d86293766e0248d07f2c7fb3","schema":"livegraph.boolean-propagation.v1","spec_hash":"0e39b653d70a8564594808e287bde39e04c23d4237025ca8000e02dc200517b1","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v16\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-a57e83f157b57f469017","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --supports--> v18\n- v11 --supports--> v07\n- v12 --supports--> v0f\n- v18 --refutes--> v03\n- v0f --supports--> v04\n- v11 --supports--> v05\n- v18 --refutes--> v09\n- v10 --supports--> v14\n- v0f --refutes--> v0e\n- v03 --supports--> v0a\n- v0d --refutes--> v13\n- v15 --supports--> v0c\n- v0a --supports--> v11\n- v02 --refutes--> v0d\n- v16 --refutes--> v15\n- v16 --supports--> v01\n- v14 --refutes--> v12\n- v0c --refutes--> v06\n- v02 --supports--> v08\n- v15 --refutes--> v17\n- v00 --refutes--> v02\n- v0c --refutes--> v10\n- v04 --supports--> v00\n- v11 --supports--> v0b\n\n[VERIFIED ROOT]\nv16 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6241e78b34494035ab3e339f87e69300541ab774da0eac2c5b98c066f5da804a","schema":"livegraph.boolean-propagation.v1","spec_hash":"a57e83f157b57f46901719ff8faa1feb6425acbb540d8dab01c47e009dc44094","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v16\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-023bd88fc75a1b532eba","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v16\n- v0f --supports--> v13\n- v11 --supports--> v18\n- v11 --supports--> v17\n- v09 --supports--> v02\n- v00 --supports--> v0c\n- v11 --refutes--> v08\n- v01 --supports--> v09\n- v03 --supports--> v0f\n- v10 --supports--> v07\n- v0e --supports--> v00\n- v16 --supports--> v06\n- v06 --refutes--> v0e\n- v05 --supports--> v15\n- v0f --refutes--> v0d\n- v0c --refutes--> v11\n- v15 --refutes--> v12\n- v0b --supports--> v14\n- v10 --supports--> v03\n- v11 --refutes--> v0b\n- v11 --refutes--> v04\n- v0d --refutes--> v01\n- v0b --supports--> v10\n- v10 --refutes--> v0a\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f216adb7e0a888f71ee1d69e8f4ade6610f7675a85728a86d6b1893212b30502","schema":"livegraph.boolean-propagation.v1","spec_hash":"023bd88fc75a1b532eba2e37f55dcb1e0b4a993892837c562ab7e6cf189a5f21","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":16,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-10383f998328894201b0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v01\n- v13 --refutes--> v0b\n- v0c --refutes--> v12\n- v10 --refutes--> v11\n- v01 --refutes--> v03\n- v11 --supports--> v15\n- v17 --supports--> v00\n- v03 --refutes--> v06\n- v0c --refutes--> v0e\n- v02 --supports--> v08\n- v0d --refutes--> v17\n- v11 --refutes--> v13\n- v02 --refutes--> v0d\n- v0f --refutes--> v10\n- v12 --refutes--> v05\n- v17 --refutes--> v09\n- v13 --refutes--> v02\n- v12 --refutes--> v18\n- v03 --refutes--> v14\n- v18 --refutes--> v16\n- v14 --refutes--> v0f\n- v07 --supports--> v04\n- v09 --refutes--> v0c\n- v07 --supports--> v0a\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv16\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5da2dd968e2ad060bc671f8c458f89c6dd56f25cf1bac84a0ac30cd0fb80fc34","schema":"livegraph.boolean-propagation.v1","spec_hash":"10383f998328894201b0ff1ab5a58bef7a5f387540879fb65fea419dcfa1e425","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v16\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-b761522b2d5126767143","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v06\n- v14 --refutes--> v17\n- v00 --refutes--> v18\n- v07 --supports--> v10\n- v0e --supports--> v02\n- v11 --supports--> v08\n- v00 --supports--> v0c\n- v11 --refutes--> v13\n- v0f --refutes--> v0d\n- v08 --supports--> v05\n- v16 --refutes--> v0e\n- v0a --supports--> v07\n- v18 --refutes--> v03\n- v14 --supports--> v04\n- v02 --refutes--> v01\n- v08 --refutes--> v09\n- v08 --refutes--> v0b\n- v17 --refutes--> v16\n- v0d --refutes--> v0a\n- v08 --refutes--> v12\n- v03 --refutes--> v0f\n- v0a --supports--> v15\n- v0b --refutes--> v14\n- v01 --supports--> v00\n\n[VERIFIED ROOT]\nv11 = FALSE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"705a7059038cc5021babb4cd5539e53004e55b0cfdb31ab9d958e511495f3d04","schema":"livegraph.boolean-propagation.v1","spec_hash":"b761522b2d512676714342352767b1318957345771973fae8af01e8cefd0ed21","spec_json":"{\"edges\":[{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-422adcc27b639246f461","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v05\n- v03 --refutes--> v17\n- v18 --supports--> v07\n- v00 --refutes--> v16\n- v17 --refutes--> v08\n- v0b --supports--> v15\n- v01 --supports--> v0c\n- v02 --supports--> v12\n- v0d --supports--> v02\n- v08 --refutes--> v09\n- v18 --refutes--> v14\n- v05 --refutes--> v0d\n- v01 --refutes--> v0f\n- v12 --supports--> v13\n- v0d --supports--> v06\n- v11 --refutes--> v0a\n- v17 --refutes--> v04\n- v15 --refutes--> v18\n- v09 --refutes--> v0e\n- v11 --refutes--> v10\n- v0f --supports--> v0b\n- v00 --supports--> v03\n- v0a --supports--> v01\n- v12 --supports--> v11\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b0fb2ace7b9574652ca50be5d551b416057fd01e2892cbdbe870cc1f038032f2","schema":"livegraph.boolean-propagation.v1","spec_hash":"422adcc27b639246f461272f23f3d0b4e86a51f1fb15731295278e832eb12e6c","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-064ae338a69b1931c7df","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v02\n- v13 --refutes--> v0f\n- v0a --refutes--> v00\n- v0d --supports--> v15\n- v15 --refutes--> v0c\n- v12 --refutes--> v14\n- v0b --refutes--> v08\n- v09 --refutes--> v18\n- v06 --refutes--> v09\n- v12 --supports--> v04\n- v10 --refutes--> v11\n- v0b --supports--> v17\n- v10 --refutes--> v0d\n- v05 --refutes--> v07\n- v0c --supports--> v05\n- v0a --refutes--> v01\n- v0b --supports--> v0a\n- v04 --supports--> v0e\n- v0f --supports--> v03\n- v00 --supports--> v10\n- v03 --supports--> v12\n- v0c --supports--> v16\n- v05 --supports--> v06\n- v18 --supports--> v13\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"eff2eb1475603bfe12a8517d16a2c658b15b39dc854c2085cd3c8b6834b05947","schema":"livegraph.boolean-propagation.v1","spec_hash":"064ae338a69b1931c7df60c4782ad955ba48eb3262c5bd41ddc4137794cc5cb1","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-af9621b7dbcac28edff9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --supports--> v06\n- v18 --refutes--> v01\n- v04 --refutes--> v0e\n- v02 --refutes--> v00\n- v03 --refutes--> v12\n- v0c --supports--> v03\n- v05 --refutes--> v18\n- v10 --refutes--> v11\n- v03 --refutes--> v08\n- v02 --refutes--> v0c\n- v07 --refutes--> v02\n- v0d --supports--> v0f\n- v16 --supports--> v10\n- v17 --supports--> v09\n- v09 --supports--> v0b\n- v13 --refutes--> v15\n- v0e --refutes--> v16\n- v11 --supports--> v17\n- v09 --refutes--> v13\n- v02 --refutes--> v0a\n- v0f --refutes--> v14\n- v18 --refutes--> v0d\n- v0f --refutes--> v07\n- v03 --refutes--> v04\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv15\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5e2488f1ecec116047dd81324543e2cb1af415400ba52f01adde7a6db96da4f1","schema":"livegraph.boolean-propagation.v1","spec_hash":"af9621b7dbcac28edff90abc0aeeea0845042edea281b8aeb6cda0adee01d5e3","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-f63bacc57d95291209a2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v03\n- v0e --supports--> v16\n- v0c --refutes--> v15\n- v01 --refutes--> v0b\n- v04 --supports--> v0c\n- v0d --refutes--> v06\n- v0f --refutes--> v05\n- v03 --supports--> v12\n- v09 --refutes--> v13\n- v18 --supports--> v0a\n- v18 --refutes--> v10\n- v14 --supports--> v0f\n- v0e --supports--> v09\n- v08 --supports--> v14\n- v11 --refutes--> v0e\n- v0c --supports--> v0d\n- v0f --refutes--> v18\n- v13 --supports--> v07\n- v03 --refutes--> v04\n- v01 --refutes--> v02\n- v14 --supports--> v00\n- v10 --refutes--> v01\n- v02 --supports--> v11\n- v14 --refutes--> v17\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c04b4af404eb7399f883cae68562c21a39fe81ef91bbb5ab580ab0fe1ab9894f","schema":"livegraph.boolean-propagation.v1","spec_hash":"f63bacc57d95291209a28daf81dc9e5074f331266d2faa6a4c24efc4ae120053","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-e04c36c27a64f24e399e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v06\n- v18 --supports--> v13\n- v16 --supports--> v0a\n- v10 --supports--> v16\n- v0a --supports--> v15\n- v14 --refutes--> v08\n- v0f --supports--> v07\n- v16 --refutes--> v02\n- v03 --refutes--> v18\n- v00 --supports--> v0e\n- v10 --supports--> v11\n- v01 --supports--> v17\n- v15 --refutes--> v0c\n- v00 --refutes--> v09\n- v0c --refutes--> v0f\n- v06 --refutes--> v12\n- v15 --refutes--> v04\n- v07 --refutes--> v01\n- v17 --supports--> v00\n- v0c --supports--> v05\n- v14 --supports--> v0d\n- v10 --refutes--> v0b\n- v12 --supports--> v14\n- v18 --refutes--> v10\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f104dc801cfe87f6ec659bd963c24a7c277b606a464acd946660e0d355edd402","schema":"livegraph.boolean-propagation.v1","spec_hash":"e04c36c27a64f24e399ef77cc65205da16a96dfaf783b1bbd74c28dadaabead0","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-0958ed65af06f54975f3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v05\n- v13 --refutes--> v02\n- v09 --refutes--> v11\n- v0a --supports--> v17\n- v03 --supports--> v14\n- v16 --supports--> v07\n- v0b --supports--> v04\n- v11 --supports--> v0a\n- v0b --refutes--> v0d\n- v12 --supports--> v08\n- v12 --supports--> v18\n- v15 --refutes--> v09\n- v17 --refutes--> v0f\n- v06 --supports--> v16\n- v00 --supports--> v0b\n- v14 --supports--> v00\n- v14 --supports--> v0e\n- v03 --refutes--> v10\n- v13 --supports--> v06\n- v16 --refutes--> v01\n- v17 --refutes--> v12\n- v18 --supports--> v03\n- v01 --supports--> v15\n- v0a --supports--> v0c\n\n[VERIFIED ROOT]\nv13 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"34e4cafe05e21f0379e8d9049b13cdc302acd108f95cd1af30a32e33486e1b17","schema":"livegraph.boolean-propagation.v1","spec_hash":"0958ed65af06f54975f3e2a1804ca451aa3332d3eb76b45706ee46fa5ce48644","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v13\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-dbf727b2e9e813a6c6a3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v02\n- v10 --refutes--> v16\n- v0d --supports--> v07\n- v0d --supports--> v01\n- v15 --refutes--> v0e\n- v04 --refutes--> v0f\n- v0f --supports--> v09\n- v04 --refutes--> v18\n- v12 --refutes--> v0c\n- v13 --refutes--> v04\n- v14 --refutes--> v03\n- v08 --supports--> v13\n- v14 --refutes--> v0a\n- v14 --supports--> v05\n- v10 --refutes--> v00\n- v03 --refutes--> v06\n- v0e --refutes--> v14\n- v0d --supports--> v12\n- v11 --supports--> v17\n- v12 --refutes--> v08\n- v02 --supports--> v0b\n- v09 --refutes--> v11\n- v11 --supports--> v10\n- v0b --supports--> v0d\n\n[VERIFIED ROOT]\nv15 = FALSE\n\n[TARGET]\nv16\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"479ba5328e07d55cd7f9c43484070c3d2af86acc0de208b16f8bc4e8af92f302","schema":"livegraph.boolean-propagation.v1","spec_hash":"dbf727b2e9e813a6c6a3bc242ca6bc0c0761124089ba84ec14df0e192ed311d5","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":16,\"root\":\"v15\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v16\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-e60f2a4fb65cd6120123","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v11\n- v17 --refutes--> v0e\n- v17 --supports--> v06\n- v15 --supports--> v04\n- v18 --refutes--> v0a\n- v16 --refutes--> v0d\n- v15 --refutes--> v02\n- v16 --refutes--> v0b\n- v03 --refutes--> v09\n- v14 --supports--> v03\n- v13 --refutes--> v05\n- v06 --supports--> v13\n- v09 --refutes--> v00\n- v00 --refutes--> v01\n- v04 --supports--> v08\n- v04 --refutes--> v07\n- v15 --supports--> v10\n- v0b --supports--> v14\n- v01 --supports--> v15\n- v14 --supports--> v0c\n- v18 --refutes--> v12\n- v12 --supports--> v16\n- v0f --refutes--> v18\n- v05 --supports--> v0f\n\n[VERIFIED ROOT]\nv17 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"99669485bf1b3e4217c8ddb041eed226d2bc897db90be9405a4e6e55db26b593","schema":"livegraph.boolean-propagation.v1","spec_hash":"e60f2a4fb65cd6120123cd392968c556cd8eaf6090e8004702f5fad09a1c2c76","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v17\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-dd5d2815f6508fde24f3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v16 --refutes--> v10\n- v03 --refutes--> v08\n- v0e --supports--> v16\n- v02 --refutes--> v15\n- v13 --refutes--> v04\n- v09 --refutes--> v07\n- v15 --supports--> v01\n- v13 --refutes--> v03\n- v06 --supports--> v0f\n- v05 --refutes--> v0a\n- v0d --refutes--> v18\n- v03 --supports--> v14\n- v09 --supports--> v13\n- v10 --supports--> v09\n- v0a --supports--> v12\n- v12 --refutes--> v00\n- v00 --supports--> v06\n- v13 --supports--> v17\n- v01 --refutes--> v05\n- v06 --refutes--> v0e\n- v05 --supports--> v0c\n- v08 --supports--> v0d\n- v0d --refutes--> v11\n- v0d --supports--> v0b\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"771ccc14ffe6fd094a84e4284dc6e89a80707dca3d4f3986703b080ad9df7d47","schema":"livegraph.boolean-propagation.v1","spec_hash":"dd5d2815f6508fde24f37ecb9b5751d1008d76d3a40617af2079751de057ff29","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":16,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-a3bd290feb720702c4a6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v13\n- v18 --refutes--> v0e\n- v15 --supports--> v11\n- v12 --refutes--> v02\n- v00 --supports--> v17\n- v07 --refutes--> v03\n- v07 --refutes--> v0b\n- v17 --refutes--> v07\n- v10 --refutes--> v0a\n- v0c --supports--> v00\n- v00 --refutes--> v09\n- v11 --supports--> v12\n- v12 --supports--> v01\n- v0f --supports--> v04\n- v0d --refutes--> v08\n- v04 --refutes--> v18\n- v0e --refutes--> v10\n- v07 --refutes--> v15\n- v02 --refutes--> v16\n- v0c --refutes--> v14\n- v08 --supports--> v05\n- v16 --refutes--> v0f\n- v10 --supports--> v0d\n- v00 --refutes--> v06\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e42bf9e1442453dc4139e743e948234ab099d47bc0eabc1f0abca3193da8675f","schema":"livegraph.boolean-propagation.v1","spec_hash":"a3bd290feb720702c4a60ad93b9203b6660838610fffbe35b5afae086b8f208c","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-64a6d9306fb6fb52e855","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v05\n- v10 --supports--> v0c\n- v18 --refutes--> v0f\n- v17 --refutes--> v16\n- v17 --refutes--> v00\n- v03 --refutes--> v0b\n- v13 --refutes--> v0d\n- v11 --refutes--> v14\n- v17 --refutes--> v02\n- v0e --refutes--> v08\n- v13 --supports--> v12\n- v04 --supports--> v18\n- v15 --refutes--> v04\n- v02 --refutes--> v06\n- v01 --refutes--> v07\n- v02 --supports--> v03\n- v14 --supports--> v13\n- v03 --supports--> v0a\n- v18 --refutes--> v10\n- v0c --supports--> v01\n- v01 --refutes--> v11\n- v04 --supports--> v09\n- v08 --refutes--> v17\n- v0b --refutes--> v15\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d30285fae3e6bcba53de469f27f998f4eef19185f4f6b248f0761f387fac4ba2","schema":"livegraph.boolean-propagation.v1","spec_hash":"64a6d9306fb6fb52e855ed4103564af161b78efba7b83aa506e567869643f5b4","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-a820dd6d7e2e38c31077","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --supports--> v07\n- v18 --refutes--> v0d\n- v06 --refutes--> v14\n- v14 --supports--> v18\n- v08 --refutes--> v05\n- v10 --supports--> v17\n- v13 --supports--> v12\n- v03 --refutes--> v13\n- v05 --supports--> v16\n- v05 --refutes--> v0c\n- v04 --supports--> v01\n- v08 --refutes--> v0b\n- v0e --supports--> v10\n- v03 --supports--> v0f\n- v18 --refutes--> v08\n- v10 --supports--> v11\n- v0c --supports--> v04\n- v07 --refutes--> v15\n- v09 --refutes--> v02\n- v15 --supports--> v06\n- v07 --refutes--> v0a\n- v11 --supports--> v09\n- v13 --refutes--> v00\n- v04 --supports--> v0e\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2467ca2c34d4511047d8c3ba1a2f2cc093477cee3ef3e213c3f821debc9b4b43","schema":"livegraph.boolean-propagation.v1","spec_hash":"a820dd6d7e2e38c310772e46bdfb3371e62daffffae69f359cfddf407d5e64e1","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-7f39c5862121db380344","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v08\n- v0f --refutes--> v00\n- v00 --supports--> v04\n- v14 --refutes--> v05\n- v12 --supports--> v0d\n- v00 --supports--> v09\n- v10 --refutes--> v11\n- v06 --refutes--> v14\n- v06 --supports--> v07\n- v10 --supports--> v17\n- v0e --supports--> v12\n- v16 --refutes--> v0f\n- v05 --refutes--> v02\n- v12 --supports--> v01\n- v0d --refutes--> v16\n- v0b --refutes--> v0e\n- v0c --supports--> v15\n- v08 --supports--> v18\n- v10 --supports--> v03\n- v02 --refutes--> v13\n- v09 --refutes--> v0c\n- v10 --supports--> v0b\n- v02 --refutes--> v0a\n- v0a --supports--> v10\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv18\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d07dd21c75ec9994bfb4ac68aee4a89e08290fbd51fc529cc9723f318c565dc4","schema":"livegraph.boolean-propagation.v1","spec_hash":"7f39c5862121db380344c62e30d3bf1a80443f7e01fb772b976d23d623c7d37d","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":16,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v18\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-87a912aaa3ce55e08bb8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --supports--> v12\n- v06 --supports--> v15\n- v16 --refutes--> v10\n- v0a --refutes--> v17\n- v0e --refutes--> v13\n- v05 --refutes--> v00\n- v11 --refutes--> v04\n- v09 --refutes--> v05\n- v17 --refutes--> v11\n- v02 --refutes--> v06\n- v0e --supports--> v0b\n- v06 --refutes--> v0e\n- v10 --refutes--> v14\n- v0c --refutes--> v0a\n- v17 --refutes--> v01\n- v0b --refutes--> v18\n- v0b --supports--> v09\n- v05 --supports--> v16\n- v14 --refutes--> v0c\n- v04 --supports--> v0d\n- v0d --refutes--> v0f\n- v0f --supports--> v07\n- v0d --refutes--> v03\n- v05 --supports--> v08\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f85cfc52856c1f41c0ee8c9fc01b2b04669f254136ef3e86e8698c9b6620e5bf","schema":"livegraph.boolean-propagation.v1","spec_hash":"87a912aaa3ce55e08bb850038705fb71c51eabff283679fe2dc62ed3196436fe","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-1fed7e8463b7111befc0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v0b\n- v09 --supports--> v15\n- v16 --supports--> v02\n- v12 --supports--> v0f\n- v10 --supports--> v0d\n- v0a --supports--> v16\n- v03 --supports--> v11\n- v18 --supports--> v0a\n- v10 --supports--> v01\n- v15 --refutes--> v06\n- v0e --refutes--> v0c\n- v06 --refutes--> v08\n- v06 --supports--> v17\n- v07 --refutes--> v03\n- v11 --refutes--> v12\n- v00 --supports--> v18\n- v0b --refutes--> v07\n- v18 --refutes--> v05\n- v10 --refutes--> v00\n- v02 --refutes--> v0e\n- v11 --refutes--> v14\n- v0b --supports--> v13\n- v03 --supports--> v04\n- v06 --refutes--> v10\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"be5fea27fa83410a02e67cc7b350f5486e7ab6a49f2f782e90c6868301504154","schema":"livegraph.boolean-propagation.v1","spec_hash":"1fed7e8463b7111befc01e8ab81bd0070c26bbbc76215abab24b5d3c1354e484","spec_json":"{\"edges\":[{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-4ad4b6f13daf376cf37a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v14\n- v0b --supports--> v01\n- v0c --refutes--> v0b\n- v04 --refutes--> v06\n- v13 --supports--> v0c\n- v09 --supports--> v07\n- v18 --supports--> v02\n- v12 --supports--> v11\n- v12 --refutes--> v10\n- v00 --refutes--> v0e\n- v0d --refutes--> v18\n- v16 --refutes--> v15\n- v17 --supports--> v16\n- v12 --refutes--> v03\n- v14 --supports--> v0a\n- v02 --supports--> v12\n- v16 --refutes--> v0f\n- v07 --refutes--> v04\n- v14 --supports--> v00\n- v00 --supports--> v17\n- v06 --refutes--> v0d\n- v03 --supports--> v13\n- v06 --refutes--> v05\n- v04 --refutes--> v08\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv15\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9fb14b8b6607c912f8d3c25ef836e072a05a4dccaf7503bc5a0b4d45932edd44","schema":"livegraph.boolean-propagation.v1","spec_hash":"4ad4b6f13daf376cf37a6c0f875493391fb63346c16443fbe8c36f71c9689d69","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-6374e6cd2c0388e9ed9e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v08\n- v18 --supports--> v06\n- v14 --refutes--> v0c\n- v12 --refutes--> v15\n- v13 --refutes--> v18\n- v01 --supports--> v14\n- v0d --refutes--> v12\n- v03 --supports--> v02\n- v06 --refutes--> v07\n- v05 --supports--> v03\n- v0a --supports--> v10\n- v08 --supports--> v16\n- v0e --supports--> v11\n- v03 --supports--> v0a\n- v0e --supports--> v13\n- v08 --supports--> v01\n- v03 --supports--> v17\n- v07 --supports--> v00\n- v0b --refutes--> v09\n- v0e --supports--> v0f\n- v12 --supports--> v04\n- v09 --supports--> v05\n- v12 --supports--> v0b\n- v0c --supports--> v0e\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c6a8dbd03930bc0de7935bfd20de2a2d6678035cebe625b3004c77bfdc735403","schema":"livegraph.boolean-propagation.v1","spec_hash":"6374e6cd2c0388e9ed9ef6a380ef92afbf089272d85882ba57089a1873fdf98f","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-756d966b22cd15b90436","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v02\n- v11 --supports--> v0b\n- v16 --refutes--> v00\n- v0e --supports--> v10\n- v0a --supports--> v0d\n- v0c --supports--> v08\n- v13 --refutes--> v11\n- v13 --supports--> v0f\n- v03 --refutes--> v0e\n- v02 --refutes--> v17\n- v02 --refutes--> v0c\n- v12 --refutes--> v15\n- v12 --refutes--> v14\n- v18 --supports--> v07\n- v0d --supports--> v01\n- v12 --supports--> v06\n- v05 --supports--> v09\n- v0d --supports--> v18\n- v02 --supports--> v04\n- v18 --refutes--> v16\n- v09 --refutes--> v0a\n- v08 --supports--> v05\n- v00 --refutes--> v03\n- v10 --supports--> v13\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0fb822126d60b8854323a5ce08905839cf9eddd9181e8b2ff06c8718db5eefba","schema":"livegraph.boolean-propagation.v1","spec_hash":"756d966b22cd15b90436b80e7a27fb719519b5b9dc5496e151288972820a6a9d","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-0e6f489d12a9acbe1a22","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v00\n- v15 --refutes--> v08\n- v0c --refutes--> v0a\n- v0c --refutes--> v06\n- v0a --supports--> v02\n- v0f --refutes--> v01\n- v12 --supports--> v11\n- v17 --refutes--> v12\n- v12 --refutes--> v18\n- v07 --refutes--> v0b\n- v17 --refutes--> v13\n- v02 --refutes--> v03\n- v0b --supports--> v0e\n- v10 --supports--> v17\n- v04 --refutes--> v14\n- v14 --refutes--> v07\n- v04 --refutes--> v0d\n- v12 --refutes--> v0f\n- v0e --refutes--> v10\n- v02 --supports--> v05\n- v05 --refutes--> v04\n- v14 --refutes--> v16\n- v08 --supports--> v0c\n- v15 --supports--> v09\n\n[VERIFIED ROOT]\nv15 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5c1cd40ddab8a7a64b36ab31624d733ceabb9c5b21aa8ac8d472bc492341513f","schema":"livegraph.boolean-propagation.v1","spec_hash":"0e6f489d12a9acbe1a2298321a4c45dacb1ddc22bdef1b8e524e545703c34e14","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v15\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-3c405be81fa591823c32","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --refutes--> v07\n- v14 --supports--> v0b\n- v17 --refutes--> v09\n- v18 --refutes--> v13\n- v15 --supports--> v12\n- v13 --refutes--> v14\n- v16 --refutes--> v04\n- v00 --supports--> v06\n- v0f --supports--> v02\n- v10 --supports--> v0c\n- v0d --refutes--> v16\n- v0b --supports--> v00\n- v0f --refutes--> v01\n- v0f --supports--> v05\n- v02 --refutes--> v0e\n- v17 --refutes--> v10\n- v0f --refutes--> v03\n- v08 --supports--> v17\n- v02 --supports--> v0a\n- v0a --supports--> v08\n- v16 --refutes--> v11\n- v11 --supports--> v18\n- v15 --supports--> v0f\n- v06 --supports--> v15\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b132dc5d69ad2b27df89a49492f528adb49ae8d7ec44bfe4028d97fa3e57c9b8","schema":"livegraph.boolean-propagation.v1","spec_hash":"3c405be81fa591823c325cd847ca19aac7bf2648924fd7dd398ad46bd13f4a1e","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-7c111108860e2d083d71","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v17\n- v08 --supports--> v07\n- v11 --refutes--> v0b\n- v0f --supports--> v14\n- v0c --refutes--> v09\n- v0a --refutes--> v0f\n- v08 --refutes--> v06\n- v11 --supports--> v15\n- v06 --supports--> v11\n- v11 --supports--> v01\n- v0f --supports--> v02\n- v17 --supports--> v10\n- v05 --refutes--> v12\n- v0e --refutes--> v0d\n- v05 --refutes--> v13\n- v0f --refutes--> v08\n- v01 --refutes--> v0c\n- v16 --supports--> v18\n- v10 --refutes--> v0a\n- v03 --supports--> v0e\n- v16 --refutes--> v05\n- v01 --refutes--> v04\n- v13 --refutes--> v03\n- v09 --supports--> v16\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"88e0f1c4d28217cb332434fb98fe9721510b1978025d6ae7a8fee1c3c3645da9","schema":"livegraph.boolean-propagation.v1","spec_hash":"7c111108860e2d083d711f55e8649cba71ddd70a318bc4d6fc384ab3fc893ed5","spec_json":"{\"edges\":[{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-40e18a94a08d79969ffa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v12\n- v0f --refutes--> v10\n- v02 --refutes--> v0b\n- v0d --refutes--> v08\n- v09 --refutes--> v07\n- v08 --refutes--> v13\n- v03 --refutes--> v0d\n- v16 --refutes--> v14\n- v10 --refutes--> v02\n- v17 --supports--> v05\n- v0b --supports--> v0c\n- v18 --refutes--> v0a\n- v0a --refutes--> v0e\n- v04 --refutes--> v00\n- v05 --supports--> v09\n- v05 --refutes--> v15\n- v16 --supports--> v11\n- v08 --refutes--> v04\n- v16 --supports--> v01\n- v08 --refutes--> v06\n- v0c --refutes--> v03\n- v0a --supports--> v17\n- v07 --refutes--> v16\n- v04 --refutes--> v18\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv14\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0dba767ac59e64ab3f138f2fa08ccee960e695ef9e43fc1b3cca28f87c0bb2f5","schema":"livegraph.boolean-propagation.v1","spec_hash":"40e18a94a08d79969ffa3c7e87f54ea15b29cb08c6dc9395c27585f0c4159608","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-44d2487dffdea389b9c0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v01\n- v0b --refutes--> v14\n- v0d --supports--> v0f\n- v0d --supports--> v0e\n- v16 --refutes--> v07\n- v02 --supports--> v05\n- v07 --refutes--> v02\n- v04 --supports--> v17\n- v0c --supports--> v10\n- v18 --supports--> v12\n- v0d --supports--> v00\n- v02 --supports--> v15\n- v09 --refutes--> v16\n- v0d --refutes--> v0c\n- v08 --supports--> v0b\n- v11 --supports--> v08\n- v06 --refutes--> v09\n- v13 --refutes--> v18\n- v11 --refutes--> v0a\n- v12 --supports--> v04\n- v05 --refutes--> v11\n- v10 --refutes--> v06\n- v04 --refutes--> v0d\n- v05 --refutes--> v03\n\n[VERIFIED ROOT]\nv13 = FALSE\n\n[TARGET]\nv14\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"158e75492efd331f019e54f3b70b58d790fbf0a8fcfb0c57d78eb58e8d8a1581","schema":"livegraph.boolean-propagation.v1","spec_hash":"44d2487dffdea389b9c00e88b813128d563b2a70bf78fed7107634c716eca942","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v13\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-f6cee4a4b5009187b046","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --supports--> v0c\n- v07 --refutes--> v11\n- v05 --refutes--> v15\n- v00 --refutes--> v0d\n- v07 --supports--> v0a\n- v17 --refutes--> v04\n- v04 --refutes--> v02\n- v10 --supports--> v09\n- v15 --refutes--> v08\n- v13 --refutes--> v06\n- v08 --refutes--> v17\n- v16 --supports--> v0f\n- v07 --refutes--> v18\n- v02 --supports--> v12\n- v13 --refutes--> v0e\n- v18 --refutes--> v0b\n- v0f --refutes--> v05\n- v05 --refutes--> v03\n- v02 --refutes--> v07\n- v0d --refutes--> v10\n- v09 --refutes--> v14\n- v02 --refutes--> v01\n- v14 --refutes--> v16\n- v18 --refutes--> v13\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ab51e3714341b311cd0496ad925f73b7e1053ebd142c7e040212e0db9d92893c","schema":"livegraph.boolean-propagation.v1","spec_hash":"f6cee4a4b5009187b0465addbb480edface2b136c793b1f4d9ddcfbc24a6f4f8","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-bcd58a6a2fd48fa2c2f2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --refutes--> v09\n- v00 --supports--> v0d\n- v0b --refutes--> v0f\n- v0e --supports--> v07\n- v10 --supports--> v0b\n- v0d --supports--> v06\n- v17 --supports--> v12\n- v06 --refutes--> v10\n- v09 --supports--> v0a\n- v04 --supports--> v01\n- v0f --refutes--> v08\n- v0e --refutes--> v0c\n- v0d --refutes--> v11\n- v16 --supports--> v17\n- v04 --refutes--> v18\n- v12 --refutes--> v15\n- v0e --supports--> v14\n- v0f --supports--> v03\n- v15 --supports--> v05\n- v15 --supports--> v00\n- v01 --supports--> v16\n- v03 --refutes--> v0e\n- v17 --refutes--> v13\n- v06 --supports--> v02\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e2864d6e4a2c017a208256c53b0638be23bd0c4a9b5ca71dbafe1653e94e8ded","schema":"livegraph.boolean-propagation.v1","spec_hash":"bcd58a6a2fd48fa2c2f2e89d3c033cc919e61a667b262b25744f30246ea29226","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-5586c96bd48fd59f2f4e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --refutes--> v12\n- v12 --refutes--> v0e\n- v17 --supports--> v07\n- v07 --refutes--> v0b\n- v04 --refutes--> v0f\n- v09 --supports--> v16\n- v15 --supports--> v13\n- v15 --refutes--> v0d\n- v0a --refutes--> v09\n- v04 --supports--> v10\n- v01 --supports--> v05\n- v17 --supports--> v02\n- v0c --supports--> v11\n- v14 --refutes--> v08\n- v0d --refutes--> v03\n- v07 --supports--> v06\n- v01 --supports--> v14\n- v00 --supports--> v18\n- v0c --refutes--> v0a\n- v0e --refutes--> v04\n- v16 --refutes--> v17\n- v0f --refutes--> v15\n- v18 --refutes--> v0c\n- v07 --supports--> v01\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3c6bcf64518987139a176c26a419dcbd0920cbf6f5043d950f5b9e1c8700cc64","schema":"livegraph.boolean-propagation.v1","spec_hash":"5586c96bd48fd59f2f4ee5ebe2e53386728300ffbc8dd419e20ba2584ddd2a7d","spec_json":"{\"edges\":[{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-25ca3ca7326790753dc6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --refutes--> v18\n- v0b --refutes--> v02\n- v00 --refutes--> v09\n- v18 --supports--> v12\n- v17 --supports--> v11\n- v08 --supports--> v14\n- v12 --refutes--> v06\n- v12 --supports--> v0f\n- v0a --supports--> v15\n- v00 --refutes--> v0b\n- v10 --supports--> v17\n- v13 --refutes--> v10\n- v0f --supports--> v05\n- v05 --refutes--> v0e\n- v14 --refutes--> v00\n- v16 --supports--> v03\n- v03 --refutes--> v0a\n- v16 --supports--> v04\n- v17 --supports--> v16\n- v0e --supports--> v08\n- v08 --supports--> v0c\n- v08 --supports--> v0d\n- v08 --refutes--> v01\n- v18 --refutes--> v07\n\n[VERIFIED ROOT]\nv13 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"efa8c540a790e06b4e147e7f782977f39363ba29ec5730b5f66e8dce75a9dbdc","schema":"livegraph.boolean-propagation.v1","spec_hash":"25ca3ca7326790753dc629b87135daab3b34c2a4a8df60b19aba527ac2fed48a","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-bcd46e83ddba2ac64526","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v16 --refutes--> v0f\n- v18 --supports--> v08\n- v0b --supports--> v0d\n- v0b --refutes--> v05\n- v01 --supports--> v17\n- v02 --supports--> v12\n- v09 --refutes--> v00\n- v06 --supports--> v01\n- v0f --supports--> v18\n- v08 --supports--> v06\n- v00 --refutes--> v11\n- v0e --supports--> v10\n- v0b --supports--> v02\n- v15 --refutes--> v16\n- v09 --supports--> v04\n- v11 --refutes--> v0a\n- v0c --supports--> v0b\n- v12 --supports--> v14\n- v0a --supports--> v15\n- v00 --refutes--> v13\n- v01 --refutes--> v07\n- v01 --supports--> v0c\n- v00 --supports--> v03\n- v0e --refutes--> v09\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv14\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"deb5a6dc0fe2c5060c56032e2d3f5cfcab046f85b0d9651ee450ff4599e8c4c7","schema":"livegraph.boolean-propagation.v1","spec_hash":"bcd46e83ddba2ac645265e9d59140221c92dbf4cfece15b0324706f5cddd4c9b","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-7f1e71d964df7cbaa08e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v16 --supports--> v11\n- v12 --refutes--> v13\n- v03 --refutes--> v0e\n- v17 --supports--> v14\n- v10 --supports--> v0c\n- v0a --refutes--> v08\n- v00 --refutes--> v16\n- v01 --refutes--> v09\n- v0e --supports--> v17\n- v14 --refutes--> v18\n- v13 --supports--> v10\n- v00 --refutes--> v07\n- v18 --supports--> v02\n- v16 --refutes--> v03\n- v02 --supports--> v0a\n- v0a --supports--> v0b\n- v01 --refutes--> v06\n- v00 --supports--> v04\n- v01 --supports--> v05\n- v09 --supports--> v12\n- v13 --refutes--> v0f\n- v0b --supports--> v01\n- v0c --supports--> v15\n- v14 --refutes--> v0d\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv15\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6343e9985e8243c3ef7140468b29c6d017900f472334a9241e287ae6eac313ff","schema":"livegraph.boolean-propagation.v1","spec_hash":"7f1e71d964df7cbaa08e6b6e4a27a3d74c3466b11297c0cde36a10ef3b4e06cb","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-38e0057aecb7b862ea55","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --supports--> v15\n- v0a --supports--> v12\n- v15 --refutes--> v0f\n- v0e --refutes--> v0c\n- v0e --refutes--> v08\n- v0c --supports--> v13\n- v0e --refutes--> v0d\n- v16 --refutes--> v00\n- v05 --refutes--> v11\n- v0f --supports--> v07\n- v11 --supports--> v0e\n- v13 --refutes--> v04\n- v12 --refutes--> v17\n- v0f --supports--> v01\n- v05 --supports--> v02\n- v14 --supports--> v03\n- v0b --refutes--> v14\n- v06 --refutes--> v0a\n- v10 --supports--> v06\n- v12 --refutes--> v16\n- v00 --supports--> v0b\n- v04 --refutes--> v10\n- v00 --supports--> v09\n- v13 --supports--> v18\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"305b9711b92f7c7d7f9e1f17e010fb0f26a67510a45510b2d512a0df8da7d467","schema":"livegraph.boolean-propagation.v1","spec_hash":"38e0057aecb7b862ea5551dd8903f9f1ea86ee28775cfdd4388fa941ebe22ba4","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":16,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-3e3346d529c89ecbc4df","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v14\n- v01 --supports--> v10\n- v18 --supports--> v07\n- v02 --supports--> v00\n- v02 --supports--> v0d\n- v03 --supports--> v0f\n- v09 --refutes--> v0b\n- v0d --refutes--> v01\n- v18 --refutes--> v02\n- v14 --refutes--> v06\n- v0b --supports--> v18\n- v0e --refutes--> v0c\n- v04 --refutes--> v09\n- v12 --supports--> v13\n- v12 --supports--> v08\n- v13 --refutes--> v0e\n- v15 --supports--> v04\n- v0f --refutes--> v17\n- v14 --refutes--> v11\n- v16 --supports--> v12\n- v0b --supports--> v0a\n- v14 --refutes--> v15\n- v10 --supports--> v03\n- v18 --supports--> v05\n\n[VERIFIED ROOT]\nv16 = FALSE\n\n[TARGET]\nv17\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2468247222f9a77a6eb8f090b7af84aaf50db7841e995ab1ce5599f2873ae361","schema":"livegraph.boolean-propagation.v1","spec_hash":"3e3346d529c89ecbc4df3258a7b8da906286bbc9a92a89beef5b68fb21312fcb","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v16\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v17\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-5a6acb947e6e234285ae","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v0e\n- v09 --supports--> v06\n- v01 --refutes--> v0c\n- v03 --refutes--> v0d\n- v15 --supports--> v04\n- v0c --supports--> v12\n- v11 --supports--> v07\n- v06 --refutes--> v14\n- v06 --supports--> v0b\n- v03 --supports--> v02\n- v12 --refutes--> v18\n- v13 --supports--> v01\n- v14 --refutes--> v16\n- v08 --refutes--> v03\n- v0f --supports--> v0a\n- v0d --refutes--> v15\n- v12 --refutes--> v00\n- v11 --supports--> v05\n- v05 --refutes--> v09\n- v06 --supports--> v10\n- v16 --supports--> v13\n- v0a --refutes--> v11\n- v12 --supports--> v08\n- v15 --refutes--> v17\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a7cf2ef9dc5f5bbff79562bb754f38758a543801373cd507ce01c9b769ce6080","schema":"livegraph.boolean-propagation.v1","spec_hash":"5a6acb947e6e234285aeb7bb12f63dcc631472d6d79c025646840e94f98f44ec","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":16,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-f337739c80a96444a121","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v0e\n- v02 --supports--> v0b\n- v0e --supports--> v14\n- v07 --supports--> v0c\n- v18 --supports--> v03\n- v00 --refutes--> v11\n- v08 --supports--> v00\n- v01 --supports--> v02\n- v13 --supports--> v15\n- v10 --supports--> v16\n- v08 --refutes--> v17\n- v02 --supports--> v10\n- v0c --supports--> v09\n- v16 --supports--> v07\n- v0c --refutes--> v06\n- v0c --supports--> v04\n- v18 --refutes--> v12\n- v0d --supports--> v08\n- v00 --supports--> v13\n- v0d --supports--> v0a\n- v16 --refutes--> v0f\n- v03 --supports--> v05\n- v14 --refutes--> v0d\n- v15 --refutes--> v01\n\n[VERIFIED ROOT]\nv18 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9486ff00fb66c4d49327a5164653efa5cbfda0edbeb71680e24eb8366e65799f","schema":"livegraph.boolean-propagation.v1","spec_hash":"f337739c80a96444a1210b66f5c6b3c019d3e40bdc14b9c111baf8a10af07e42","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v18\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-9007831ce253297347e9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v07\n- v02 --supports--> v01\n- v02 --refutes--> v10\n- v14 --refutes--> v0c\n- v14 --supports--> v16\n- v0b --supports--> v15\n- v00 --supports--> v13\n- v08 --supports--> v06\n- v0d --supports--> v05\n- v0d --refutes--> v11\n- v01 --supports--> v17\n- v14 --supports--> v09\n- v01 --supports--> v12\n- v0a --supports--> v00\n- v03 --supports--> v0f\n- v11 --refutes--> v08\n- v09 --supports--> v03\n- v06 --supports--> v04\n- v08 --supports--> v0e\n- v17 --refutes--> v14\n- v04 --refutes--> v02\n- v08 --refutes--> v18\n- v15 --refutes--> v0a\n- v13 --refutes--> v0d\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c613e2aac1f6737296bdf415b797d8cd296e28f51c12b5792fa3b80339b933c7","schema":"livegraph.boolean-propagation.v1","spec_hash":"9007831ce253297347e91a106506f02d75a53f7bcfa7362e6d659be6288d78ed","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-d21f3e53cadc8821736e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v04\n- v05 --refutes--> v0f\n- v04 --refutes--> v14\n- v07 --refutes--> v00\n- v16 --supports--> v11\n- v07 --supports--> v15\n- v06 --supports--> v0d\n- v0c --supports--> v13\n- v11 --supports--> v18\n- v17 --supports--> v07\n- v11 --supports--> v0c\n- v0c --supports--> v12\n- v00 --refutes--> v10\n- v01 --refutes--> v08\n- v12 --supports--> v02\n- v12 --refutes--> v03\n- v12 --refutes--> v0a\n- v06 --refutes--> v0b\n- v01 --supports--> v05\n- v00 --refutes--> v0e\n- v09 --refutes--> v01\n- v0f --refutes--> v16\n- v14 --supports--> v17\n- v0e --supports--> v06\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"42bcf891627e4d1b013df9446665ec6e0cd7a1a7e8f55eddf0c7fafbcbf9b5bb","schema":"livegraph.boolean-propagation.v1","spec_hash":"d21f3e53cadc8821736e4e83bea7e3e78a830f376beca43920b55a456d76d870","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-dc487f430045d25ff14a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v10\n- v0a --refutes--> v14\n- v0d --supports--> v03\n- v08 --supports--> v18\n- v06 --supports--> v16\n- v03 --refutes--> v00\n- v0e --supports--> v08\n- v00 --refutes--> v0a\n- v15 --refutes--> v0e\n- v03 --refutes--> v04\n- v0c --supports--> v12\n- v05 --refutes--> v15\n- v18 --refutes--> v06\n- v09 --refutes--> v0f\n- v06 --refutes--> v0d\n- v0a --refutes--> v17\n- v0f --refutes--> v02\n- v0a --refutes--> v11\n- v05 --supports--> v0b\n- v11 --supports--> v09\n- v12 --supports--> v01\n- v0c --refutes--> v13\n- v07 --refutes--> v05\n- v12 --refutes--> v07\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cac1938c7db6f9fdbd83db25d9f2968012f9b2b982bbde515aac2a61d777125f","schema":"livegraph.boolean-propagation.v1","spec_hash":"dc487f430045d25ff14ad93a4abd4f17a380940ed29c7b826608d67ff217ee6c","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-bde3865c3eb22ac181dc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v0d\n- v15 --refutes--> v0e\n- v09 --supports--> v0b\n- v15 --refutes--> v07\n- v11 --supports--> v13\n- v01 --supports--> v05\n- v0f --supports--> v0a\n- v03 --refutes--> v12\n- v06 --supports--> v14\n- v00 --refutes--> v09\n- v08 --supports--> v11\n- v05 --supports--> v0c\n- v09 --refutes--> v04\n- v11 --refutes--> v02\n- v13 --refutes--> v06\n- v12 --supports--> v18\n- v00 --refutes--> v10\n- v00 --supports--> v16\n- v17 --refutes--> v00\n- v07 --supports--> v01\n- v18 --supports--> v0f\n- v06 --supports--> v15\n- v0f --supports--> v08\n- v05 --refutes--> v17\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"13043bdae31c5a5c1697d2b1025f168365d1eb04a8501d6399648eab76b3ce93","schema":"livegraph.boolean-propagation.v1","spec_hash":"bde3865c3eb22ac181dc517ac44d39cbce8d1f3ff6a7622a2fa898fed029b14d","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-b56a5b3485bf2a007c06","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v07\n- v09 --supports--> v15\n- v07 --refutes--> v01\n- v06 --supports--> v17\n- v17 --refutes--> v16\n- v07 --supports--> v0d\n- v17 --supports--> v03\n- v0d --refutes--> v0f\n- v0e --supports--> v11\n- v14 --refutes--> v0b\n- v02 --supports--> v0c\n- v17 --refutes--> v14\n- v08 --supports--> v05\n- v14 --supports--> v00\n- v00 --refutes--> v02\n- v13 --supports--> v0e\n- v0f --refutes--> v12\n- v13 --supports--> v0a\n- v11 --supports--> v09\n- v05 --refutes--> v13\n- v0c --refutes--> v08\n- v12 --refutes--> v04\n- v13 --supports--> v18\n- v17 --refutes--> v10\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ecddb64aa4bf7674adbc72db47d62b171cbb10e0bfb289a4800a242918e62eec","schema":"livegraph.boolean-propagation.v1","spec_hash":"b56a5b3485bf2a007c061b877dab0ad17ce854432492469b148a3dc9e83c527f","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-1870297660923abdcbdb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v01\n- v10 --refutes--> v11\n- v03 --refutes--> v16\n- v16 --refutes--> v04\n- v00 --supports--> v07\n- v0a --supports--> v08\n- v14 --supports--> v13\n- v13 --supports--> v03\n- v07 --supports--> v15\n- v06 --supports--> v0f\n- v11 --refutes--> v0e\n- v04 --supports--> v02\n- v11 --supports--> v0a\n- v11 --refutes--> v12\n- v0a --supports--> v14\n- v0b --refutes--> v00\n- v0f --refutes--> v17\n- v02 --supports--> v05\n- v0f --supports--> v18\n- v15 --supports--> v10\n- v0f --supports--> v09\n- v18 --supports--> v0b\n- v03 --supports--> v0c\n- v15 --refutes--> v0d\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cb99fdad1da10f0c2e90fd146fed860b3ae9983126657848ae263d2302122289","schema":"livegraph.boolean-propagation.v1","spec_hash":"1870297660923abdcbdb065f4cd48e3d8b4201c2ea321576f759da6f9a84c08b","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-65b28fe8e2b28768bb4c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v17 --refutes--> v16\n- v11 --refutes--> v17\n- v08 --refutes--> v11\n- v06 --refutes--> v0a\n- v10 --supports--> v04\n- v01 --supports--> v15\n- v06 --supports--> v0f\n- v04 --refutes--> v14\n- v11 --refutes--> v05\n- v10 --refutes--> v07\n- v14 --supports--> v18\n- v18 --refutes--> v0b\n- v11 --supports--> v12\n- v09 --refutes--> v02\n- v01 --refutes--> v00\n- v0e --refutes--> v01\n- v15 --refutes--> v10\n- v15 --supports--> v13\n- v04 --refutes--> v0c\n- v0f --refutes--> v0e\n- v03 --supports--> v09\n- v16 --supports--> v03\n- v02 --supports--> v06\n- v16 --supports--> v0d\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e7dfdc5e60c894539d4e4a53bf9eb87c963abb1622144192bc0b93954c2fccf4","schema":"livegraph.boolean-propagation.v1","spec_hash":"65b28fe8e2b28768bb4cc5e7245fe126f0acc8dbd370f3650b11917eb3682304","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-7cc724575ec7402cf5eb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v18 --supports--> v14\n- v0a --refutes--> v08\n- v05 --supports--> v07\n- v11 --refutes--> v12\n- v0d --refutes--> v09\n- v02 --supports--> v0e\n- v11 --refutes--> v06\n- v06 --supports--> v05\n- v0a --supports--> v0c\n- v00 --supports--> v18\n- v0a --supports--> v04\n- v10 --supports--> v00\n- v0e --refutes--> v0b\n- v03 --refutes--> v17\n- v09 --refutes--> v11\n- v17 --refutes--> v0a\n- v09 --supports--> v13\n- v0f --supports--> v01\n- v09 --supports--> v15\n- v07 --supports--> v02\n- v01 --refutes--> v10\n- v18 --supports--> v0d\n- v11 --refutes--> v16\n- v0e --supports--> v03\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"194db4a6735c33be26bd28a833e81bd93436e1b2db3e0988e7d8326d236e96aa","schema":"livegraph.boolean-propagation.v1","spec_hash":"7cc724575ec7402cf5ebd8f4e21a2ab9a573f9a2414963c2ed831004d5b8f10a","spec_json":"{\"edges\":[{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-3ce3f75d609158f17338","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v0f\n- v09 --refutes--> v11\n- v11 --supports--> v12\n- v01 --refutes--> v04\n- v16 --supports--> v18\n- v0d --supports--> v03\n- v08 --supports--> v0e\n- v11 --supports--> v14\n- v07 --refutes--> v15\n- v07 --refutes--> v13\n- v00 --refutes--> v0b\n- v10 --refutes--> v0c\n- v17 --refutes--> v01\n- v00 --supports--> v10\n- v13 --refutes--> v00\n- v14 --supports--> v16\n- v09 --supports--> v02\n- v18 --supports--> v08\n- v13 --refutes--> v06\n- v0e --refutes--> v0a\n- v0d --refutes--> v05\n- v01 --refutes--> v09\n- v05 --supports--> v07\n- v0c --supports--> v17\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f85e6bdff42af3e1155aa713afb5662dd38b76e90e6d35404972e08832f6a45d","schema":"livegraph.boolean-propagation.v1","spec_hash":"3ce3f75d609158f17338725555149619065699acf3f4231685db14b1d1547e6a","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-37b914d12641c7286c65","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v06\n- v0e --supports--> v0a\n- v16 --supports--> v17\n- v17 --refutes--> v01\n- v12 --refutes--> v11\n- v04 --supports--> v05\n- v0b --refutes--> v04\n- v01 --refutes--> v03\n- v05 --supports--> v15\n- v09 --refutes--> v10\n- v01 --supports--> v13\n- v01 --refutes--> v09\n- v02 --supports--> v14\n- v12 --supports--> v0f\n- v11 --supports--> v0e\n- v0a --refutes--> v02\n- v07 --supports--> v0b\n- v0c --supports--> v16\n- v05 --refutes--> v08\n- v18 --supports--> v07\n- v09 --supports--> v12\n- v16 --supports--> v0d\n- v14 --supports--> v18\n- v14 --supports--> v00\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"23039767b12fd5797dc18b761f386f8482b8e0e1535b974c1ff39e2c69bf8994","schema":"livegraph.boolean-propagation.v1","spec_hash":"37b914d12641c7286c659a61ab04977b17b4aa373374726b73e79cfba4b8aef1","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-137c68f2c7e174e86ea8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --refutes--> v10\n- v0a --supports--> v11\n- v17 --supports--> v0d\n- v05 --refutes--> v0a\n- v0f --refutes--> v16\n- v0e --refutes--> v09\n- v0b --supports--> v17\n- v02 --refutes--> v0c\n- v12 --supports--> v03\n- v0e --refutes--> v00\n- v00 --refutes--> v05\n- v07 --refutes--> v0e\n- v08 --supports--> v07\n- v04 --refutes--> v13\n- v15 --supports--> v12\n- v15 --refutes--> v18\n- v12 --refutes--> v04\n- v16 --refutes--> v15\n- v0c --refutes--> v0b\n- v04 --refutes--> v14\n- v0a --refutes--> v0f\n- v14 --refutes--> v02\n- v02 --refutes--> v06\n- v02 --refutes--> v01\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1a0f7831edaad072e359c3c76a17a36dd7405ee1775a504fdcb82b6ff3b9c680","schema":"livegraph.boolean-propagation.v1","spec_hash":"137c68f2c7e174e86ea87355e1c344d883f25cb7066075f5655351f0e3c2126b","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-9e2b31a2a6c1f54a1f15","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v11\n- v09 --supports--> v0e\n- v12 --supports--> v06\n- v11 --refutes--> v17\n- v05 --supports--> v12\n- v17 --refutes--> v0f\n- v17 --supports--> v0c\n- v0d --refutes--> v00\n- v0e --supports--> v15\n- v06 --refutes--> v02\n- v06 --refutes--> v01\n- v12 --refutes--> v18\n- v0a --supports--> v16\n- v10 --supports--> v04\n- v16 --supports--> v10\n- v01 --supports--> v03\n- v13 --refutes--> v0a\n- v15 --supports--> v07\n- v17 --supports--> v14\n- v10 --refutes--> v09\n- v07 --supports--> v0d\n- v13 --supports--> v08\n- v12 --refutes--> v0b\n- v14 --supports--> v05\n\n[VERIFIED ROOT]\nv13 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"423e5bdc5ecdef1509dfd0d8dfea4fca24b22fab499761e0897174cfaeb18995","schema":"livegraph.boolean-propagation.v1","spec_hash":"9e2b31a2a6c1f54a1f15ae31763a70a0de191a668a394b8843bab02193fc5c25","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-a03fae4647f4691fc659","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v0e\n- v05 --refutes--> v03\n- v08 --refutes--> v10\n- v11 --refutes--> v16\n- v0f --refutes--> v15\n- v04 --refutes--> v02\n- v12 --supports--> v18\n- v10 --supports--> v04\n- v07 --supports--> v08\n- v17 --refutes--> v09\n- v09 --refutes--> v05\n- v13 --supports--> v0f\n- v07 --supports--> v06\n- v05 --supports--> v07\n- v11 --refutes--> v00\n- v14 --refutes--> v0b\n- v15 --refutes--> v01\n- v14 --supports--> v12\n- v11 --supports--> v0d\n- v16 --refutes--> v0a\n- v18 --refutes--> v13\n- v02 --supports--> v14\n- v15 --supports--> v11\n- v17 --supports--> v0c\n\n[VERIFIED ROOT]\nv17 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6d7188470268137c5d382c1e3f5bac22f247100f488124bb3b4935e116d608a8","schema":"livegraph.boolean-propagation.v1","spec_hash":"a03fae4647f4691fc659ace1ca880ed431ca7da39382bed7cb73f6df2fc80e08","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v17\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-bfa19a28d60a2ab7e93f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v17 --refutes--> v0c\n- v15 --supports--> v04\n- v09 --refutes--> v17\n- v10 --supports--> v07\n- v09 --refutes--> v13\n- v02 --refutes--> v16\n- v18 --refutes--> v0e\n- v08 --refutes--> v0a\n- v06 --refutes--> v0b\n- v0b --supports--> v11\n- v10 --supports--> v0f\n- v0b --refutes--> v0d\n- v03 --refutes--> v06\n- v09 --supports--> v00\n- v18 --supports--> v01\n- v0d --supports--> v15\n- v12 --supports--> v10\n- v0f --supports--> v05\n- v04 --supports--> v09\n- v0e --refutes--> v03\n- v02 --supports--> v18\n- v0d --refutes--> v14\n- v0c --refutes--> v12\n- v08 --refutes--> v02\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0f234bdcf6d6f3d0dda3a3778c203c4a02f4bf11426b79057719515a739f6de0","schema":"livegraph.boolean-propagation.v1","spec_hash":"bfa19a28d60a2ab7e93f3e0c005f6575be2353f4e8801478305a3b72002cf260","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-0b417cb9a4e2c9b3bb14","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v16 --refutes--> v15\n- v15 --supports--> v18\n- v06 --refutes--> v0a\n- v16 --supports--> v14\n- v12 --supports--> v17\n- v0d --refutes--> v10\n- v09 --supports--> v03\n- v04 --refutes--> v0d\n- v05 --refutes--> v01\n- v0e --supports--> v11\n- v08 --supports--> v06\n- v00 --supports--> v0b\n- v0d --refutes--> v0f\n- v12 --supports--> v0c\n- v00 --supports--> v02\n- v0f --refutes--> v00\n- v01 --refutes--> v0e\n- v11 --refutes--> v09\n- v05 --supports--> v13\n- v17 --refutes--> v16\n- v09 --supports--> v07\n- v15 --supports--> v04\n- v02 --refutes--> v05\n- v09 --supports--> v08\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"258119f2be9b19a5c608f89853232271c1366d2cbaff83b9d756d7a3af3d89fa","schema":"livegraph.boolean-propagation.v1","spec_hash":"0b417cb9a4e2c9b3bb14bb5d0e175d45df9c12e06fe8b858c3b64b7885eb0bfa","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-b785e1aded1aa033171f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v08\n- v05 --refutes--> v01\n- v06 --supports--> v07\n- v0c --refutes--> v13\n- v05 --refutes--> v0d\n- v0e --refutes--> v14\n- v14 --supports--> v0c\n- v18 --refutes--> v02\n- v01 --refutes--> v0e\n- v03 --supports--> v10\n- v18 --supports--> v17\n- v11 --supports--> v0f\n- v07 --refutes--> v0b\n- v0c --supports--> v04\n- v05 --supports--> v0a\n- v11 --refutes--> v09\n- v08 --refutes--> v15\n- v00 --refutes--> v03\n- v07 --supports--> v05\n- v10 --supports--> v06\n- v0f --supports--> v12\n- v18 --refutes--> v00\n- v12 --refutes--> v18\n- v05 --supports--> v16\n\n[VERIFIED ROOT]\nv11 = FALSE\n\n[TARGET]\nv15\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"61642e047c6ca4ae51efb35498365e964625c769b3e688e5b0b2b3147c852290","schema":"livegraph.boolean-propagation.v1","spec_hash":"b785e1aded1aa033171f6acb18bfa1bbbec8140e44c00fbdc74e4724e737dc8b","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-b563ff23b8f00e8edece","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v16 --refutes--> v18\n- v01 --supports--> v00\n- v0e --supports--> v01\n- v08 --refutes--> v0e\n- v0b --refutes--> v13\n- v10 --refutes--> v14\n- v0e --refutes--> v09\n- v0a --refutes--> v03\n- v0f --refutes--> v11\n- v11 --refutes--> v05\n- v0a --supports--> v04\n- v00 --refutes--> v0a\n- v0b --supports--> v0f\n- v0a --refutes--> v0d\n- v17 --supports--> v0b\n- v11 --supports--> v06\n- v02 --refutes--> v15\n- v10 --refutes--> v12\n- v05 --supports--> v0c\n- v05 --refutes--> v07\n- v04 --supports--> v17\n- v16 --supports--> v10\n- v15 --refutes--> v16\n- v14 --supports--> v08\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0a32bc38394232bb7537c522b1d0c47b4e6c1c458e5ab6f598aa1da13168a03b","schema":"livegraph.boolean-propagation.v1","spec_hash":"b563ff23b8f00e8edece07a685f06e24b6665439d04893c3f4d1c4cf7c048bee","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-42c848d5dab06e4ba763","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v00\n- v0f --refutes--> v0c\n- v17 --supports--> v04\n- v15 --supports--> v11\n- v06 --refutes--> v05\n- v0c --refutes--> v16\n- v0a --supports--> v02\n- v01 --supports--> v09\n- v06 --supports--> v14\n- v16 --refutes--> v01\n- v12 --supports--> v03\n- v0a --supports--> v12\n- v11 --refutes--> v0f\n- v0e --supports--> v10\n- v0e --refutes--> v18\n- v07 --supports--> v06\n- v0d --supports--> v07\n- v10 --supports--> v17\n- v05 --supports--> v08\n- v17 --refutes--> v0d\n- v01 --refutes--> v0e\n- v05 --refutes--> v13\n- v12 --refutes--> v0b\n- v08 --supports--> v0a\n\n[VERIFIED ROOT]\nv15 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3c0678ee5d512c00c8c219571d44140893524a9c41752a657a9c6a5a1c78ebe8","schema":"livegraph.boolean-propagation.v1","spec_hash":"42c848d5dab06e4ba763d10ece3a0b9aa5682c40fcfcde1741ea8c806502033f","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v15\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-e7ff9fade97756bf5cec","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v01\n- v10 --refutes--> v05\n- v0c --supports--> v18\n- v06 --refutes--> v0a\n- v14 --refutes--> v0f\n- v0a --refutes--> v17\n- v16 --refutes--> v15\n- v0b --refutes--> v14\n- v01 --supports--> v03\n- v0d --refutes--> v16\n- v13 --supports--> v08\n- v12 --refutes--> v06\n- v02 --supports--> v12\n- v15 --refutes--> v11\n- v17 --refutes--> v0c\n- v15 --refutes--> v02\n- v0f --refutes--> v0d\n- v02 --supports--> v09\n- v16 --refutes--> v07\n- v10 --supports--> v00\n- v17 --supports--> v0e\n- v13 --refutes--> v10\n- v01 --supports--> v0b\n- v0c --supports--> v04\n\n[VERIFIED ROOT]\nv13 = TRUE\n\n[TARGET]\nv18\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"33076d14c364ded4ca48dbf96af2d8fc9a3b3f8b98844ab052fa8e68dc9745f9","schema":"livegraph.boolean-propagation.v1","spec_hash":"e7ff9fade97756bf5cec2c93b371c9661ebd5d62cfa677c32825e31f062f14bd","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v18\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-5d007d88830cda7db871","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v0c\n- v0b --supports--> v02\n- v0d --refutes--> v05\n- v15 --refutes--> v0b\n- v00 --supports--> v01\n- v0e --refutes--> v13\n- v11 --refutes--> v16\n- v0c --refutes--> v18\n- v18 --supports--> v00\n- v18 --supports--> v06\n- v01 --supports--> v0a\n- v12 --supports--> v07\n- v14 --supports--> v0e\n- v05 --refutes--> v03\n- v11 --refutes--> v10\n- v0c --refutes--> v08\n- v12 --refutes--> v11\n- v09 --supports--> v0f\n- v11 --supports--> v15\n- v01 --refutes--> v14\n- v14 --refutes--> v04\n- v00 --supports--> v17\n- v02 --supports--> v09\n- v13 --refutes--> v0d\n\n[VERIFIED ROOT]\nv12 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f79cd3424ce9d8d4dcae0160f21d41707b70108fef88e4a91379098dc265adb7","schema":"livegraph.boolean-propagation.v1","spec_hash":"5d007d88830cda7db8714c8922692a1a3aed7a4aecb61ddfa17f598bfac9a495","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v12\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-6f439e5ec595fe798644","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v02\n- v0b --refutes--> v13\n- v03 --supports--> v0c\n- v0f --refutes--> v00\n- v03 --refutes--> v15\n- v0e --refutes--> v01\n- v15 --supports--> v07\n- v09 --supports--> v0f\n- v12 --refutes--> v05\n- v17 --supports--> v06\n- v11 --refutes--> v14\n- v05 --refutes--> v0b\n- v07 --refutes--> v0a\n- v00 --refutes--> v0d\n- v03 --supports--> v04\n- v08 --refutes--> v11\n- v01 --supports--> v03\n- v07 --supports--> v17\n- v0b --supports--> v10\n- v00 --supports--> v0e\n- v07 --supports--> v18\n- v10 --supports--> v08\n- v06 --refutes--> v12\n- v0b --refutes--> v16\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv14\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5ae7754f722e73ef690adf6c53067a4dfeb437ee12b93ef3947fcef3cea5cc5e","schema":"livegraph.boolean-propagation.v1","spec_hash":"6f439e5ec595fe79864436884bf8cc5bd10b8411f8bed5b325a37ba12696b84f","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-09e2c14c47a9188fb2be","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v15\n- v18 --refutes--> v11\n- v01 --refutes--> v17\n- v0a --refutes--> v0b\n- v08 --refutes--> v16\n- v0c --refutes--> v06\n- v0c --refutes--> v13\n- v0d --supports--> v0e\n- v12 --refutes--> v07\n- v02 --supports--> v03\n- v16 --supports--> v18\n- v18 --refutes--> v0f\n- v15 --supports--> v08\n- v10 --refutes--> v14\n- v17 --refutes--> v05\n- v12 --supports--> v09\n- v12 --supports--> v00\n- v0b --supports--> v12\n- v11 --supports--> v0c\n- v05 --supports--> v02\n- v05 --supports--> v04\n- v0d --supports--> v10\n- v02 --refutes--> v0a\n- v0c --supports--> v0d\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv14\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8da088e14a5bd18a07b767157a5be3eadf4c11d7d6be986a5b431ac03aab07ff","schema":"livegraph.boolean-propagation.v1","spec_hash":"09e2c14c47a9188fb2bebf1d69bfc5729b7aa1aab7d9a2ea8e9812fe47f74d09","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-e7c549a3178910888bf5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --supports--> v05\n- v0f --supports--> v0d\n- v01 --supports--> v0a\n- v09 --supports--> v16\n- v13 --refutes--> v15\n- v13 --refutes--> v0f\n- v0d --refutes--> v00\n- v17 --refutes--> v08\n- v06 --refutes--> v12\n- v07 --supports--> v18\n- v12 --refutes--> v0e\n- v17 --supports--> v13\n- v09 --refutes--> v07\n- v18 --supports--> v14\n- v00 --supports--> v06\n- v0e --supports--> v0c\n- v0c --refutes--> v09\n- v01 --supports--> v17\n- v09 --supports--> v03\n- v02 --supports--> v11\n- v11 --refutes--> v10\n- v10 --supports--> v01\n- v17 --refutes--> v04\n- v12 --supports--> v0b\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv14\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9eb197464b7458b25e0fd08bc231cc47944c2380577e83ae0ab8e98a14b6a08e","schema":"livegraph.boolean-propagation.v1","spec_hash":"e7c549a3178910888bf596c1ccb7d0b16a70d1b5b53603a57af7dc4d35a9f4b4","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-51022313d3bac96e58a7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v11\n- v12 --refutes--> v13\n- v0b --refutes--> v10\n- v06 --refutes--> v14\n- v14 --refutes--> v09\n- v13 --supports--> v07\n- v17 --refutes--> v15\n- v0d --refutes--> v18\n- v02 --supports--> v12\n- v07 --refutes--> v0c\n- v08 --supports--> v0d\n- v0b --refutes--> v08\n- v17 --refutes--> v0e\n- v12 --refutes--> v0a\n- v0e --refutes--> v06\n- v0b --refutes--> v03\n- v00 --refutes--> v16\n- v04 --supports--> v02\n- v0d --supports--> v04\n- v16 --supports--> v17\n- v00 --supports--> v0f\n- v05 --refutes--> v00\n- v0c --refutes--> v05\n- v0b --supports--> v01\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7e8d2f1916d3a38af9841c91ad492b6a7aa9ab95dfd61ba771d191f0fc92eb8a","schema":"livegraph.boolean-propagation.v1","spec_hash":"51022313d3bac96e58a75455b925fafddb578a28e0786ac50e24e02ca3382c80","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-b1fe383594b53704f975","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v14\n- v00 --refutes--> v01\n- v0b --refutes--> v10\n- v07 --supports--> v09\n- v0f --supports--> v03\n- v14 --refutes--> v00\n- v0c --refutes--> v15\n- v0c --supports--> v08\n- v09 --refutes--> v0e\n- v0b --supports--> v0d\n- v17 --supports--> v06\n- v02 --supports--> v17\n- v15 --supports--> v18\n- v0d --refutes--> v13\n- v11 --supports--> v07\n- v09 --supports--> v05\n- v16 --refutes--> v04\n- v06 --supports--> v12\n- v05 --refutes--> v0f\n- v11 --supports--> v0a\n- v15 --supports--> v11\n- v04 --refutes--> v0b\n- v00 --supports--> v16\n- v0f --refutes--> v02\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1cb86102bfcd15948ffde647e47a9e5f0436d0e755fceadd89529e92c1c81bca","schema":"livegraph.boolean-propagation.v1","spec_hash":"b1fe383594b53704f975adb45fc6ae55b7b019eda0f2259eb440347818d9f1cc","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-04ca78c141f3551ee322","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --refutes--> v06\n- v08 --refutes--> v0b\n- v14 --refutes--> v0a\n- v07 --refutes--> v03\n- v0a --refutes--> v17\n- v0e --supports--> v07\n- v08 --refutes--> v01\n- v13 --supports--> v14\n- v15 --refutes--> v18\n- v04 --refutes--> v02\n- v03 --refutes--> v0f\n- v01 --supports--> v0e\n- v15 --supports--> v0d\n- v03 --supports--> v04\n- v11 --refutes--> v12\n- v18 --supports--> v10\n- v13 --supports--> v00\n- v18 --supports--> v0c\n- v02 --refutes--> v15\n- v12 --refutes--> v05\n- v05 --refutes--> v13\n- v13 --refutes--> v09\n- v08 --refutes--> v16\n- v17 --refutes--> v08\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9e4fb425da799cee87f0096b8d72eaf0fde5c744161dce4eadb7364fcf5c17e1","schema":"livegraph.boolean-propagation.v1","spec_hash":"04ca78c141f3551ee322792de8c2f3bd3e7ca429d3175ebc4b21d7174cd79c43","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-8e95641e4bd7599efcfb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --refutes--> v0e\n- v00 --supports--> v15\n- v06 --refutes--> v10\n- v17 --refutes--> v09\n- v16 --refutes--> v07\n- v0f --refutes--> v05\n- v0e --refutes--> v0f\n- v0d --refutes--> v02\n- v01 --refutes--> v13\n- v0a --supports--> v18\n- v0b --refutes--> v0d\n- v0f --supports--> v04\n- v04 --refutes--> v11\n- v03 --supports--> v16\n- v0d --refutes--> v06\n- v06 --supports--> v14\n- v17 --refutes--> v03\n- v04 --supports--> v01\n- v18 --refutes--> v00\n- v01 --supports--> v0c\n- v13 --refutes--> v0b\n- v10 --supports--> v17\n- v06 --refutes--> v08\n- v16 --refutes--> v12\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6295d50b5b26cd27e4a5f9a5053abbd9b8c0bdaa9ed5737d502ffdc96c5c1787","schema":"livegraph.boolean-propagation.v1","spec_hash":"8e95641e4bd7599efcfbf437fc07f53c8a301dc575dca37d20a090d2bb3aca7d","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-fab702f4b617bd8d7f4e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v08\n- v0d --supports--> v10\n- v00 --supports--> v0a\n- v03 --refutes--> v0e\n- v0c --supports--> v15\n- v16 --refutes--> v03\n- v03 --supports--> v06\n- v02 --refutes--> v07\n- v02 --refutes--> v12\n- v0e --supports--> v17\n- v11 --supports--> v13\n- v02 --supports--> v11\n- v00 --refutes--> v04\n- v01 --refutes--> v14\n- v09 --refutes--> v16\n- v17 --supports--> v0f\n- v13 --supports--> v18\n- v17 --refutes--> v0c\n- v05 --refutes--> v00\n- v14 --refutes--> v0b\n- v00 --supports--> v0d\n- v0b --refutes--> v05\n- v0d --refutes--> v09\n- v15 --refutes--> v02\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv18\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f25c45606e66a18f3a4ab68c385d4c646d6373ea595524fbd3befd2e0c520ad5","schema":"livegraph.boolean-propagation.v1","spec_hash":"fab702f4b617bd8d7f4e13b97dacfefd607e0a272cf7f91d72c2247febfa3e59","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v18\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-a934060e12ca0e7e4792","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v0b\n- v0b --refutes--> v02\n- v00 --refutes--> v0e\n- v0d --supports--> v06\n- v11 --supports--> v09\n- v05 --refutes--> v11\n- v12 --supports--> v0f\n- v0d --refutes--> v00\n- v17 --supports--> v16\n- v0e --supports--> v03\n- v04 --refutes--> v08\n- v14 --supports--> v15\n- v04 --supports--> v14\n- v02 --refutes--> v13\n- v0c --refutes--> v18\n- v12 --supports--> v04\n- v0d --refutes--> v07\n- v13 --supports--> v0a\n- v0b --refutes--> v10\n- v09 --refutes--> v17\n- v14 --supports--> v05\n- v17 --supports--> v01\n- v16 --refutes--> v0d\n- v13 --refutes--> v0c\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv18\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"62db6d0e8aecddff27cee9dd6fe4cca66754847ea66a335c198d151cbd714e60","schema":"livegraph.boolean-propagation.v1","spec_hash":"a934060e12ca0e7e479275fe3f21aaab83e527bb2542efbf006e5877728affc2","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v18\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-7b9817f76287eb328a8c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v0d\n- v05 --refutes--> v07\n- v0d --supports--> v05\n- v16 --refutes--> v10\n- v15 --refutes--> v08\n- v15 --refutes--> v06\n- v0c --refutes--> v12\n- v16 --refutes--> v0f\n- v00 --supports--> v16\n- v09 --refutes--> v01\n- v07 --supports--> v18\n- v13 --refutes--> v04\n- v18 --supports--> v0b\n- v16 --supports--> v02\n- v11 --supports--> v00\n- v09 --supports--> v14\n- v0b --supports--> v0a\n- v09 --supports--> v03\n- v08 --supports--> v0e\n- v0e --supports--> v0c\n- v02 --supports--> v17\n- v01 --supports--> v13\n- v0b --refutes--> v09\n- v13 --supports--> v15\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9fd6c238fd665a030c75977f3f937ee5485a78f88222756d9c622bb53b9d60d8","schema":"livegraph.boolean-propagation.v1","spec_hash":"7b9817f76287eb328a8cb063184628e38540bf88b324d982f795ec40d1202056","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-8beeaf5a6a6f17adcc7b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --refutes--> v00\n- v00 --refutes--> v02\n- v02 --refutes--> v13\n- v09 --refutes--> v10\n- v15 --refutes--> v08\n- v11 --supports--> v0d\n- v09 --refutes--> v07\n- v16 --refutes--> v0a\n- v11 --refutes--> v12\n- v17 --refutes--> v0f\n- v18 --refutes--> v04\n- v05 --supports--> v16\n- v0b --supports--> v03\n- v0b --supports--> v05\n- v18 --supports--> v14\n- v05 --refutes--> v01\n- v08 --refutes--> v18\n- v16 --refutes--> v06\n- v0e --refutes--> v17\n- v11 --refutes--> v09\n- v13 --supports--> v0b\n- v0a --refutes--> v11\n- v10 --supports--> v0e\n- v17 --refutes--> v0c\n\n[VERIFIED ROOT]\nv15 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4e7ed4cf9e9db84a31f433310bf8000d6bead2caca573ff5b0f8301f1254a0bd","schema":"livegraph.boolean-propagation.v1","spec_hash":"8beeaf5a6a6f17adcc7b90ba233513ec536e2a8278405d7ba06b909c4a70ce85","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v15\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-36425e09d9f95546dfa9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v0c\n- v0b --refutes--> v11\n- v06 --supports--> v0e\n- v03 --refutes--> v09\n- v0e --supports--> v03\n- v05 --refutes--> v12\n- v09 --refutes--> v00\n- v16 --refutes--> v08\n- v05 --refutes--> v13\n- v12 --supports--> v07\n- v14 --refutes--> v05\n- v17 --refutes--> v0b\n- v0f --supports--> v0d\n- v08 --refutes--> v04\n- v00 --supports--> v0f\n- v12 --supports--> v01\n- v04 --refutes--> v06\n- v16 --refutes--> v02\n- v04 --supports--> v15\n- v01 --supports--> v10\n- v16 --supports--> v18\n- v10 --refutes--> v17\n- v06 --supports--> v0a\n- v0b --refutes--> v16\n\n[VERIFIED ROOT]\nv14 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"65160dd224d1450b673d4819f899ff9ee10947bb5cab1d47c6b41dbff92f6056","schema":"livegraph.boolean-propagation.v1","spec_hash":"36425e09d9f95546dfa9fe6a129e0168c7e18f01fda4d84867ffb5962985a1e1","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v14\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-7f6df4c9b63b62714d06","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v11\n- v09 --refutes--> v0b\n- v17 --refutes--> v0d\n- v00 --supports--> v14\n- v0a --refutes--> v08\n- v14 --supports--> v18\n- v10 --supports--> v0c\n- v0d --refutes--> v05\n- v17 --refutes--> v01\n- v16 --refutes--> v00\n- v06 --refutes--> v12\n- v05 --supports--> v07\n- v10 --refutes--> v0e\n- v15 --refutes--> v0a\n- v0c --supports--> v17\n- v10 --refutes--> v13\n- v0b --refutes--> v10\n- v12 --refutes--> v16\n- v09 --supports--> v03\n- v09 --supports--> v02\n- v05 --refutes--> v0f\n- v0f --supports--> v15\n- v0a --supports--> v04\n- v08 --refutes--> v06\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv18\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"73bccc41a0ddfa1478b68f57ea30b4df5a60d7d38bdaa687e7cb60ee48dec2e8","schema":"livegraph.boolean-propagation.v1","spec_hash":"7f6df4c9b63b62714d06d2e3c9731a6dddeec0a4d2548f127dd0d7eb3a8e4f0c","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v18\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-78572f3e082cbd659aaf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v0f\n- v15 --supports--> v0a\n- v02 --refutes--> v13\n- v0e --refutes--> v06\n- v0b --refutes--> v00\n- v0f --supports--> v10\n- v00 --supports--> v0e\n- v11 --refutes--> v0b\n- v10 --refutes--> v09\n- v0a --supports--> v18\n- v17 --supports--> v11\n- v0c --refutes--> v15\n- v02 --supports--> v07\n- v15 --refutes--> v05\n- v10 --supports--> v03\n- v0a --supports--> v14\n- v0d --refutes--> v12\n- v04 --supports--> v02\n- v0d --refutes--> v16\n- v16 --refutes--> v04\n- v17 --supports--> v08\n- v03 --supports--> v0d\n- v0a --refutes--> v17\n- v16 --supports--> v01\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fc40529291ce776d043b3790575adc6b2e2311cbc9cb7958212dda991aef3df8","schema":"livegraph.boolean-propagation.v1","spec_hash":"78572f3e082cbd659aaf7bdb01b35f0eb36842ac143176833193e2d6c389ba1c","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-d698603b1a3448f6be0a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --refutes--> v06\n- v11 --supports--> v08\n- v0c --supports--> v18\n- v0e --refutes--> v10\n- v12 --refutes--> v0e\n- v08 --refutes--> v16\n- v08 --supports--> v17\n- v02 --refutes--> v15\n- v17 --refutes--> v13\n- v04 --supports--> v02\n- v09 --supports--> v14\n- v17 --refutes--> v03\n- v05 --supports--> v12\n- v0e --supports--> v0a\n- v10 --refutes--> v11\n- v12 --refutes--> v0b\n- v06 --supports--> v09\n- v17 --refutes--> v0f\n- v13 --refutes--> v0c\n- v07 --refutes--> v04\n- v0c --supports--> v07\n- v06 --supports--> v00\n- v14 --refutes--> v0d\n- v06 --refutes--> v01\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7ba630ce2ed4c02a92d0c41df894b6276409414748f8f2a1e1ce8e6950cb8a36","schema":"livegraph.boolean-propagation.v1","spec_hash":"d698603b1a3448f6be0afb9109f7cedce865795e99156d2b35a91ca3b9b6afe2","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-5b965a943e48586ee490","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --supports--> v08\n- v03 --refutes--> v0b\n- v0c --supports--> v04\n- v0d --refutes--> v16\n- v17 --refutes--> v12\n- v16 --supports--> v03\n- v0a --refutes--> v14\n- v0b --supports--> v0a\n- v00 --supports--> v09\n- v08 --refutes--> v0e\n- v07 --supports--> v05\n- v10 --refutes--> v0f\n- v16 --supports--> v13\n- v08 --supports--> v00\n- v12 --refutes--> v11\n- v0a --refutes--> v15\n- v05 --refutes--> v17\n- v06 --refutes--> v0c\n- v00 --refutes--> v06\n- v15 --supports--> v02\n- v11 --supports--> v10\n- v07 --supports--> v01\n- v10 --refutes--> v0d\n- v0c --supports--> v18\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7f4e737da81c328b980f6a99cc944dec4a6b8752c92d3bac1744c0ded5b02f2f","schema":"livegraph.boolean-propagation.v1","spec_hash":"5b965a943e48586ee490dbba5b759ab74436427496415c4cc4ac34cd10775183","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-9a2e4ce512aa023e72ce","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --refutes--> v09\n- v11 --supports--> v15\n- v06 --refutes--> v13\n- v0c --supports--> v00\n- v0f --supports--> v01\n- v07 --supports--> v03\n- v11 --refutes--> v0e\n- v0e --supports--> v14\n- v14 --refutes--> v04\n- v14 --refutes--> v08\n- v01 --supports--> v16\n- v10 --refutes--> v07\n- v12 --supports--> v0c\n- v07 --refutes--> v02\n- v18 --supports--> v06\n- v16 --supports--> v0d\n- v07 --supports--> v0a\n- v0d --refutes--> v10\n- v05 --refutes--> v12\n- v04 --refutes--> v05\n- v13 --supports--> v0f\n- v0f --supports--> v0b\n- v0c --supports--> v18\n- v0d --supports--> v17\n\n[VERIFIED ROOT]\nv11 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"270c73a7a1a1454b493460c1e434c7d7b69efd24c05222bbe9f12348e3227b49","schema":"livegraph.boolean-propagation.v1","spec_hash":"9a2e4ce512aa023e72ce9ac38d592b50dbecda3fa0a55865f1f8d4b61b07e823","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-b2238539385375717fea","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v12\n- v0c --supports--> v06\n- v10 --refutes--> v04\n- v09 --supports--> v0f\n- v06 --refutes--> v00\n- v09 --supports--> v03\n- v18 --supports--> v05\n- v02 --supports--> v11\n- v13 --supports--> v02\n- v11 --supports--> v0b\n- v11 --supports--> v0e\n- v02 --supports--> v15\n- v12 --supports--> v10\n- v08 --supports--> v0a\n- v09 --refutes--> v17\n- v04 --refutes--> v09\n- v0b --refutes--> v16\n- v05 --supports--> v13\n- v17 --supports--> v18\n- v0a --supports--> v07\n- v06 --supports--> v08\n- v05 --refutes--> v14\n- v11 --refutes--> v0d\n- v16 --supports--> v01\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9c46580c8751ea9c67b43354a1c0bfe86cd87d65abecae000fd9a71cd8ff0384","schema":"livegraph.boolean-propagation.v1","spec_hash":"b2238539385375717fea85d439197cb1c8557f8cab8f74bee44bf14edc353252","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-4a56afb9070ea6ac77f5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v07\n- v17 --supports--> v0b\n- v01 --refutes--> v02\n- v04 --supports--> v12\n- v17 --supports--> v14\n- v0e --refutes--> v05\n- v10 --refutes--> v06\n- v01 --refutes--> v17\n- v01 --supports--> v15\n- v16 --refutes--> v0c\n- v09 --refutes--> v00\n- v13 --supports--> v16\n- v10 --supports--> v0a\n- v12 --refutes--> v01\n- v10 --refutes--> v18\n- v06 --supports--> v0f\n- v11 --supports--> v04\n- v0b --refutes--> v10\n- v05 --supports--> v13\n- v09 --refutes--> v03\n- v07 --refutes--> v0e\n- v03 --supports--> v0d\n- v08 --supports--> v09\n- v06 --refutes--> v08\n\n[VERIFIED ROOT]\nv11 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fe5156c6ba1968de07cb75b4501c27127a2a57c50516c1c51477043faeeb18ad","schema":"livegraph.boolean-propagation.v1","spec_hash":"4a56afb9070ea6ac77f5894b220cbdba74b139fa2c36a98d59a6ae9289182aa8","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-b3cdfbeb0bb39d64f93a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v06\n- v07 --supports--> v13\n- v14 --supports--> v08\n- v04 --refutes--> v0e\n- v0e --supports--> v01\n- v02 --supports--> v03\n- v01 --refutes--> v05\n- v13 --supports--> v0f\n- v16 --refutes--> v15\n- v15 --supports--> v07\n- v12 --refutes--> v02\n- v0c --supports--> v0a\n- v00 --refutes--> v14\n- v17 --supports--> v12\n- v18 --refutes--> v09\n- v06 --supports--> v18\n- v00 --supports--> v11\n- v18 --refutes--> v04\n- v18 --supports--> v10\n- v0f --supports--> v00\n- v01 --supports--> v0c\n- v14 --supports--> v17\n- v01 --refutes--> v0b\n- v0f --supports--> v0d\n\n[VERIFIED ROOT]\nv16 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"df28362830190975be2b1c4db41803b1ee4b61876b75b042e2ffdf3b0a6e300f","schema":"livegraph.boolean-propagation.v1","spec_hash":"b3cdfbeb0bb39d64f93a8f5dddb3bc2c27e796138af491ee9f947307fcdc0a42","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v16\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-19c3e36e936f05ed674c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v05\n- v0f --supports--> v01\n- v00 --refutes--> v14\n- v02 --supports--> v0a\n- v03 --refutes--> v04\n- v06 --supports--> v12\n- v11 --supports--> v08\n- v15 --supports--> v13\n- v08 --supports--> v18\n- v02 --supports--> v0e\n- v0f --supports--> v02\n- v16 --refutes--> v17\n- v03 --refutes--> v11\n- v0b --refutes--> v0d\n- v0b --refutes--> v03\n- v17 --refutes--> v00\n- v13 --supports--> v09\n- v11 --refutes--> v10\n- v0a --refutes--> v15\n- v09 --refutes--> v0b\n- v00 --supports--> v07\n- v05 --refutes--> v0f\n- v02 --supports--> v0c\n- v18 --refutes--> v06\n\n[VERIFIED ROOT]\nv16 = FALSE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"93c74eb0475316fb3efdd70a1dced11ef38386b012b5693275f1ce5c86a3a7b8","schema":"livegraph.boolean-propagation.v1","spec_hash":"19c3e36e936f05ed674cdf908db36b1c15e6528b1cfeec155876252445b9f49f","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v16\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-2bbdb5debfa3421c7815","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v17 --refutes--> v06\n- v13 --refutes--> v08\n- v01 --supports--> v0b\n- v16 --refutes--> v14\n- v0b --refutes--> v0d\n- v0f --supports--> v15\n- v16 --refutes--> v18\n- v16 --refutes--> v02\n- v0f --supports--> v12\n- v0a --supports--> v07\n- v07 --supports--> v00\n- v14 --refutes--> v09\n- v14 --supports--> v05\n- v08 --refutes--> v0f\n- v00 --refutes--> v16\n- v0c --supports--> v0e\n- v0e --refutes--> v01\n- v12 --refutes--> v17\n- v10 --refutes--> v0c\n- v17 --supports--> v0a\n- v09 --supports--> v10\n- v01 --supports--> v04\n- v0c --supports--> v03\n- v00 --refutes--> v11\n\n[VERIFIED ROOT]\nv13 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3d8592a44279a45c940aa6643fec2db857d61dfc01ba8e7bc702cbab5f9e63c5","schema":"livegraph.boolean-propagation.v1","spec_hash":"2bbdb5debfa3421c78159b1b83bf4ed888d23b9ff4424580a5baef5fa80bb377","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-3343d169f8d8cb966051","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --refutes--> v16\n- v0a --supports--> v0d\n- v0e --supports--> v07\n- v15 --refutes--> v0f\n- v04 --supports--> v0c\n- v02 --refutes--> v18\n- v14 --supports--> v0a\n- v0f --refutes--> v02\n- v02 --refutes--> v05\n- v0c --supports--> v0e\n- v14 --supports--> v10\n- v0d --supports--> v01\n- v17 --supports--> v0b\n- v15 --refutes--> v00\n- v0e --supports--> v08\n- v02 --refutes--> v06\n- v06 --refutes--> v11\n- v13 --refutes--> v12\n- v17 --refutes--> v09\n- v08 --supports--> v15\n- v12 --supports--> v04\n- v01 --refutes--> v17\n- v09 --supports--> v13\n- v0a --refutes--> v03\n\n[VERIFIED ROOT]\nv14 = FALSE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e914ec0399b1d30a09bac34499a9d8f2734abad4f0b318f65285cdc8440e6438","schema":"livegraph.boolean-propagation.v1","spec_hash":"3343d169f8d8cb966051383d99518649386daf0347115bbb4910d9195c71ec9b","spec_json":"{\"edges\":[{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v14\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-7531d6cbe4fe6550811e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v16 --supports--> v17\n- v04 --refutes--> v01\n- v0e --refutes--> v09\n- v08 --supports--> v14\n- v0f --supports--> v18\n- v03 --refutes--> v0f\n- v0b --supports--> v00\n- v01 --refutes--> v16\n- v05 --refutes--> v11\n- v17 --supports--> v0b\n- v07 --refutes--> v12\n- v10 --refutes--> v0c\n- v17 --refutes--> v13\n- v0b --refutes--> v07\n- v03 --refutes--> v15\n- v09 --refutes--> v03\n- v10 --refutes--> v02\n- v14 --refutes--> v0a\n- v18 --supports--> v08\n- v17 --supports--> v06\n- v04 --refutes--> v0d\n- v05 --supports--> v0e\n- v07 --refutes--> v05\n- v0c --supports--> v04\n\n[VERIFIED ROOT]\nv10 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fe3050022fe07813bdbcb71229a0ae5decb338b29ba1e15738f811bb9c694abf","schema":"livegraph.boolean-propagation.v1","spec_hash":"7531d6cbe4fe6550811e639f5cc242a46b18f3446b3fcaf0a6c387b7c6305fd7","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-5957eb7f88dc25146764","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v11\n- v01 --refutes--> v02\n- v02 --supports--> v08\n- v0a --supports--> v0c\n- v12 --supports--> v0f\n- v03 --supports--> v0a\n- v16 --refutes--> v17\n- v12 --refutes--> v0e\n- v16 --refutes--> v0d\n- v10 --supports--> v12\n- v02 --supports--> v06\n- v11 --refutes--> v01\n- v13 --supports--> v03\n- v08 --supports--> v05\n- v0d --refutes--> v00\n- v08 --supports--> v10\n- v0d --supports--> v13\n- v11 --supports--> v04\n- v15 --refutes--> v16\n- v11 --supports--> v14\n- v07 --supports--> v09\n- v0b --refutes--> v15\n- v18 --refutes--> v0b\n- v12 --refutes--> v07\n\n[VERIFIED ROOT]\nv18 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"db80dafc3d06957dcddeb3f004d95b81664bde96f48f7775395e9be672e966db","schema":"livegraph.boolean-propagation.v1","spec_hash":"5957eb7f88dc25146764f3c9f2e4929409ad43f16774ccd96807ee816098f3bc","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v18\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-f33e54285888bec049e1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v0c\n- v0e --refutes--> v08\n- v18 --supports--> v0e\n- v0f --supports--> v07\n- v06 --refutes--> v0f\n- v02 --supports--> v13\n- v06 --supports--> v11\n- v01 --supports--> v18\n- v0d --supports--> v14\n- v0e --refutes--> v12\n- v03 --refutes--> v0b\n- v16 --supports--> v05\n- v07 --supports--> v03\n- v03 --supports--> v17\n- v0d --refutes--> v10\n- v03 --refutes--> v15\n- v04 --refutes--> v16\n- v0c --supports--> v01\n- v0d --refutes--> v0a\n- v00 --refutes--> v06\n- v0b --supports--> v0d\n- v02 --supports--> v00\n- v16 --supports--> v09\n- v08 --supports--> v04\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a5ad27b54a3e5210146b7d57662770df17820f3aed56ea52b1ccfa4cd2e3a29a","schema":"livegraph.boolean-propagation.v1","spec_hash":"f33e54285888bec049e1e91f76e034ce0c3635294dbe358266a63d740aed8493","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-d4f2f0ebc04541fbbcbf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v03\n- v13 --refutes--> v11\n- v10 --supports--> v0b\n- v08 --supports--> v02\n- v15 --supports--> v18\n- v03 --supports--> v14\n- v02 --supports--> v09\n- v03 --refutes--> v0e\n- v0d --refutes--> v05\n- v0b --refutes--> v15\n- v06 --refutes--> v07\n- v13 --refutes--> v0c\n- v0b --refutes--> v01\n- v10 --refutes--> v0f\n- v16 --supports--> v0d\n- v03 --supports--> v04\n- v03 --supports--> v0a\n- v0d --refutes--> v00\n- v06 --refutes--> v13\n- v12 --supports--> v16\n- v18 --refutes--> v08\n- v0c --supports--> v17\n- v09 --supports--> v12\n- v14 --supports--> v06\n\n[VERIFIED ROOT]\nv10 = TRUE\n\n[TARGET]\nv17\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5aad1cdc087c710b63b7369adf00df8ae4dcb025d6de020cd64fb9556829b58e","schema":"livegraph.boolean-propagation.v1","spec_hash":"d4f2f0ebc04541fbbcbf20e78a46c8dd6840dc57a5c71959d3d67442685734f6","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v17\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-43b13fd3a8a435caac97","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --refutes--> v06\n- v14 --refutes--> v00\n- v0c --refutes--> v08\n- v15 --refutes--> v07\n- v0a --refutes--> v11\n- v0d --supports--> v0b\n- v00 --supports--> v0a\n- v03 --supports--> v15\n- v03 --supports--> v10\n- v16 --refutes--> v04\n- v08 --supports--> v01\n- v11 --refutes--> v09\n- v02 --refutes--> v05\n- v18 --refutes--> v14\n- v13 --supports--> v18\n- v16 --refutes--> v02\n- v08 --refutes--> v16\n- v0d --refutes--> v13\n- v03 --refutes--> v12\n- v07 --refutes--> v0f\n- v0d --refutes--> v0e\n- v0d --supports--> v17\n- v0f --refutes--> v0c\n- v05 --refutes--> v0d\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b09c6ac34ea572ec8a81d8b845a97b845d673525b17245d7f1675549763068f2","schema":"livegraph.boolean-propagation.v1","spec_hash":"43b13fd3a8a435caac971cb711872dccbbc32598d231751b5de4046e147346c6","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-817275c6fc0309225f95","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v15\n- v10 --supports--> v16\n- v07 --refutes--> v0e\n- v09 --supports--> v08\n- v0e --supports--> v0f\n- v17 --supports--> v0b\n- v13 --refutes--> v18\n- v0d --refutes--> v05\n- v16 --refutes--> v11\n- v03 --refutes--> v07\n- v00 --refutes--> v0c\n- v11 --supports--> v14\n- v17 --refutes--> v12\n- v10 --refutes--> v0a\n- v11 --supports--> v01\n- v13 --supports--> v03\n- v0d --refutes--> v09\n- v0e --refutes--> v06\n- v00 --supports--> v10\n- v04 --refutes--> v17\n- v12 --supports--> v13\n- v08 --supports--> v02\n- v14 --supports--> v0d\n- v06 --supports--> v00\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9d2a988a5738341ff6a74793cb1bef2f3c4bb20e1a6079884e6e378aa97c81f8","schema":"livegraph.boolean-propagation.v1","spec_hash":"817275c6fc0309225f9586f3cacbaed63ae24658a043d5f3260f540ca79bd08e","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-229db44caf3b790e6598","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v01\n- v17 --refutes--> v02\n- v03 --refutes--> v00\n- v12 --refutes--> v15\n- v04 --supports--> v10\n- v06 --refutes--> v08\n- v08 --refutes--> v09\n- v17 --refutes--> v0f\n- v07 --refutes--> v12\n- v0a --refutes--> v0b\n- v17 --refutes--> v18\n- v0c --refutes--> v16\n- v12 --refutes--> v03\n- v0c --supports--> v11\n- v09 --refutes--> v07\n- v04 --supports--> v0c\n- v03 --refutes--> v13\n- v00 --refutes--> v0a\n- v16 --supports--> v0e\n- v01 --supports--> v06\n- v09 --supports--> v0d\n- v18 --supports--> v05\n- v12 --supports--> v14\n- v0b --refutes--> v04\n\n[VERIFIED ROOT]\nv17 = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"48a5dd3909c1f80ecf1db3daaca612b54618ada67af480babbf7b553f5e56ca0","schema":"livegraph.boolean-propagation.v1","spec_hash":"229db44caf3b790e65987639c8fecb84eb16f847990685ff55ee6b90ad9dbe3f","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v17\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-dbf4526d558f536dceab","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v0e\n- v0f --refutes--> v09\n- v0c --supports--> v12\n- v0c --supports--> v04\n- v18 --supports--> v0a\n- v0b --supports--> v07\n- v01 --refutes--> v18\n- v07 --refutes--> v05\n- v15 --refutes--> v0d\n- v0e --refutes--> v0c\n- v0a --supports--> v11\n- v02 --supports--> v03\n- v05 --refutes--> v08\n- v00 --refutes--> v17\n- v0f --refutes--> v14\n- v0d --supports--> v0f\n- v12 --refutes--> v02\n- v0a --supports--> v16\n- v00 --supports--> v01\n- v16 --refutes--> v10\n- v03 --supports--> v0b\n- v07 --supports--> v06\n- v0d --refutes--> v13\n- v16 --supports--> v15\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9a9b9b47f796d4613ab7e9e1f24f11140e81242fcc28690865e50c4db9dd9412","schema":"livegraph.boolean-propagation.v1","spec_hash":"dbf4526d558f536dceab8ee7f2ea2214993559c7145d808c0ed4e934b19b7236","spec_json":"{\"edges\":[{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-bbe77fa6e453e7b9d730","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v18 --refutes--> v07\n- v05 --refutes--> v0b\n- v12 --refutes--> v0f\n- v14 --supports--> v15\n- v15 --refutes--> v09\n- v09 --supports--> v05\n- v0c --refutes--> v0e\n- v0c --supports--> v0a\n- v17 --refutes--> v11\n- v16 --supports--> v01\n- v00 --supports--> v06\n- v01 --refutes--> v17\n- v18 --refutes--> v03\n- v16 --refutes--> v02\n- v04 --refutes--> v0c\n- v11 --refutes--> v14\n- v13 --supports--> v10\n- v18 --supports--> v00\n- v17 --supports--> v08\n- v13 --supports--> v16\n- v0a --supports--> v18\n- v0f --supports--> v13\n- v00 --refutes--> v0d\n- v05 --supports--> v04\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d5407e73f9236b4ef596a8c433c0fb9501a0eee2bb219c5b88e06ef2a5155954","schema":"livegraph.boolean-propagation.v1","spec_hash":"bbe77fa6e453e7b9d7309fcc77048b229e7997075e0213a2b5a53a608b87fc5c","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-3fece48472fe04011242","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v03\n- v05 --refutes--> v09\n- v09 --supports--> v11\n- v0a --refutes--> v04\n- v17 --refutes--> v05\n- v04 --refutes--> v17\n- v01 --supports--> v18\n- v14 --supports--> v0c\n- v17 --refutes--> v08\n- v13 --supports--> v02\n- v0f --refutes--> v13\n- v02 --supports--> v0b\n- v12 --refutes--> v0a\n- v10 --refutes--> v0d\n- v11 --refutes--> v01\n- v01 --supports--> v00\n- v0d --supports--> v12\n- v01 --supports--> v07\n- v11 --supports--> v06\n- v0f --supports--> v0e\n- v13 --supports--> v15\n- v16 --refutes--> v0f\n- v07 --supports--> v14\n- v14 --supports--> v16\n\n[VERIFIED ROOT]\nv10 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cbbf0496db9b5cdd48dd96a5bea7a4917624ce6227599f0fd403bb7e5c078fd2","schema":"livegraph.boolean-propagation.v1","spec_hash":"3fece48472fe040112425ecc9bbfc8e2e2a1d605be956bcf4d569b338ec9637b","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-25110afc2c2255288c03","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v0a\n- v18 --supports--> v12\n- v13 --supports--> v0b\n- v05 --supports--> v06\n- v17 --refutes--> v0c\n- v02 --supports--> v0d\n- v13 --supports--> v03\n- v10 --supports--> v18\n- v13 --refutes--> v00\n- v04 --refutes--> v14\n- v0f --refutes--> v17\n- v03 --supports--> v09\n- v15 --refutes--> v13\n- v0d --supports--> v05\n- v17 --supports--> v15\n- v0a --refutes--> v16\n- v14 --supports--> v02\n- v18 --refutes--> v08\n- v0e --supports--> v04\n- v0a --refutes--> v01\n- v01 --supports--> v10\n- v17 --supports--> v07\n- v09 --supports--> v0e\n- v02 --supports--> v11\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c1e9e5fa2d781302418293e45a5a1a4002e395379e455b390b0f6d6b638a71ab","schema":"livegraph.boolean-propagation.v1","spec_hash":"25110afc2c2255288c03e8e539598fcc0491458cd1100d9dad354995aa75e143","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-c2ce841aa724a7cc2af6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v0f\n- v08 --supports--> v01\n- v0f --supports--> v0b\n- v10 --supports--> v0c\n- v11 --supports--> v16\n- v0d --refutes--> v10\n- v0d --supports--> v12\n- v13 --supports--> v04\n- v0a --supports--> v13\n- v08 --refutes--> v17\n- v13 --supports--> v05\n- v18 --refutes--> v09\n- v14 --supports--> v15\n- v0b --refutes--> v11\n- v03 --refutes--> v0e\n- v10 --refutes--> v18\n- v14 --refutes--> v06\n- v14 --supports--> v00\n- v09 --supports--> v14\n- v03 --refutes--> v07\n- v11 --refutes--> v03\n- v15 --refutes--> v08\n- v0e --refutes--> v0a\n- v02 --refutes--> v0d\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"21bed99b2218f13cf200b7183bcac8952daa1236e99e2a37cfbff000689410ba","schema":"livegraph.boolean-propagation.v1","spec_hash":"c2ce841aa724a7cc2af650a5a011be91850c40368ce9a9130fe7ccfb76442912","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-0e99d5fd01eee6cdeb2c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v00\n- v16 --refutes--> v17\n- v10 --refutes--> v0b\n- v0c --refutes--> v08\n- v16 --refutes--> v02\n- v04 --supports--> v09\n- v0a --supports--> v0f\n- v0a --supports--> v13\n- v04 --refutes--> v15\n- v14 --refutes--> v0c\n- v08 --refutes--> v0a\n- v03 --supports--> v18\n- v16 --refutes--> v11\n- v03 --supports--> v01\n- v0b --supports--> v03\n- v07 --refutes--> v14\n- v13 --supports--> v04\n- v08 --supports--> v0e\n- v12 --refutes--> v16\n- v01 --refutes--> v06\n- v06 --refutes--> v05\n- v16 --supports--> v07\n- v09 --refutes--> v10\n- v05 --supports--> v0d\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"978ce10a634b5688deca2621080347435bfbcbf54939ead0b72b8a8f628afbc0","schema":"livegraph.boolean-propagation.v1","spec_hash":"0e99d5fd01eee6cdeb2c57f2351ad47fa4845fa813d290e3cb11e1a2b6c103f5","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":16,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-6409be2a09b36e667f66","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v06\n- v10 --refutes--> v13\n- v05 --refutes--> v17\n- v03 --refutes--> v04\n- v02 --refutes--> v15\n- v15 --refutes--> v14\n- v06 --supports--> v07\n- v15 --refutes--> v12\n- v13 --refutes--> v05\n- v04 --refutes--> v0b\n- v09 --supports--> v03\n- v0f --supports--> v08\n- v0e --refutes--> v0f\n- v15 --supports--> v18\n- v09 --refutes--> v0d\n- v11 --refutes--> v0e\n- v06 --supports--> v11\n- v0c --supports--> v01\n- v06 --supports--> v00\n- v09 --supports--> v0a\n- v0f --refutes--> v02\n- v16 --refutes--> v10\n- v18 --refutes--> v0c\n- v17 --supports--> v09\n\n[VERIFIED ROOT]\nv16 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5cccdd5d54cd81a7a441446cf4f2e4c345c6413c3a97b3ba04e26bfcf0b07106","schema":"livegraph.boolean-propagation.v1","spec_hash":"6409be2a09b36e667f663c9951068a13c970efc6483c20f6ecf778f4bf3374d9","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v16\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-e08a0e510c9088ff38a0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v14\n- v0f --refutes--> v04\n- v03 --refutes--> v17\n- v08 --refutes--> v0f\n- v13 --refutes--> v01\n- v17 --refutes--> v0a\n- v17 --refutes--> v07\n- v12 --refutes--> v13\n- v07 --supports--> v00\n- v0f --refutes--> v0b\n- v01 --refutes--> v0d\n- v18 --supports--> v11\n- v08 --refutes--> v02\n- v0f --refutes--> v05\n- v09 --supports--> v06\n- v11 --refutes--> v12\n- v10 --refutes--> v16\n- v00 --refutes--> v08\n- v10 --refutes--> v15\n- v04 --supports--> v10\n- v06 --supports--> v18\n- v0e --refutes--> v03\n- v10 --supports--> v09\n- v01 --supports--> v0c\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"41aee4ef8a8c6052e934336eb79c6ed282e8e44c6b538eed79df53ffe95e673a","schema":"livegraph.boolean-propagation.v1","spec_hash":"e08a0e510c9088ff38a0022370db821529d93809c6e76a8275c8a3d887841d2d","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-0df8d6b8aabc5ff78284","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --supports--> v09\n- v10 --supports--> v14\n- v0c --supports--> v0d\n- v03 --supports--> v0a\n- v12 --refutes--> v0b\n- v16 --supports--> v03\n- v0e --refutes--> v05\n- v13 --supports--> v06\n- v13 --refutes--> v11\n- v17 --refutes--> v0c\n- v07 --refutes--> v02\n- v12 --refutes--> v00\n- v10 --refutes--> v08\n- v07 --refutes--> v18\n- v00 --refutes--> v04\n- v0f --supports--> v07\n- v04 --refutes--> v10\n- v0e --supports--> v15\n- v08 --refutes--> v0e\n- v06 --supports--> v0f\n- v0c --refutes--> v13\n- v15 --supports--> v16\n- v0a --refutes--> v17\n- v16 --supports--> v01\n\n[VERIFIED ROOT]\nv12 = FALSE\n\n[TARGET]\nv18\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d9114ec62980216529fad95597deee2a0024decd7e3c84205a1d8c7778734100","schema":"livegraph.boolean-propagation.v1","spec_hash":"0df8d6b8aabc5ff782842d6c78258508ff8689efb57db55962db97fb4a2e2294","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v12\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v18\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-875c6b5c45bc5c6e1789","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v13\n- v05 --refutes--> v17\n- v08 --supports--> v0a\n- v17 --refutes--> v15\n- v17 --refutes--> v11\n- v16 --refutes--> v09\n- v0c --supports--> v18\n- v00 --supports--> v0f\n- v07 --supports--> v0e\n- v0d --refutes--> v16\n- v01 --refutes--> v0d\n- v0e --refutes--> v06\n- v16 --refutes--> v05\n- v03 --refutes--> v02\n- v02 --refutes--> v14\n- v14 --refutes--> v00\n- v05 --supports--> v0b\n- v13 --supports--> v08\n- v01 --supports--> v04\n- v0c --supports--> v01\n- v08 --refutes--> v12\n- v01 --supports--> v10\n- v15 --supports--> v07\n- v00 --supports--> v0c\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7813e66de698f9e848ccba88b8afe0c6d7679232763eea397aea144c64673acf","schema":"livegraph.boolean-propagation.v1","spec_hash":"875c6b5c45bc5c6e17891d0a4c8025b854e07cc9ae0e6e8552368c35bc0fc59c","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-5fe558b94d4124091fd0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v0b\n- v02 --refutes--> v01\n- v03 --supports--> v04\n- v11 --refutes--> v0e\n- v16 --refutes--> v13\n- v0e --supports--> v0a\n- v01 --refutes--> v0f\n- v03 --supports--> v18\n- v07 --refutes--> v10\n- v14 --refutes--> v05\n- v14 --supports--> v0d\n- v10 --refutes--> v11\n- v0f --refutes--> v16\n- v10 --refutes--> v00\n- v16 --supports--> v0c\n- v15 --supports--> v17\n- v05 --refutes--> v09\n- v05 --supports--> v02\n- v17 --refutes--> v07\n- v0b --supports--> v14\n- v16 --refutes--> v06\n- v0f --refutes--> v12\n- v13 --refutes--> v03\n- v05 --refutes--> v08\n\n[VERIFIED ROOT]\nv15 = FALSE\n\n[TARGET]\nv18\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3933d68e56edaf4a12ac71ecf42134abebd84d22c7ed5be3855db8408fd8cd22","schema":"livegraph.boolean-propagation.v1","spec_hash":"5fe558b94d4124091fd011a0ee2dedb4d19d4890595054b4ef7ea2c7c08388af","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v15\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v18\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-71c7c8d34f30fd60f829","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v0f\n- v16 --supports--> v05\n- v0c --supports--> v16\n- v08 --refutes--> v09\n- v09 --supports--> v15\n- v18 --supports--> v17\n- v17 --supports--> v08\n- v0f --refutes--> v07\n- v10 --refutes--> v11\n- v18 --supports--> v14\n- v0f --supports--> v00\n- v04 --supports--> v0e\n- v11 --supports--> v01\n- v11 --refutes--> v0d\n- v18 --supports--> v03\n- v06 --refutes--> v0c\n- v0d --supports--> v0b\n- v0e --refutes--> v18\n- v18 --supports--> v02\n- v15 --supports--> v06\n- v16 --refutes--> v12\n- v00 --refutes--> v04\n- v08 --refutes--> v0a\n- v16 --supports--> v13\n\n[VERIFIED ROOT]\nv10 = FALSE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"af163a3c8d49479193017ac1b85e1a16cf1cb3c26536517c4987f0c6e82e7281","schema":"livegraph.boolean-propagation.v1","spec_hash":"71c7c8d34f30fd60f829a1f87e913a2ff76d5866ae41f62da0e2b8ff2407c3a5","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-dfd885b2b570056d9ae0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v17\n- v17 --refutes--> v06\n- v13 --refutes--> v0a\n- v14 --refutes--> v0f\n- v02 --supports--> v0e\n- v00 --refutes--> v0c\n- v0f --refutes--> v16\n- v03 --refutes--> v15\n- v00 --supports--> v05\n- v16 --supports--> v01\n- v15 --supports--> v14\n- v12 --supports--> v13\n- v07 --refutes--> v08\n- v00 --refutes--> v11\n- v16 --supports--> v09\n- v04 --refutes--> v03\n- v13 --supports--> v02\n- v06 --refutes--> v0d\n- v0e --refutes--> v07\n- v0f --supports--> v10\n- v09 --refutes--> v12\n- v06 --refutes--> v04\n- v14 --supports--> v18\n- v15 --refutes--> v0b\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"eb02755673fe9c3c6bff406eb503846fe7d00fe37d6543d4da4b3fbca6a2ae16","schema":"livegraph.boolean-propagation.v1","spec_hash":"dfd885b2b570056d9ae0a21914d1bee192fc8354af6bea8d0eaa91abfe532a10","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-efc5fd0c8810cf5fdbd6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v11\n- v15 --supports--> v07\n- v11 --refutes--> v02\n- v06 --supports--> v08\n- v14 --supports--> v17\n- v15 --refutes--> v0e\n- v14 --refutes--> v01\n- v02 --refutes--> v18\n- v06 --supports--> v04\n- v11 --refutes--> v13\n- v16 --supports--> v09\n- v18 --supports--> v0c\n- v06 --refutes--> v12\n- v14 --refutes--> v05\n- v04 --refutes--> v0a\n- v10 --supports--> v0b\n- v18 --refutes--> v10\n- v0b --supports--> v0d\n- v0e --refutes--> v00\n- v0f --refutes--> v15\n- v17 --supports--> v16\n- v00 --supports--> v06\n- v18 --supports--> v03\n- v0a --supports--> v14\n\n[VERIFIED ROOT]\nv0f = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d1cefeeb7a3845a2cf086a927d75b1a2487280c93b1c63a20ed0240f921d7a29","schema":"livegraph.boolean-propagation.v1","spec_hash":"efc5fd0c8810cf5fdbd60c44a95cfa2052eee84b399d23034a5b46646298e229","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-728d5f8c1a56e157d6df","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v03\n- v13 --supports--> v15\n- v07 --supports--> v0e\n- v14 --supports--> v00\n- v0a --refutes--> v10\n- v05 --supports--> v08\n- v10 --supports--> v01\n- v0d --supports--> v02\n- v16 --refutes--> v12\n- v00 --supports--> v13\n- v08 --supports--> v0c\n- v0c --supports--> v0f\n- v0c --refutes--> v06\n- v0f --refutes--> v11\n- v0f --supports--> v14\n- v0e --supports--> v09\n- v14 --supports--> v18\n- v04 --refutes--> v0a\n- v16 --refutes--> v0d\n- v09 --refutes--> v16\n- v01 --refutes--> v07\n- v08 --supports--> v0b\n- v0d --supports--> v17\n- v13 --supports--> v04\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv17\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"320c9c9fc73be6a744381a22fec056b26d42d9704a0484b88c26e4bef4480a51","schema":"livegraph.boolean-propagation.v1","spec_hash":"728d5f8c1a56e157d6df0a7b11d592ff3b1779ab81ead37236cf2131c6c7c14c","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":16,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v17\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-d195ec12d599f164e6c8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v14\n- v00 --refutes--> v0e\n- v0c --refutes--> v01\n- v00 --supports--> v16\n- v0e --supports--> v05\n- v13 --refutes--> v12\n- v05 --refutes--> v03\n- v17 --refutes--> v07\n- v17 --supports--> v00\n- v13 --refutes--> v17\n- v17 --supports--> v10\n- v03 --refutes--> v15\n- v01 --supports--> v13\n- v18 --refutes--> v11\n- v17 --supports--> v02\n- v15 --refutes--> v0a\n- v11 --refutes--> v0f\n- v0d --refutes--> v08\n- v0f --refutes--> v0c\n- v08 --supports--> v18\n- v04 --refutes--> v06\n- v06 --refutes--> v0d\n- v0d --refutes--> v09\n- v0e --supports--> v0b\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5c2130694f53258614c0e44ede7526bebb22a57d39bfead85228f4bb994be38d","schema":"livegraph.boolean-propagation.v1","spec_hash":"d195ec12d599f164e6c87ea79bfb6b36270439bbf916cdca206a0018ea54e65a","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-cdfa89d1c647f449ef0d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --supports--> v18\n- v10 --supports--> v0c\n- v0e --refutes--> v01\n- v16 --refutes--> v09\n- v12 --supports--> v10\n- v07 --supports--> v03\n- v15 --refutes--> v07\n- v17 --supports--> v11\n- v07 --supports--> v14\n- v0b --refutes--> v13\n- v06 --refutes--> v00\n- v00 --supports--> v17\n- v05 --refutes--> v06\n- v09 --refutes--> v0d\n- v0b --refutes--> v05\n- v05 --supports--> v04\n- v0d --supports--> v0e\n- v16 --supports--> v0a\n- v06 --refutes--> v08\n- v0c --supports--> v02\n- v11 --refutes--> v15\n- v0e --refutes--> v12\n- v0c --supports--> v0b\n- v0d --refutes--> v0f\n\n[VERIFIED ROOT]\nv16 = TRUE\n\n[TARGET]\nv18\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4c3b862b0592b5121ae1cc52819574e3936480e62bd9c89ab8833ddf2afb5b9d","schema":"livegraph.boolean-propagation.v1","spec_hash":"cdfa89d1c647f449ef0dfcdca6b155f9637fa308de375abfbc3e30258582c632","spec_json":"{\"edges\":[{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v16\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v18\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-f54334f37fde27aa3209","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v12\n- v12 --supports--> v01\n- v00 --refutes--> v0a\n- v13 --supports--> v03\n- v0e --supports--> v11\n- v15 --supports--> v07\n- v01 --refutes--> v05\n- v0f --supports--> v14\n- v15 --refutes--> v09\n- v0b --refutes--> v10\n- v05 --supports--> v0b\n- v18 --refutes--> v04\n- v14 --supports--> v0d\n- v13 --supports--> v16\n- v11 --refutes--> v02\n- v14 --supports--> v00\n- v0b --refutes--> v08\n- v04 --refutes--> v17\n- v08 --supports--> v13\n- v0f --supports--> v0c\n- v13 --supports--> v15\n- v05 --supports--> v06\n- v17 --refutes--> v0e\n- v0a --supports--> v18\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"97fdbc50dd387dd46189848a0c4ae2cedd63a26af57205faaba66ad10d7bab41","schema":"livegraph.boolean-propagation.v1","spec_hash":"f54334f37fde27aa32090b9eb918f9c52218043a6a1afdd9273d783b331c2c2d","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-d6edfbb957fc4eb0e89a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v02\n- v13 --refutes--> v16\n- v17 --supports--> v00\n- v00 --refutes--> v07\n- v15 --supports--> v0e\n- v03 --refutes--> v01\n- v00 --supports--> v04\n- v0c --refutes--> v06\n- v04 --refutes--> v0f\n- v04 --supports--> v05\n- v06 --refutes--> v03\n- v0e --supports--> v10\n- v18 --supports--> v11\n- v0d --supports--> v13\n- v10 --refutes--> v09\n- v10 --supports--> v17\n- v18 --refutes--> v15\n- v0a --refutes--> v14\n- v03 --supports--> v12\n- v13 --supports--> v0a\n- v08 --refutes--> v0d\n- v14 --refutes--> v0c\n- v05 --refutes--> v08\n- v17 --refutes--> v0b\n\n[VERIFIED ROOT]\nv18 = FALSE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a6579c22bf6edb729b442956d62419e129e47e8394e52fab5d53d38097d5e5ff","schema":"livegraph.boolean-propagation.v1","spec_hash":"d6edfbb957fc4eb0e89a03cefe443660f06178b99665a0591d567baca810b3a7","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v18\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-21e6d60e42ccfd5a0796","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v14\n- v0b --refutes--> v02\n- v05 --refutes--> v08\n- v0a --supports--> v12\n- v14 --refutes--> v0d\n- v00 --refutes--> v0e\n- v00 --supports--> v15\n- v09 --refutes--> v04\n- v10 --refutes--> v0f\n- v12 --supports--> v0b\n- v0a --supports--> v11\n- v04 --refutes--> v07\n- v08 --refutes--> v03\n- v01 --refutes--> v13\n- v05 --refutes--> v16\n- v15 --refutes--> v01\n- v06 --supports--> v0a\n- v08 --supports--> v06\n- v07 --supports--> v00\n- v02 --refutes--> v09\n- v0d --refutes--> v10\n- v06 --supports--> v17\n- v06 --refutes--> v0c\n- v10 --refutes--> v18\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv18\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"35f8875993bc816cab4390fac49caac63f28dfdb27de3c2b22e3fb557cdd2ec2","schema":"livegraph.boolean-propagation.v1","spec_hash":"21e6d60e42ccfd5a0796c679cb7b195b8843fe3fe79e2ec0f974684a7f9abfc0","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":16,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v18\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-10550071efa21a251c40","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v10\n- v06 --refutes--> v09\n- v01 --supports--> v15\n- v0c --refutes--> v02\n- v00 --supports--> v06\n- v11 --supports--> v0c\n- v10 --supports--> v03\n- v06 --refutes--> v0d\n- v06 --supports--> v12\n- v11 --supports--> v0e\n- v10 --supports--> v04\n- v0c --supports--> v13\n- v05 --refutes--> v07\n- v07 --supports--> v16\n- v01 --supports--> v0b\n- v17 --supports--> v0f\n- v0f --supports--> v01\n- v18 --refutes--> v11\n- v03 --supports--> v05\n- v12 --refutes--> v18\n- v07 --supports--> v00\n- v13 --refutes--> v14\n- v15 --supports--> v0a\n- v15 --refutes--> v08\n\n[VERIFIED ROOT]\nv17 = FALSE\n\n[TARGET]\nv14\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2f1353008d00bc234e4fdc3d98bd341e35982509b2d1f0a37778449e0232afad","schema":"livegraph.boolean-propagation.v1","spec_hash":"10550071efa21a251c402b429989226687686bd47045d5604531adb1afa3c03f","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v17\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-fccccfdf1f0e943f72aa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v08\n- v0b --supports--> v0e\n- v08 --supports--> v15\n- v16 --refutes--> v0f\n- v04 --refutes--> v13\n- v03 --supports--> v00\n- v00 --supports--> v11\n- v14 --refutes--> v10\n- v0c --refutes--> v06\n- v14 --supports--> v02\n- v0b --supports--> v0c\n- v0a --refutes--> v05\n- v18 --supports--> v0d\n- v0a --refutes--> v18\n- v04 --refutes--> v01\n- v15 --supports--> v0b\n- v0f --refutes--> v14\n- v0d --refutes--> v03\n- v12 --refutes--> v07\n- v02 --supports--> v04\n- v0c --supports--> v09\n- v0c --refutes--> v16\n- v12 --supports--> v0a\n- v03 --supports--> v17\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2dfdc085fae1f8877e833a59065898b37d3071379672837e10b2bde8cc6541b8","schema":"livegraph.boolean-propagation.v1","spec_hash":"fccccfdf1f0e943f72aa6fdaf99bc915863a64b18206bf75dfe5537ac9528ff8","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-9cc64f5c15cf6db8eea4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v0c\n- v07 --supports--> v03\n- v0d --supports--> v14\n- v04 --supports--> v00\n- v16 --supports--> v05\n- v18 --supports--> v10\n- v17 --refutes--> v18\n- v03 --supports--> v0a\n- v17 --supports--> v08\n- v11 --supports--> v0e\n- v0f --supports--> v17\n- v05 --refutes--> v07\n- v00 --refutes--> v02\n- v0f --supports--> v09\n- v02 --supports--> v11\n- v11 --supports--> v16\n- v02 --supports--> v01\n- v0c --supports--> v0f\n- v18 --supports--> v12\n- v13 --supports--> v0d\n- v17 --refutes--> v06\n- v0d --supports--> v04\n- v0a --supports--> v0b\n- v13 --refutes--> v15\n\n[VERIFIED ROOT]\nv13 = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0ff092a0532a0d72899726ed692074a38fffef51c70b6f288183371f4d4b6f12","schema":"livegraph.boolean-propagation.v1","spec_hash":"9cc64f5c15cf6db8eea4da3c2ad2fc5cd8af8ea10c011559122ec0b9237ced60","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-b103ce84f47868412986","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v02\n- v04 --refutes--> v0e\n- v08 --supports--> v04\n- v09 --refutes--> v17\n- v13 --supports--> v09\n- v16 --supports--> v06\n- v0d --supports--> v00\n- v0e --supports--> v0b\n- v12 --supports--> v07\n- v0b --supports--> v16\n- v03 --refutes--> v13\n- v06 --supports--> v14\n- v14 --supports--> v15\n- v0d --refutes--> v03\n- v06 --refutes--> v0f\n- v01 --refutes--> v12\n- v07 --refutes--> v0d\n- v05 --supports--> v11\n- v14 --supports--> v0a\n- v08 --refutes--> v0c\n- v0a --refutes--> v10\n- v0a --supports--> v01\n- v0b --supports--> v18\n- v09 --supports--> v05\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a681b72cf4555f8594def47a1b92d4e0439fa584b278939dae3e4381bd4ee2a7","schema":"livegraph.boolean-propagation.v1","spec_hash":"b103ce84f47868412986a1b3c2455a6d756d9a64110114ccb7897556ae56a075","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":16,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-65126e56342911ac3448","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --supports--> v00\n- v18 --refutes--> v0a\n- v08 --supports--> v17\n- v01 --refutes--> v15\n- v04 --supports--> v08\n- v14 --supports--> v0f\n- v16 --refutes--> v0e\n- v00 --supports--> v0b\n- v07 --supports--> v09\n- v15 --refutes--> v04\n- v06 --refutes--> v05\n- v05 --supports--> v16\n- v00 --supports--> v13\n- v18 --refutes--> v12\n- v17 --supports--> v18\n- v0f --refutes--> v11\n- v0e --refutes--> v01\n- v0e --supports--> v02\n- v00 --supports--> v0c\n- v08 --supports--> v03\n- v00 --refutes--> v14\n- v09 --supports--> v10\n- v0f --supports--> v06\n- v15 --refutes--> v0d\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"128f7462d5130e844d1732bc3aa644028d5ce67c5c77b88c8ce014a2d7a0089c","schema":"livegraph.boolean-propagation.v1","spec_hash":"65126e56342911ac3448bd320cabacb58524553880c68788a5c12db7d7b5a0ee","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-e15411d349f54afece22","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v05\n- v16 --supports--> v17\n- v01 --refutes--> v11\n- v13 --refutes--> v00\n- v0a --supports--> v0c\n- v17 --supports--> v0d\n- v01 --supports--> v12\n- v0a --refutes--> v18\n- v14 --supports--> v08\n- v10 --supports--> v15\n- v11 --supports--> v16\n- v0d --supports--> v02\n- v16 --supports--> v04\n- v0a --refutes--> v07\n- v09 --supports--> v0e\n- v02 --supports--> v0f\n- v10 --supports--> v13\n- v13 --supports--> v0a\n- v05 --supports--> v01\n- v06 --refutes--> v09\n- v14 --supports--> v10\n- v0e --refutes--> v14\n- v0c --refutes--> v0b\n- v0a --supports--> v03\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f8b273afe4d186ee374640d7e78f709412d41a448a25c761514650123a46c2b6","schema":"livegraph.boolean-propagation.v1","spec_hash":"e15411d349f54afece224a2d239e38700c3feac02dca24665eec829ba427dc52","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-488f3a17a4bd7803293f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v0c\n- v06 --supports--> v0d\n- v11 --supports--> v06\n- v0c --refutes--> v09\n- v02 --supports--> v07\n- v17 --supports--> v0a\n- v05 --supports--> v13\n- v16 --supports--> v11\n- v0a --refutes--> v01\n- v0a --supports--> v16\n- v08 --supports--> v03\n- v0d --supports--> v18\n- v02 --refutes--> v0b\n- v05 --supports--> v12\n- v0f --supports--> v15\n- v0c --refutes--> v02\n- v0e --refutes--> v05\n- v13 --refutes--> v04\n- v0b --supports--> v14\n- v04 --refutes--> v0f\n- v0f --supports--> v00\n- v00 --refutes--> v08\n- v08 --supports--> v17\n- v0b --supports--> v10\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv14\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"04609956dcfd69a99eddbdc4608a89ae2c52b7012c47a94570821bb00ecf3d37","schema":"livegraph.boolean-propagation.v1","spec_hash":"488f3a17a4bd7803293f2128faf92bc8de1369d9c1d2aab027484b67e4c87fc0","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-9c7b3f22714d20555f81","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v0c\n- v01 --refutes--> v08\n- v0d --supports--> v18\n- v0c --refutes--> v14\n- v0c --refutes--> v0e\n- v16 --supports--> v0d\n- v0a --supports--> v03\n- v0b --refutes--> v00\n- v06 --supports--> v13\n- v07 --supports--> v01\n- v02 --refutes--> v0a\n- v0c --supports--> v15\n- v05 --supports--> v16\n- v0e --supports--> v11\n- v08 --refutes--> v05\n- v03 --refutes--> v06\n- v0b --supports--> v09\n- v0d --supports--> v12\n- v02 --refutes--> v04\n- v17 --supports--> v02\n- v06 --supports--> v0b\n- v11 --refutes--> v17\n- v01 --refutes--> v10\n- v0d --supports--> v0f\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"588e67f0eb94e5be285286f743e97e1e6fe700c0af81ce65544183f04da65a6f","schema":"livegraph.boolean-propagation.v1","spec_hash":"9c7b3f22714d20555f81fba15c842f9e3bd25b255dde7b8acf465aab5248b22e","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-fabd03112bcef2f55b49","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v03\n- v02 --refutes--> v00\n- v09 --supports--> v0b\n- v09 --refutes--> v16\n- v0c --refutes--> v04\n- v13 --refutes--> v0d\n- v12 --refutes--> v0a\n- v12 --refutes--> v02\n- v14 --supports--> v01\n- v17 --supports--> v07\n- v04 --supports--> v09\n- v14 --refutes--> v0e\n- v08 --supports--> v18\n- v18 --supports--> v14\n- v11 --refutes--> v17\n- v04 --refutes--> v0f\n- v00 --refutes--> v08\n- v16 --supports--> v15\n- v0e --refutes--> v0c\n- v0e --supports--> v05\n- v11 --supports--> v06\n- v07 --refutes--> v10\n- v10 --refutes--> v12\n- v16 --supports--> v13\n\n[VERIFIED ROOT]\nv11 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9b50f2dee003eef4b5085c7a9ab9f1c4dd76f6cec01dde91a75be61b0ee94570","schema":"livegraph.boolean-propagation.v1","spec_hash":"fabd03112bcef2f55b49859d69f9fc13ae2879ea6f1de000c1d5183c03ef267a","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-ede025ed08ae8baa1a78","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v01\n- v18 --refutes--> v14\n- v13 --refutes--> v11\n- v11 --refutes--> v0c\n- v0a --supports--> v0d\n- v17 --supports--> v0b\n- v0e --supports--> v00\n- v0b --supports--> v02\n- v0c --supports--> v0f\n- v0b --refutes--> v10\n- v07 --refutes--> v03\n- v01 --refutes--> v06\n- v16 --refutes--> v0e\n- v04 --refutes--> v05\n- v07 --supports--> v13\n- v0b --supports--> v08\n- v05 --refutes--> v0a\n- v14 --refutes--> v15\n- v0f --refutes--> v16\n- v0e --refutes--> v17\n- v14 --refutes--> v04\n- v07 --refutes--> v09\n- v05 --refutes--> v12\n- v01 --refutes--> v07\n\n[VERIFIED ROOT]\nv18 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4e438a642bc5b55a096d16c4495cb6b286338e8fdf76cfe191019f01e03e441e","schema":"livegraph.boolean-propagation.v1","spec_hash":"ede025ed08ae8baa1a78fabf76fa2524377c23c4a12596fe0cad21164d0785fe","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v18\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-03e4fa88556850fb0061","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v17\n- v02 --supports--> v11\n- v10 --supports--> v12\n- v07 --supports--> v13\n- v06 --supports--> v07\n- v0b --supports--> v06\n- v0a --supports--> v01\n- v04 --supports--> v0a\n- v05 --refutes--> v14\n- v09 --refutes--> v15\n- v09 --refutes--> v18\n- v0e --refutes--> v09\n- v0e --supports--> v0d\n- v14 --supports--> v0f\n- v07 --supports--> v0e\n- v15 --refutes--> v16\n- v16 --refutes--> v00\n- v05 --supports--> v08\n- v10 --refutes--> v0b\n- v02 --refutes--> v04\n- v0a --refutes--> v0c\n- v03 --refutes--> v10\n- v17 --supports--> v05\n- v14 --refutes--> v02\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a09a628e1dfa73582c4275e49d6e2e7af2c563203339b6dc33796753d4c99f59","schema":"livegraph.boolean-propagation.v1","spec_hash":"03e4fa88556850fb0061ae82ee00a019b25b3c8339fc89041e980ea1ff291aa2","spec_json":"{\"edges\":[{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-a6b65e6feaf45cf0e955","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v17\n- v08 --supports--> v0f\n- v14 --refutes--> v0b\n- v0d --supports--> v12\n- v09 --supports--> v0d\n- v03 --supports--> v0e\n- v18 --supports--> v14\n- v14 --refutes--> v05\n- v0d --supports--> v0a\n- v04 --supports--> v00\n- v02 --supports--> v18\n- v14 --refutes--> v07\n- v0a --refutes--> v0c\n- v04 --refutes--> v15\n- v0d --refutes--> v06\n- v10 --refutes--> v11\n- v16 --refutes--> v02\n- v0c --refutes--> v10\n- v0e --supports--> v16\n- v13 --supports--> v09\n- v08 --supports--> v01\n- v07 --refutes--> v04\n- v04 --refutes--> v08\n- v0f --supports--> v13\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4526ede1f77df7a6c721d14d3b035eedf9191b8a66b15f96b923354b1844b8af","schema":"livegraph.boolean-propagation.v1","spec_hash":"a6b65e6feaf45cf0e955dd14c0e30c24349ac6df692408536a5daf5a2ca8585b","spec_json":"{\"edges\":[{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-e3168c1893c7a42a958f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v07\n- v0f --supports--> v04\n- v11 --supports--> v0d\n- v07 --refutes--> v12\n- v13 --supports--> v0a\n- v09 --supports--> v14\n- v02 --refutes--> v18\n- v0f --refutes--> v09\n- v14 --supports--> v0b\n- v06 --refutes--> v15\n- v17 --refutes--> v08\n- v03 --supports--> v06\n- v06 --supports--> v16\n- v0b --refutes--> v02\n- v12 --refutes--> v13\n- v18 --supports--> v11\n- v17 --refutes--> v00\n- v11 --refutes--> v10\n- v07 --refutes--> v05\n- v06 --refutes--> v17\n- v08 --supports--> v0f\n- v0a --supports--> v03\n- v07 --supports--> v0e\n- v0f --supports--> v01\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5a24d9a6e5dcd8878f0e864704bb98318f0a6ea79029fa384104a81cfd790a9c","schema":"livegraph.boolean-propagation.v1","spec_hash":"e3168c1893c7a42a958f398a3c7ef68aeca1edf383bfc8ddc859fcffd4fdc828","spec_json":"{\"edges\":[{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-ae529d0e017dca0d8be9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v0c\n- v08 --refutes--> v0e\n- v0d --supports--> v10\n- v13 --refutes--> v14\n- v11 --supports--> v0b\n- v12 --supports--> v0d\n- v13 --supports--> v16\n- v08 --supports--> v0f\n- v0c --supports--> v07\n- v0a --refutes--> v05\n- v0d --supports--> v18\n- v0b --refutes--> v01\n- v0b --supports--> v04\n- v10 --supports--> v02\n- v0a --refutes--> v15\n- v06 --refutes--> v09\n- v09 --supports--> v13\n- v17 --supports--> v06\n- v07 --refutes--> v0a\n- v04 --refutes--> v03\n- v15 --supports--> v00\n- v02 --supports--> v17\n- v15 --refutes--> v08\n- v13 --supports--> v11\n\n[VERIFIED ROOT]\nv12 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"87f0ceff3be727c3c0659b7fad586abb0686188930d48a6b65e64295fde315cc","schema":"livegraph.boolean-propagation.v1","spec_hash":"ae529d0e017dca0d8be90df3a506775670cf383520cd3cac694d54e21ec67bb1","spec_json":"{\"edges\":[{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v12\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-b6413447f024ba5ae80e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v17 --refutes--> v08\n- v06 --refutes--> v0d\n- v14 --refutes--> v0c\n- v18 --refutes--> v03\n- v0b --refutes--> v00\n- v0f --refutes--> v04\n- v14 --refutes--> v01\n- v0c --refutes--> v07\n- v08 --supports--> v18\n- v07 --refutes--> v05\n- v09 --refutes--> v0f\n- v11 --supports--> v02\n- v15 --supports--> v0a\n- v00 --refutes--> v16\n- v11 --supports--> v17\n- v03 --refutes--> v06\n- v0a --supports--> v14\n- v03 --supports--> v0e\n- v14 --refutes--> v10\n- v00 --supports--> v09\n- v0b --refutes--> v13\n- v05 --supports--> v0b\n- v07 --supports--> v12\n- v06 --supports--> v15\n\n[VERIFIED ROOT]\nv11 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2e6ff4d595e9d780cf47c1b99a0f271ad712a4b84320034e7c95fa6859290fdb","schema":"livegraph.boolean-propagation.v1","spec_hash":"b6413447f024ba5ae80e2f597ad1945cc9b7bdc5a42c19e0369146b182d536f3","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-1fd5c14a551dfe40ea3e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --supports--> v17\n- v07 --refutes--> v0e\n- v0d --supports--> v18\n- v15 --supports--> v09\n- v06 --supports--> v03\n- v12 --refutes--> v0a\n- v13 --refutes--> v01\n- v16 --refutes--> v02\n- v14 --supports--> v07\n- v12 --refutes--> v04\n- v18 --supports--> v13\n- v0f --supports--> v12\n- v12 --refutes--> v11\n- v03 --supports--> v0b\n- v04 --refutes--> v00\n- v09 --supports--> v05\n- v07 --supports--> v16\n- v01 --supports--> v0f\n- v08 --refutes--> v0d\n- v16 --supports--> v06\n- v03 --refutes--> v10\n- v14 --supports--> v0c\n- v03 --supports--> v15\n- v00 --refutes--> v14\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"852de39f69c1b8127172035f617e4850028398fbc590f45a16e5726e6ef27f25","schema":"livegraph.boolean-propagation.v1","spec_hash":"1fd5c14a551dfe40ea3eb3ed07e1c0476288100810a8adbc34eb3c30c993166d","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-3b9a8e66cefa27ea9b01","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --supports--> v05\n- v01 --supports--> v17\n- v00 --supports--> v16\n- v0f --refutes--> v0c\n- v0c --supports--> v06\n- v0e --refutes--> v18\n- v00 --refutes--> v13\n- v07 --refutes--> v0d\n- v15 --refutes--> v01\n- v13 --supports--> v07\n- v12 --supports--> v11\n- v06 --supports--> v00\n- v07 --supports--> v10\n- v05 --refutes--> v0a\n- v01 --refutes--> v08\n- v17 --refutes--> v0b\n- v05 --refutes--> v0e\n- v0b --supports--> v0f\n- v12 --supports--> v03\n- v01 --supports--> v04\n- v10 --supports--> v09\n- v18 --supports--> v15\n- v06 --refutes--> v14\n- v17 --refutes--> v02\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2c86df2c6d4cbac0f585480c553685d5ebfdf49e279814fb214c28f6624044ae","schema":"livegraph.boolean-propagation.v1","spec_hash":"3b9a8e66cefa27ea9b01c3a7ec49e52f27391378924472aefd932ed08de3afb9","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-8d9d7f7b08e1aa49d24f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v0e\n- v12 --refutes--> v13\n- v0d --supports--> v00\n- v18 --refutes--> v06\n- v15 --refutes--> v0d\n- v10 --supports--> v01\n- v13 --refutes--> v18\n- v10 --supports--> v15\n- v06 --supports--> v0f\n- v0e --refutes--> v02\n- v05 --supports--> v04\n- v08 --supports--> v07\n- v02 --refutes--> v14\n- v03 --refutes--> v05\n- v07 --supports--> v03\n- v02 --supports--> v12\n- v05 --supports--> v0c\n- v11 --supports--> v10\n- v02 --refutes--> v17\n- v04 --supports--> v09\n- v08 --refutes--> v0b\n- v00 --supports--> v08\n- v10 --supports--> v0a\n- v15 --refutes--> v16\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"414eae6b8e2b6c2c15ce45390105876ca88f79fe130dda5bd9fe632c7303ea69","schema":"livegraph.boolean-propagation.v1","spec_hash":"8d9d7f7b08e1aa49d24f3fb89265bf899e727abc07eed8dda5a0620e6a7c241f","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-28f3a15c8cbc8c92cabd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v07\n- v10 --supports--> v11\n- v17 --refutes--> v16\n- v0d --refutes--> v02\n- v08 --refutes--> v0a\n- v00 --supports--> v18\n- v03 --supports--> v0e\n- v06 --supports--> v10\n- v01 --refutes--> v05\n- v16 --supports--> v03\n- v12 --refutes--> v17\n- v04 --refutes--> v13\n- v13 --supports--> v0c\n- v0c --supports--> v0b\n- v08 --refutes--> v01\n- v05 --refutes--> v04\n- v0e --supports--> v06\n- v02 --supports--> v14\n- v08 --refutes--> v0f\n- v02 --supports--> v09\n- v10 --supports--> v00\n- v13 --supports--> v15\n- v00 --supports--> v08\n- v0c --refutes--> v0d\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"724f500a81b5d96f658bd8a92bb64a577e6644dfed0ddb74898377a0fc299c59","schema":"livegraph.boolean-propagation.v1","spec_hash":"28f3a15c8cbc8c92cabda130cad4a646b41c72b35467aafb8319ec196d7ddde3","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-653304449395ea05d658","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v09\n- v00 --supports--> v08\n- v0e --supports--> v0a\n- v09 --supports--> v07\n- v15 --refutes--> v00\n- v09 --supports--> v13\n- v03 --supports--> v0e\n- v12 --refutes--> v0c\n- v04 --supports--> v0d\n- v18 --refutes--> v03\n- v15 --supports--> v0f\n- v02 --supports--> v11\n- v11 --refutes--> v0b\n- v0a --supports--> v04\n- v08 --refutes--> v05\n- v04 --supports--> v06\n- v04 --supports--> v14\n- v09 --supports--> v16\n- v0d --supports--> v12\n- v08 --refutes--> v17\n- v02 --refutes--> v01\n- v07 --supports--> v02\n- v11 --supports--> v10\n- v17 --supports--> v18\n\n[VERIFIED ROOT]\nv15 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"90a0c9ad010b7a79b22a6d332b623abc9fc8eef34324cdafd97928dc50d51c39","schema":"livegraph.boolean-propagation.v1","spec_hash":"653304449395ea05d6588df9cb5cfbb171ef572cffde643c169dae283662fa82","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v15\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-4c1dab504c1e470f5b35","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --supports--> v0d\n- v0d --supports--> v15\n- v11 --supports--> v16\n- v12 --supports--> v17\n- v11 --supports--> v00\n- v14 --supports--> v06\n- v08 --supports--> v0f\n- v07 --supports--> v12\n- v0e --supports--> v10\n- v18 --supports--> v03\n- v12 --supports--> v13\n- v03 --refutes--> v0b\n- v07 --supports--> v05\n- v08 --supports--> v0a\n- v04 --refutes--> v18\n- v09 --refutes--> v0e\n- v10 --refutes--> v02\n- v03 --supports--> v09\n- v17 --supports--> v11\n- v10 --supports--> v14\n- v15 --refutes--> v07\n- v0f --supports--> v04\n- v09 --refutes--> v01\n- v00 --refutes--> v0c\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4fba7552d52541cb49f8308a7eca549670759f51b68934dcf087434004f20a5d","schema":"livegraph.boolean-propagation.v1","spec_hash":"4c1dab504c1e470f5b3566c8006d56be9be7c08a461267cdbd2b03c059c0ac65","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-6ea2d41c18f4bfa97b14","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --supports--> v00\n- v0c --supports--> v11\n- v17 --refutes--> v0c\n- v0e --refutes--> v09\n- v18 --refutes--> v03\n- v07 --supports--> v18\n- v03 --supports--> v12\n- v02 --refutes--> v05\n- v0b --supports--> v02\n- v05 --refutes--> v0a\n- v06 --refutes--> v01\n- v08 --refutes--> v0e\n- v16 --supports--> v07\n- v09 --supports--> v16\n- v13 --refutes--> v17\n- v09 --refutes--> v0f\n- v15 --supports--> v13\n- v17 --supports--> v14\n- v04 --supports--> v08\n- v06 --refutes--> v04\n- v0c --supports--> v06\n- v03 --refutes--> v0b\n- v05 --refutes--> v0d\n- v0b --refutes--> v10\n\n[VERIFIED ROOT]\nv15 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dafb25616848027a8383e8b4a77554090e93898fdc7be1d0a8a1fc96f9be139b","schema":"livegraph.boolean-propagation.v1","spec_hash":"6ea2d41c18f4bfa97b14c01f16ddaca5f00c8d7214dc40aba5a27d6d2f7314a5","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v15\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-d5ee8e537f798c336f4f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v0f\n- v09 --refutes--> v18\n- v10 --refutes--> v17\n- v07 --refutes--> v05\n- v18 --refutes--> v14\n- v11 --refutes--> v10\n- v08 --refutes--> v15\n- v13 --refutes--> v08\n- v15 --refutes--> v0e\n- v17 --supports--> v01\n- v01 --refutes--> v0a\n- v0e --supports--> v0b\n- v0f --refutes--> v02\n- v01 --refutes--> v03\n- v0f --supports--> v06\n- v18 --supports--> v16\n- v10 --supports--> v0c\n- v07 --supports--> v0d\n- v0d --supports--> v13\n- v00 --supports--> v12\n- v16 --refutes--> v00\n- v0f --refutes--> v09\n- v0a --supports--> v07\n- v17 --refutes--> v04\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"67a168cd2835481ff54d0a38ecb29fdf663f052d1bd59f55de6d30173f27372e","schema":"livegraph.boolean-propagation.v1","spec_hash":"d5ee8e537f798c336f4f2dfe6cbb1155b3aa9807ab95f89e6028c67d01e47d82","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-bcf206c0458da444c249","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v12\n- v03 --supports--> v02\n- v0c --refutes--> v16\n- v16 --refutes--> v00\n- v09 --refutes--> v01\n- v0c --refutes--> v08\n- v04 --supports--> v03\n- v0f --refutes--> v17\n- v02 --supports--> v15\n- v16 --supports--> v0f\n- v05 --supports--> v14\n- v11 --supports--> v18\n- v05 --supports--> v0a\n- v01 --refutes--> v0b\n- v01 --refutes--> v06\n- v18 --supports--> v0c\n- v0a --refutes--> v11\n- v13 --refutes--> v05\n- v04 --supports--> v07\n- v10 --refutes--> v04\n- v15 --refutes--> v13\n- v0a --supports--> v0d\n- v06 --refutes--> v10\n- v05 --supports--> v0e\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv17\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8fdede33ff08fdadf64eaba74c6e44b363d2358e1e97729fabaf6996399f9c88","schema":"livegraph.boolean-propagation.v1","spec_hash":"bcf206c0458da444c24967baaf419039bfc3906ad8489c5cd08b2af84fc5e8c6","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v17\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-47c1dbca7fd89d4766c5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v15\n- v07 --refutes--> v0c\n- v00 --refutes--> v11\n- v18 --supports--> v12\n- v0b --supports--> v08\n- v11 --refutes--> v09\n- v10 --refutes--> v0f\n- v09 --supports--> v03\n- v0b --refutes--> v02\n- v0a --supports--> v06\n- v16 --refutes--> v10\n- v0d --supports--> v04\n- v04 --refutes--> v07\n- v0d --supports--> v17\n- v15 --supports--> v05\n- v0d --refutes--> v14\n- v03 --refutes--> v13\n- v0a --refutes--> v01\n- v07 --refutes--> v18\n- v03 --supports--> v0a\n- v05 --supports--> v0b\n- v10 --refutes--> v0d\n- v12 --supports--> v00\n- v15 --refutes--> v0e\n\n[VERIFIED ROOT]\nv16 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a8c31bd1a6a0e5ac549026ae4e316581a4a9917e374d80aa4184cfaef8723450","schema":"livegraph.boolean-propagation.v1","spec_hash":"47c1dbca7fd89d4766c503923536fc8ae916e276568acad985865857bf635803","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v16\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-ba20cdb71ffc43a7c526","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v13\n- v0b --supports--> v10\n- v03 --refutes--> v01\n- v05 --refutes--> v03\n- v15 --supports--> v0b\n- v00 --refutes--> v17\n- v11 --refutes--> v0a\n- v01 --refutes--> v12\n- v08 --supports--> v11\n- v0a --refutes--> v16\n- v02 --supports--> v00\n- v0a --refutes--> v09\n- v06 --refutes--> v07\n- v10 --refutes--> v06\n- v05 --refutes--> v0f\n- v09 --supports--> v15\n- v14 --supports--> v0e\n- v07 --refutes--> v04\n- v02 --supports--> v18\n- v00 --refutes--> v05\n- v0b --supports--> v0d\n- v07 --refutes--> v0c\n- v0e --refutes--> v08\n- v07 --refutes--> v02\n\n[VERIFIED ROOT]\nv14 = TRUE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"895b72376862ea999a85006f06b895335e6d30f8d6ddeb3dbbb48a5e2962af70","schema":"livegraph.boolean-propagation.v1","spec_hash":"ba20cdb71ffc43a7c526f5dff6fab3085654ebb7fcdc6588a1a0d330c3528bec","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":16,\"root\":\"v14\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-ca20ebbb066a51565749","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v0a\n- v02 --supports--> v11\n- v0a --supports--> v15\n- v11 --supports--> v03\n- v10 --supports--> v09\n- v03 --refutes--> v10\n- v0d --supports--> v17\n- v13 --refutes--> v0d\n- v08 --refutes--> v02\n- v0d --refutes--> v14\n- v0a --supports--> v0e\n- v05 --supports--> v12\n- v0c --refutes--> v06\n- v07 --supports--> v04\n- v10 --refutes--> v05\n- v04 --refutes--> v00\n- v0c --supports--> v0f\n- v0a --supports--> v16\n- v0b --refutes--> v07\n- v0b --supports--> v01\n- v04 --refutes--> v08\n- v12 --supports--> v13\n- v11 --refutes--> v18\n- v15 --supports--> v0b\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv14\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d1308e5cd7f41143868a2fafb4f1c265c70808c9aef477d7dd6b3c0de369b771","schema":"livegraph.boolean-propagation.v1","spec_hash":"ca20ebbb066a5156574976b6fafcae2dfe6a0eed3ef484085238feab34a032ef","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":16,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-e288f925c4609633f504","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v02\n- v14 --supports--> v12\n- v07 --supports--> v10\n- v08 --refutes--> v18\n- v04 --supports--> v0d\n- v09 --refutes--> v0a\n- v06 --supports--> v15\n- v17 --refutes--> v13\n- v11 --refutes--> v01\n- v17 --refutes--> v06\n- v0d --supports--> v0f\n- v0a --refutes--> v03\n- v09 --supports--> v0b\n- v12 --supports--> v17\n- v02 --refutes--> v0c\n- v06 --supports--> v09\n- v17 --refutes--> v16\n- v0f --refutes--> v11\n- v08 --refutes--> v05\n- v05 --refutes--> v00\n- v02 --supports--> v04\n- v07 --refutes--> v0e\n- v05 --supports--> v07\n- v03 --refutes--> v08\n\n[VERIFIED ROOT]\nv14 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5156082affd3e06bf90b2a511e84e4634781fd0de5f7833fe2fdaea7cc6b7014","schema":"livegraph.boolean-propagation.v1","spec_hash":"e288f925c4609633f5041dd37daeef81c0c3ba3fe8d502fed8ffdad412ca4863","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v14\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-d3701a3b84b22760df47","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v0d\n- v00 --refutes--> v0a\n- v0f --refutes--> v08\n- v14 --supports--> v11\n- v14 --supports--> v01\n- v17 --refutes--> v14\n- v0e --refutes--> v02\n- v12 --refutes--> v0e\n- v18 --supports--> v13\n- v0f --supports--> v07\n- v10 --supports--> v12\n- v0c --supports--> v15\n- v00 --refutes--> v0c\n- v09 --supports--> v10\n- v14 --refutes--> v06\n- v08 --refutes--> v05\n- v0d --refutes--> v03\n- v02 --refutes--> v0b\n- v09 --refutes--> v16\n- v0b --supports--> v00\n- v0d --supports--> v09\n- v12 --supports--> v04\n- v05 --refutes--> v18\n- v0c --refutes--> v0f\n\n[VERIFIED ROOT]\nv17 = TRUE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d432eb3da7316e7e77b8f0fcd5cb8a5249aab5a59640ed6d5ad8c7558d6d6b1b","schema":"livegraph.boolean-propagation.v1","spec_hash":"d3701a3b84b22760df47195c68c71034215936012acb6c2c86fc476b8074276b","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v17\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-0a650a67ae796ee956e6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v17 --refutes--> v12\n- v06 --supports--> v13\n- v07 --refutes--> v11\n- v07 --supports--> v0e\n- v14 --supports--> v05\n- v13 --supports--> v07\n- v0b --supports--> v00\n- v10 --supports--> v0c\n- v0b --supports--> v02\n- v18 --refutes--> v17\n- v12 --supports--> v0d\n- v0c --supports--> v0f\n- v09 --refutes--> v04\n- v03 --supports--> v14\n- v14 --supports--> v15\n- v0b --refutes--> v08\n- v0b --refutes--> v16\n- v0d --refutes--> v01\n- v01 --refutes--> v10\n- v14 --refutes--> v0a\n- v11 --refutes--> v0b\n- v09 --refutes--> v03\n- v02 --supports--> v18\n- v0c --supports--> v09\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv15\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0101987e5c851aa706193ce1210d34783d0d03fa19b2242bef97224b17e34602","schema":"livegraph.boolean-propagation.v1","spec_hash":"0a650a67ae796ee956e6b53297686bcf7f15c3e820ab2927956f9b8817948421","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-d4420845fea6586aeacb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v02\n- v0a --refutes--> v16\n- v13 --refutes--> v0c\n- v13 --refutes--> v00\n- v07 --refutes--> v0b\n- v11 --refutes--> v0f\n- v06 --supports--> v11\n- v0f --supports--> v10\n- v0b --refutes--> v01\n- v10 --supports--> v08\n- v0a --supports--> v17\n- v15 --refutes--> v18\n- v0e --refutes--> v07\n- v05 --supports--> v13\n- v0a --supports--> v0d\n- v0b --supports--> v0a\n- v06 --refutes--> v14\n- v0d --supports--> v15\n- v02 --supports--> v09\n- v0e --refutes--> v12\n- v02 --refutes--> v05\n- v05 --supports--> v03\n- v18 --refutes--> v06\n- v04 --supports--> v0e\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2349ff0b5905a90ee9fd74a7d174abf4988bc849c652ba41d5942e2b02938e26","schema":"livegraph.boolean-propagation.v1","spec_hash":"d4420845fea6586aeacb12eb30b30f11c93b82372b51256a498d09e6b9941775","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-d60ba4437958c0c50dca","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v18\n- v06 --supports--> v11\n- v0f --refutes--> v01\n- v0d --supports--> v17\n- v0c --refutes--> v03\n- v03 --refutes--> v0d\n- v06 --supports--> v13\n- v17 --supports--> v07\n- v09 --supports--> v0a\n- v15 --supports--> v00\n- v04 --supports--> v0f\n- v0d --refutes--> v10\n- v07 --refutes--> v06\n- v01 --refutes--> v12\n- v01 --refutes--> v0b\n- v0e --supports--> v02\n- v14 --supports--> v09\n- v11 --supports--> v15\n- v01 --refutes--> v0c\n- v0a --supports--> v04\n- v00 --refutes--> v0e\n- v0e --refutes--> v05\n- v0a --refutes--> v08\n- v14 --refutes--> v16\n\n[VERIFIED ROOT]\nv14 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2f4a7032805a70a9d52cdcdf20bfe75209e8f9d09f84d7db0f7a5545d55f7767","schema":"livegraph.boolean-propagation.v1","spec_hash":"d60ba4437958c0c50dca50538340282154d24bc1510561c14f939c1d8cda2ca7","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v14\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-eebc167c34824411cba8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --refutes--> v10\n- v0d --refutes--> v02\n- v13 --refutes--> v0c\n- v11 --refutes--> v0e\n- v03 --supports--> v00\n- v17 --refutes--> v13\n- v08 --refutes--> v09\n- v12 --supports--> v14\n- v03 --refutes--> v05\n- v0e --refutes--> v04\n- v0c --refutes--> v16\n- v02 --supports--> v0f\n- v08 --refutes--> v12\n- v18 --supports--> v17\n- v0a --refutes--> v01\n- v0a --refutes--> v03\n- v15 --refutes--> v0d\n- v0c --supports--> v08\n- v0f --refutes--> v11\n- v02 --supports--> v07\n- v05 --refutes--> v0b\n- v12 --supports--> v06\n- v0e --refutes--> v0a\n- v06 --supports--> v15\n\n[VERIFIED ROOT]\nv18 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"35dcd036203fcd912c132595d52b60ea34926b084ad1482133a59746c2e46246","schema":"livegraph.boolean-propagation.v1","spec_hash":"eebc167c34824411cba8a92a5b222e6a822d63d0912ef0474505dcbb95843775","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v18\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-7a28f9ba483d6f6fb7d7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --refutes--> v0c\n- v0f --refutes--> v16\n- v0f --refutes--> v07\n- v0a --refutes--> v0f\n- v0d --supports--> v15\n- v06 --refutes--> v12\n- v0b --refutes--> v04\n- v14 --refutes--> v0b\n- v18 --supports--> v17\n- v10 --supports--> v0a\n- v0a --refutes--> v11\n- v06 --refutes--> v10\n- v02 --supports--> v14\n- v17 --supports--> v02\n- v0b --refutes--> v00\n- v09 --supports--> v18\n- v02 --supports--> v08\n- v13 --supports--> v06\n- v16 --refutes--> v03\n- v03 --refutes--> v05\n- v02 --refutes--> v01\n- v15 --supports--> v09\n- v05 --supports--> v0d\n- v05 --supports--> v0e\n\n[VERIFIED ROOT]\nv13 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d3a09d655da27f6ce03a36cdce96cb3b06ac383c67617b9e18b13dc374283492","schema":"livegraph.boolean-propagation.v1","spec_hash":"7a28f9ba483d6f6fb7d70ba1926c3936db2d11a06587273cbde4bb9b129f214a","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"train","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-2a6a1978ab41fe735b80","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v16\n- v18 --supports--> v0a\n- v0e --supports--> v14\n- v11 --refutes--> v0b\n- v18 --refutes--> v04\n- v11 --refutes--> v15\n- v14 --refutes--> v03\n- v0f --refutes--> v01\n- v15 --refutes--> v0c\n- v16 --supports--> v12\n- v09 --refutes--> v13\n- v0c --supports--> v05\n- v14 --supports--> v0f\n- v15 --refutes--> v00\n- v17 --supports--> v11\n- v06 --supports--> v18\n- v16 --supports--> v06\n- v01 --refutes--> v17\n- v02 --refutes--> v08\n- v18 --supports--> v10\n- v04 --supports--> v0e\n- v02 --supports--> v07\n- v02 --supports--> v09\n- v13 --supports--> v0d\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8c58c0d39c4a5ee0286f096f5d8f7b53c0eedd6cd0d57b31296582382f0b995b","schema":"livegraph.boolean-propagation.v1","spec_hash":"2a6a1978ab41fe735b8037bccbacaa5807d5067ec949a0e9515332645ce14219","spec_json":"{\"edges\":[{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"train","view":"graph"}