{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-720dfa59360c3feb57ae","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --supports--> v05\n- v05 --refutes--> v06\n- v06 --supports--> v00\n- v00 --supports--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v04\n- v06 --refutes--> 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":"f037a6ef0571ca925225d1d13728abb68859f615cec8cb1dfddf09d5957e5487","schema":"livegraph.boolean-propagation.v1","spec_hash":"720dfa59360c3feb57aeba45e1f17c96546b92916dd0ee74daf705382e2310b0","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-c851a7c78b850b677178","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --refutes--> v03\n- v03 --refutes--> v04\n- v04 --supports--> v05\n- v05 --supports--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v01\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":"05a38efd030c954ebf38714b177acecb5d6cd87c822126206cedc9a4a13090fd","schema":"livegraph.boolean-propagation.v1","spec_hash":"c851a7c78b850b6771787922835ad383204dd3130a4b3c1b09fd977712b01ada","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-b2f459210c35bc18c00e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v01\n- v01 --refutes--> v06\n- v06 --supports--> v05\n- v05 --supports--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v02\n- v01 --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":"4b3cb996bf72524e6c62a0f2069e5d6ab726640bf594a63b26d7fb41a6c844a9","schema":"livegraph.boolean-propagation.v1","spec_hash":"b2f459210c35bc18c00ed13b230033e3cbefc07169f54f4bfe5e5030a4821d66","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-f92472930f5e55a01788","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --supports--> v06\n- v06 --supports--> v01\n- v01 --supports--> v02\n- v02 --supports--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v03\n- v06 --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":"a0e60db2f4acd21d3a485e77977e392c14e7688e3cffbea4cfd689f2b60cbbb7","schema":"livegraph.boolean-propagation.v1","spec_hash":"f92472930f5e55a01788602f22bf42cff509c9ec40c3f8f9e145f3858e5c2606","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-e09679f069f6ab395744","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v05\n- v05 --refutes--> v04\n- v04 --supports--> v00\n- v00 --refutes--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v02\n- v06 --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":"2ba76055f502626ee792bee0d5dec41f992a8cb7fa7d27c398a589e4621c06d8","schema":"livegraph.boolean-propagation.v1","spec_hash":"e09679f069f6ab395744b4026ffb47aea9a317342508694461c1474676e2c9e6","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-09cc9e79c3635578d269","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v04\n- v04 --supports--> v00\n- v00 --supports--> v02\n- v02 --supports--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v03\n- v00 --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":"780735f061e547d951f12300d112dbf76ee17cece378da46388e18debc5a8214","schema":"livegraph.boolean-propagation.v1","spec_hash":"09cc9e79c3635578d2699740b8c5197fc68284a71faf000e58642f4a190380a7","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-e97ea3a21484dba34595","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --refutes--> v03\n- v03 --refutes--> v04\n- v04 --supports--> v01\n- v01 --supports--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v06\n- v03 --refutes--> v02\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":"ad26c51dad7dd1ec7f577aa2b049bda7f220adf0788a129f68ebe04605d5163d","schema":"livegraph.boolean-propagation.v1","spec_hash":"e97ea3a21484dba3459567b40543db925970f4f325231a2f057ac849b19ee8ee","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-853bff18bd4c709a8b53","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --refutes--> v01\n- v01 --refutes--> v00\n- v00 --refutes--> v02\n- v02 --supports--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v06\n- v03 --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":"5860c6390ae669d84457806f7f710bf3fdf25dca83106e1770e5872316bc5c0f","schema":"livegraph.boolean-propagation.v1","spec_hash":"853bff18bd4c709a8b5331fb9e4d10043a60450d6eb5c5c203e067e9d576ff58","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-3793ffcdbb65d2a40a48","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v05\n- v05 --supports--> v03\n- v03 --refutes--> v02\n- v02 --supports--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v06\n- v05 --supports--> v01\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":"de5b0df6e6a581dffb4759c5d678d0ba50cb47ec13ba4cbb4302308640f8f976","schema":"livegraph.boolean-propagation.v1","spec_hash":"3793ffcdbb65d2a40a487a97f67627b92939ea6f9f91ea01c3295b12750f5e95","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-da51a9b1f89e836461f7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --refutes--> v06\n- v06 --refutes--> v04\n- v04 --refutes--> v02\n- v02 --refutes--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v00\n- v06 --supports--> v03\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":"3578793bd9b3c990d503b972be87b809d00176e41588f772632152dbcc5a6568","schema":"livegraph.boolean-propagation.v1","spec_hash":"da51a9b1f89e836461f7d1d704aec9f4862f009647ffe863f68f0525790e9068","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"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\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-0dac0506cc2a94949f67","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --supports--> v00\n- v00 --supports--> v04\n- v04 --refutes--> v02\n- v02 --supports--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v01\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":"b51b97e4b0b7955547b7541f78d3ce6856843ecdb2d3af51f5b5dd04cd5039f9","schema":"livegraph.boolean-propagation.v1","spec_hash":"0dac0506cc2a94949f671689e119d545e21c0cdc19ff271ebf15afeb6f7d7719","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-a406ab2000d5fc25a69e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --refutes--> v06\n- v06 --supports--> v01\n- v01 --refutes--> v05\n- v05 --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v04\n- v06 --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":"3727feb43c01a43fe09b8b6bcde75f3b17d2543b2e9939cfea9405921272966d","schema":"livegraph.boolean-propagation.v1","spec_hash":"a406ab2000d5fc25a69e1bb3aa967911184c4a172ca51aa3f01a74a20f670f8f","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-888d2efa719f2a57c114","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --refutes--> v03\n- v03 --supports--> v06\n- v06 --refutes--> v04\n- v04 --refutes--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v05\n- v06 --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":"b86b826941d2946c9906bbce4d6d0ed5463b582e9cf1b56eabcd4420314a1bab","schema":"livegraph.boolean-propagation.v1","spec_hash":"888d2efa719f2a57c1142b96bad89f2c45035d9a0c0733c363255bcd4c2dd745","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-544cbf1f40ff64716835","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --supports--> v00\n- v00 --supports--> v03\n- v03 --supports--> v02\n- v02 --refutes--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v05\n- v00 --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":"6a96ff0f917b86499225ea7c28103e05362df678401b6133d66da462bc3a481e","schema":"livegraph.boolean-propagation.v1","spec_hash":"544cbf1f40ff64716835d04896f71d755ec2954228d1f1311dbb8db5ef4cc2d1","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-38142cfa2680db795afc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v01\n- v01 --refutes--> v02\n- v02 --supports--> v00\n- v00 --supports--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v06\n- v02 --supports--> v03\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":"43dac65899f4fa989a2332cbbf6ee6ea17eef9c516c0ba07dc64bf826b7853df","schema":"livegraph.boolean-propagation.v1","spec_hash":"38142cfa2680db795afc3ab3ca4d10c9534bea76613f76544aa173a970d81f44","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"}],\"hop\":4,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-de9b19541bc8868406f4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --supports--> v00\n- v00 --supports--> v04\n- v04 --refutes--> v05\n- v05 --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v01\n- v02 --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":"438fa66f1d608ae0f772786d7a34d7b0ead7c50ee8522ba6cb62c85c8a70459c","schema":"livegraph.boolean-propagation.v1","spec_hash":"de9b19541bc8868406f4b217ea8909adb007932c4d75c245e4499f00527b025b","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-7a16a0e6dcd8a812b444","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --supports--> v01\n- v01 --refutes--> v05\n- v05 --refutes--> v04\n- v04 --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v06\n- v01 --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":"d26e5c36699ff7bd62e310fb8bdb52a1be7d6fab077ad3872a011405242be3e6","schema":"livegraph.boolean-propagation.v1","spec_hash":"7a16a0e6dcd8a812b444f1c283e8449fcfb097b9c671947c2472347907a978c6","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-5086853a4ec8cff6654b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --refutes--> v01\n- v01 --supports--> v05\n- v05 --refutes--> v00\n- v00 --supports--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v04\n- v00 --supports--> v06\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":"9f75cd83a556367a2a991ad12dcd985263e32da8c2185b97de2b045dcd21bb92","schema":"livegraph.boolean-propagation.v1","spec_hash":"5086853a4ec8cff6654be19aa2b9f62f3595dd792ec2183f08a2bef2dfc0f2c2","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-1fadc933ca3eabe7e8e6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --refutes--> v00\n- v00 --supports--> v06\n- v06 --refutes--> v01\n- v01 --supports--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v02\n- v05 --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":"260acad5480d1ee006aa216118f526afff4f57182b6207858bd4bf67235ca823","schema":"livegraph.boolean-propagation.v1","spec_hash":"1fadc933ca3eabe7e8e614dffe87e88402a8b5baf8980e57b5efac126fdd57f1","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-9c28be46b37a5eb9640e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v05\n- v05 --supports--> v02\n- v02 --supports--> v01\n- v01 --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v06\n- v01 --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":"499019323afc67730813567f72c43730ce75723c3ddee703ed165f0247912902","schema":"livegraph.boolean-propagation.v1","spec_hash":"9c28be46b37a5eb9640e79c1bb8677a5f64fddeba28fbbc73b478a5a174dfb24","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-a529f5ce532ad852aad3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --refutes--> v05\n- v05 --supports--> v02\n- v02 --supports--> v04\n- v04 --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v00\n- v05 --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":"557e5f142dd2ebfe0f1adc69bd6881c71953db55c9fdd99ee9cec7316ac638b9","schema":"livegraph.boolean-propagation.v1","spec_hash":"a529f5ce532ad852aad3db6687c9432ad1720d0fbbdd99b4332f493e2b7bb4fc","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-b3288e8bd7374bba6e8c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v05\n- v05 --refutes--> v00\n- v00 --refutes--> v04\n- v04 --refutes--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v03\n- v06 --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":"2a66a14e7dfb59fd195950338c413b2f1b1927d1c9aaade7847fb87b06b982a4","schema":"livegraph.boolean-propagation.v1","spec_hash":"b3288e8bd7374bba6e8c7145d8ae3e043cac7fb34b734a74d03bdc228c56afb0","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-a28f360084a54158206b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v00\n- v00 --refutes--> v05\n- v05 --supports--> v03\n- v03 --supports--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v02\n- v06 --refutes--> v01\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":"ff1c2d7941e0966546ca6d768e161afbdcf601316d7f702970c40ef4aebe47f6","schema":"livegraph.boolean-propagation.v1","spec_hash":"a28f360084a54158206b4941a88bf82c80f7c6af6a8756df98f6eb9176993cbc","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-f4efe6f2a30ab91199c0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --supports--> v05\n- v05 --refutes--> v02\n- v02 --supports--> v03\n- v03 --supports--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v04\n- v05 --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":"95fbde90a80e305d08fcda7db2e316dbd700447fc615aba34002f114bfdc8aea","schema":"livegraph.boolean-propagation.v1","spec_hash":"f4efe6f2a30ab91199c08ce8ea23534cffc86e741e5fa7ca11d5dc6efa7db8c5","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-c58e2b7db53988188639","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v03\n- v03 --refutes--> v00\n- v00 --supports--> v05\n- v05 --refutes--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v02\n- v05 --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":"82d6cd8dcd725c6c704d5f1ce56a0895c706b75bf8568d66179905b0d802b439","schema":"livegraph.boolean-propagation.v1","spec_hash":"c58e2b7db539881886396165955d4e081f443555832af0ed94b795c1f0e86e9f","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-aef2896da941021af92b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --supports--> v01\n- v01 --supports--> v06\n- v06 --supports--> v00\n- v00 --refutes--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v03\n- v01 --refutes--> v04\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":"8a3cc81205b5f937420af140e25495a7ea35a1e7ee5bd0042564d1721c53eed7","schema":"livegraph.boolean-propagation.v1","spec_hash":"aef2896da941021af92b8b6607ef946a7bb1472256f8cba002564face9a763c8","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-35a52e848f61abaf443d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --supports--> v02\n- v02 --refutes--> v03\n- v03 --refutes--> v01\n- v01 --supports--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --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":"ac3fec517b7fc495db84bd3a91ab35f7c38723e3722cf86927e206ab4594d46e","schema":"livegraph.boolean-propagation.v1","spec_hash":"35a52e848f61abaf443d6588ba6fc270a92f1f4f582a8254cecdb81306fe96ee","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-5f98c156aa99a210c072","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v05\n- v05 --refutes--> v04\n- v04 --supports--> v03\n- v03 --supports--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v01\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":"57b687569a84e107e3a313b941886e74e31b16e6991cdc857c2ccbbff62abb36","schema":"livegraph.boolean-propagation.v1","spec_hash":"5f98c156aa99a210c0723bb0f066e73965539da15f287421b1ce6d72cddc2894","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-8b77cf24d46dfd7c4172","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --supports--> v05\n- v05 --supports--> v03\n- v03 --refutes--> v01\n- v01 --supports--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v00\n- v02 --refutes--> v04\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":"f266ef79717b128f01aa871910c58b5c64f40a05cca8bd73d17b2c83afb528bc","schema":"livegraph.boolean-propagation.v1","spec_hash":"8b77cf24d46dfd7c4172fc6b54937c20ffd3cc31748518e95e312bd9f0e34b6e","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-8cf309a4acee08a4622e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --supports--> v02\n- v02 --supports--> v03\n- v03 --supports--> v00\n- v00 --refutes--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v06\n- v05 --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":"87eeec33bae7eedd5291b5c2c0b5a2520290ecd818cd0566d1c7a1242fa06219","schema":"livegraph.boolean-propagation.v1","spec_hash":"8cf309a4acee08a4622ed5362f122fb14da464dc8ae9041b24df130c220f39c5","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-5a21ca5565aab55ceb2b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --supports--> v04\n- v04 --refutes--> v05\n- v05 --supports--> v06\n- v06 --supports--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v00\n- v06 --supports--> 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":"7e9f69ed465524cc0f4cdaaa9c3cdbd21a51aa48693b1e0993197912db39b58c","schema":"livegraph.boolean-propagation.v1","spec_hash":"5a21ca5565aab55ceb2bf0f14e579432ff1e799a157a622defe77eec687e5b96","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"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\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-036a4008bb538c862d2c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --supports--> v00\n- v00 --refutes--> v03\n- v03 --supports--> v04\n- v04 --supports--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v01\n- v06 --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":"78891aeba7cd2e16b03fd7e4ca932fe7b18f5cfa0f7cb28b58fecf157aa02eb4","schema":"livegraph.boolean-propagation.v1","spec_hash":"036a4008bb538c862d2c28452e27ce999985b54b2776ecf36d5d577399cb3dd6","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-315021aad120a02e37ab","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v01\n- v01 --refutes--> v02\n- v02 --refutes--> v05\n- v05 --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v00\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":"cfdc87a603a69833799836542b5321ba6229f8bfb75a6acf5e93a917699da3cc","schema":"livegraph.boolean-propagation.v1","spec_hash":"315021aad120a02e37ab121b65c9e949f23a40b106f76a1ddfba68d5d9d92266","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-d9f72e832f3aacfe009a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --supports--> v00\n- v00 --refutes--> v03\n- v03 --refutes--> v02\n- v02 --supports--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v05\n- v02 --refutes--> v04\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":"eae66012fdac2e5cca07c5d906bf37e3b252c143370124bad1002f375eb549da","schema":"livegraph.boolean-propagation.v1","spec_hash":"d9f72e832f3aacfe009a04e00dcd11420f913aa0131489970d3b4d7c5ad909fe","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-0eea1ef5a822c7de8b6f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v02\n- v02 --supports--> v03\n- v03 --refutes--> v04\n- v04 --supports--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v06\n- v04 --refutes--> 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":"e03b37fc80b60c2464864d99b13379c1ac4e7977289037107bb70b251ab6c8e7","schema":"livegraph.boolean-propagation.v1","spec_hash":"0eea1ef5a822c7de8b6f54692123a51bc94ca28710775cbd179e403233c8d76c","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-e75623871cac761f495d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --refutes--> v01\n- v01 --supports--> v00\n- v00 --supports--> v04\n- v04 --refutes--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v03\n- v00 --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":"27b0ee85e9f174d64634506815ffc3a4207a8f8a2e399673313c3fba1ef5e90f","schema":"livegraph.boolean-propagation.v1","spec_hash":"e75623871cac761f495d3a22d10ce43aef2014acd771869c7bc77faf1ac3f879","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-dfdcd4b9e7032986cb0d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --supports--> v04\n- v04 --refutes--> v03\n- v03 --supports--> v06\n- v06 --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v02\n- v06 --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":"e802b5fc0c5061a02c34e94ac7b8ee6fce24ebc899bd2e18ad213c5a67b54f02","schema":"livegraph.boolean-propagation.v1","spec_hash":"dfdcd4b9e7032986cb0d7fc804e9d3ac901ea3696aea1ffbdd566dcf06030056","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-6f0dbab50143866ce705","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v04\n- v04 --supports--> v05\n- v05 --refutes--> v00\n- v00 --supports--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v03\n- v06 --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":"897e40ea753a50843c229842d10fdf0dd17729953eaf0a900b06486086748b4c","schema":"livegraph.boolean-propagation.v1","spec_hash":"6f0dbab50143866ce7058a883b0abac225dbf355ac453042c8dd59b6a8eed1f3","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-fa676353d157110a3562","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v03\n- v03 --refutes--> v00\n- v00 --supports--> v01\n- v01 --supports--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --supports--> v04\n- v06 --supports--> 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":"df75b8f21f0dba288aec1e886dce4c671b78e7dad3819a2236f44beb6a3887f4","schema":"livegraph.boolean-propagation.v1","spec_hash":"fa676353d157110a356266e54176f85e5aafe53466eb40892f9f3bb31ed47cf0","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-c155620d4683073a5ec5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --refutes--> v03\n- v03 --supports--> v02\n- v02 --refutes--> v06\n- v06 --refutes--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v04\n- v06 --refutes--> 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":"659be8b339e4edf4f9a67bbd8a497fa3bc7aa4facef6d98a7b69f15111e0f9b4","schema":"livegraph.boolean-propagation.v1","spec_hash":"c155620d4683073a5ec5fafb9b9e7862d37d3f5b08f5b1980b25beb1c5ae4cb0","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-2bdca4662f8a622f7e4a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --supports--> v02\n- v02 --supports--> v03\n- v03 --supports--> v01\n- v01 --refutes--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v00\n- v03 --refutes--> 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":"db0145ec64ba856d99b221ff34b0d66c760cf36698a612cc3257d0f8b96eb3b8","schema":"livegraph.boolean-propagation.v1","spec_hash":"2bdca4662f8a622f7e4ab124a662268dfc3c04f069eb10bb8e99bec73a98f676","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"}],\"hop\":4,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-2c8f92082a8ca45f9126","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v03\n- v03 --refutes--> v02\n- v02 --refutes--> v00\n- v00 --refutes--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v01\n- v02 --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":"592f8d0fb8a4f357716b99a050543efb5ddfc1166c244d8bd13a5693ffbff235","schema":"livegraph.boolean-propagation.v1","spec_hash":"2c8f92082a8ca45f9126ab57b7982e7d975b0070bae0e88f12a6a8c4a79e10bd","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-c1a05bee79ace42e05e0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v02\n- v02 --supports--> v00\n- v00 --supports--> v06\n- v06 --supports--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v01\n- v04 --refutes--> 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":"5a4a2575a6fe25834e87c4662e32b9a8d36c849c855941d73ecc4d1640ff554f","schema":"livegraph.boolean-propagation.v1","spec_hash":"c1a05bee79ace42e05e0944bfbb66cf53e7f438fb904828c461f3723d4d4c762","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-4acbd6567e37bdfafa9b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --supports--> v02\n- v02 --supports--> v05\n- v05 --refutes--> v04\n- v04 --supports--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v03\n- v04 --supports--> 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":"3614b993b9e4dcfedd00401337758513538139424f9aa7e017c00b4fb6af150a","schema":"livegraph.boolean-propagation.v1","spec_hash":"4acbd6567e37bdfafa9bf26aa97249618ffc963ed6f5eea6620fb48f3b173ee1","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-238a63e97544e5833d94","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --refutes--> v02\n- v02 --refutes--> v00\n- v00 --refutes--> v05\n- v05 --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v06\n- v02 --supports--> v04\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":"87f60e995f5179a702490ea0fa45b6f90f89b7ef489087ad5cc4546e38c72728","schema":"livegraph.boolean-propagation.v1","spec_hash":"238a63e97544e5833d94764e0e48d9eb4b4fad600dc3d4f636bc593c422499ae","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-34052a80062fe952269c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --supports--> v02\n- v02 --refutes--> v00\n- v00 --supports--> v03\n- v03 --supports--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v04\n- v06 --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":"c30458ec7cb2562e4ed7bd845f0b3013ca33317a8ac5184ed9334e9fd590cb50","schema":"livegraph.boolean-propagation.v1","spec_hash":"34052a80062fe952269c53a4beaa6e8f146073abed85c933899c9e4d8c4d553d","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-c04bfd727f74c7710ba6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --refutes--> v01\n- v01 --supports--> v06\n- v06 --refutes--> v00\n- v00 --supports--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v03\n- v05 --refutes--> 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":"685e3068438a97f5e7087c99ac4ab2a6cfa98d62f5fdec3d095b4eb1c9ebe394","schema":"livegraph.boolean-propagation.v1","spec_hash":"c04bfd727f74c7710ba6bd18b9132cd556fa92b52dead13345552e19b1741ae2","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-a0b19e0166d837cec93a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --supports--> v00\n- v00 --supports--> v02\n- v02 --supports--> v04\n- v04 --supports--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v03\n- v04 --supports--> v01\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":"cbe41b7261e6b6c08bd5678e4acdf76badf96d5230589cb4bfd546901feae935","schema":"livegraph.boolean-propagation.v1","spec_hash":"a0b19e0166d837cec93a9bd9771156e25b490a1a2eb99b073aa644bffed25938","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-15abbbef2f0b56c65fe2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v02\n- v02 --refutes--> v00\n- v00 --refutes--> v01\n- v01 --supports--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v05\n- v02 --refutes--> v03\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":"b72857770ff14174b5d8ac1d68cf225c13c1d0348e3911630573d9b3c5a8a114","schema":"livegraph.boolean-propagation.v1","spec_hash":"15abbbef2f0b56c65fe2708bc7eec68ca4330f509078eac3fb225e1afc82957a","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\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-f28340e26e7677f4b25c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v05\n- v05 --supports--> v03\n- v03 --refutes--> v04\n- v04 --supports--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v01\n- v04 --supports--> v02\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":"9773c2386465262d99ef2844d4e21f084913a3f157e8852122b2eed8fc53f681","schema":"livegraph.boolean-propagation.v1","spec_hash":"f28340e26e7677f4b25c9a59f93240fa630ef6124d3feda2153ac421dcbabcff","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"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\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-9a641f4d0c0907651fd1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v06\n- v06 --refutes--> v05\n- v05 --refutes--> v01\n- v01 --refutes--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v03\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":"c52cfea2e562d2f6cb3641d6665e7e40448b89c06e2a4e70db091e17bb9f19ee","schema":"livegraph.boolean-propagation.v1","spec_hash":"9a641f4d0c0907651fd130e120652414a6ae3c1d49a47fc9d7637065b2458b56","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-dc6d64c54c8d7d87328b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --supports--> v02\n- v02 --refutes--> v01\n- v01 --supports--> v04\n- v04 --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v05\n- v02 --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":"7a86d288bce4e06c1716c86543a653ceec72f4c6bda4bfa861e60dad9d4ee3d3","schema":"livegraph.boolean-propagation.v1","spec_hash":"dc6d64c54c8d7d87328b9d0eec198086560186e3aa5adc8a10814452b059d89f","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"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\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-91a8d68e8b09105ca94b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --refutes--> v06\n- v06 --supports--> v03\n- v03 --supports--> v04\n- v04 --supports--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v01\n- v06 --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":"942169d57a2d9460afcbb52fa2a16b85245faaab99ebd771a1a6d57f680900d1","schema":"livegraph.boolean-propagation.v1","spec_hash":"91a8d68e8b09105ca94b70b55f9a0b0bd964e820b74880d6d9346eb19bcd249c","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-0c59cee8ebb629c838cb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --supports--> v03\n- v03 --refutes--> v00\n- v00 --refutes--> v02\n- v02 --refutes--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v06\n- v04 --supports--> 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":"e6ff0f4fc29e8b31b77fabac8955a41b5ffd27880bf79b24c0275a79972156ec","schema":"livegraph.boolean-propagation.v1","spec_hash":"0c59cee8ebb629c838cbfb2a6e82569a35e45f90252dc7d6d57026548e54ac16","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"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\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-95acb1d4853154b82fa6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v03\n- v03 --refutes--> v04\n- v04 --supports--> v02\n- v02 --supports--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v01\n- v00 --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":"a23479a5e4d3574bcd671474e844e6aaec00e54332aee4449b542147c3a8156f","schema":"livegraph.boolean-propagation.v1","spec_hash":"95acb1d4853154b82fa6126d18aac24b5f57f1dca875e534b47bbdef5c0de722","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-40e83e382fe8bbda6bcb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --supports--> v06\n- v06 --refutes--> v04\n- v04 --supports--> v00\n- v00 --refutes--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v03\n- v04 --supports--> 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":"4dde9fa9e65aead9d842bce2db4b2b95919effc68efd1356d048f8361948ef60","schema":"livegraph.boolean-propagation.v1","spec_hash":"40e83e382fe8bbda6bcb4946db2a027ff940e823a520f3483167b0f35fb58493","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"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\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-a2508bee281f9dd82e60","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v05\n- v05 --refutes--> v03\n- v03 --supports--> v00\n- v00 --refutes--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v02\n- v05 --refutes--> v01\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":"12769a4bc169fa5db271bb84a68b6c10131e4134d292d2be35bbfcf0b35b8ddc","schema":"livegraph.boolean-propagation.v1","spec_hash":"a2508bee281f9dd82e602276bdec36fb74ec1dde63542a216609530949e81500","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-0877004ac7af85096038","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --refutes--> v02\n- v02 --supports--> v05\n- v05 --refutes--> v06\n- v06 --supports--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v01\n- v06 --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":"8bebef386729c49ff9415625d143299b01858e8594a361a8021176b186793514","schema":"livegraph.boolean-propagation.v1","spec_hash":"0877004ac7af8509603870c1ab1a77b3261d4e09fd39fc670a91ec37b6b5064d","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-ae3dc77a9098b40b1e7b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --supports--> v00\n- v00 --refutes--> v05\n- v05 --refutes--> v04\n- v04 --refutes--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v03\n- v04 --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":"ae5592ac7cb764c0a1b6e08e53bddfd52edeeaf3da0f96f50a201c7f5caba5cb","schema":"livegraph.boolean-propagation.v1","spec_hash":"ae3dc77a9098b40b1e7b6719495786302433d6e7ac694709a18fced1c2c54c31","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-8bb4d1039b8819f359a6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --refutes--> v02\n- v02 --supports--> v00\n- v00 --supports--> v04\n- v04 --supports--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v06\n- v04 --supports--> 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":"8fe88a3f504e14d73065915239f0dbe29a321f4180847397934f742dfc69c0d2","schema":"livegraph.boolean-propagation.v1","spec_hash":"8bb4d1039b8819f359a6e36cc7fd0a04af07631a552f69a4d2e469efa14b48a5","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-c55cb0bfd720a4c14117","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --refutes--> v04\n- v04 --refutes--> v03\n- v03 --supports--> v05\n- v05 --supports--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v06\n- v04 --refutes--> v01\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":"f7ca7587f7619a48cb45ba11073660a1db13c0ad1081356bb93ca656ec0b7c6e","schema":"livegraph.boolean-propagation.v1","spec_hash":"c55cb0bfd720a4c1411750bc79ad53a7e97feb5ac0292bccb5ecc6284d6384fd","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":4,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-3bb4eb9d5c168765f9f6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --supports--> v06\n- v06 --supports--> v03\n- v03 --refutes--> v02\n- v02 --supports--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v05\n- v02 --supports--> v01\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":"29fbeb05e0b7c90a697fd7cb379e2e5954e09fcabaaa1fb1f8971f275a919894","schema":"livegraph.boolean-propagation.v1","spec_hash":"3bb4eb9d5c168765f9f657b86e1646a86128490ae095a44a5f7176580c225f19","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":4,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":4,"item_id":"lggrpo1-eabee8aed40f40c06390","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v03\n- v03 --supports--> v01\n- v01 --supports--> v00\n- v00 --supports--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v02\n- v03 --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":"822d1d12486a47dd0c96ad2733679d83bdfca13c4ad117424634142bd64d6234","schema":"livegraph.boolean-propagation.v1","spec_hash":"eabee8aed40f40c063901f0c636250662e5d7ec48a08945dc4a21d93f42919f3","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"}],\"hop\":4,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":4,"item_id":"lggrpo1-2e4d1b5d97361fc10569","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v00\n- v00 --refutes--> v02\n- v02 --refutes--> v05\n- v05 --refutes--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v03\n- v05 --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":"067d1491cc5d72c63b4297974668bb176af70320da494470305fb1c31f6561fb","schema":"livegraph.boolean-propagation.v1","spec_hash":"2e4d1b5d97361fc1056902ca0b35c7bf2be7877294c71f01252ac22fd405bba5","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"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":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-5d1e54f8ebcf7c4e2808","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --supports--> v03\n- v03 --supports--> v02\n- v02 --refutes--> v04\n- v04 --refutes--> v05\n- v05 --refutes--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v00\n- v05 --supports--> 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":"2b9f030c3ff80a5ba9b1b0b12bea68987bcdd144a2437f7109e36f065e8e1858","schema":"livegraph.boolean-propagation.v1","spec_hash":"5d1e54f8ebcf7c4e2808bb4af25e2adc7d2c791ac24ead990b56a342ea163af1","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-c04221e8f0a046d87054","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v05\n- v05 --supports--> v01\n- v01 --refutes--> v03\n- v03 --refutes--> v02\n- v02 --supports--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v06\n- v03 --supports--> 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":"ad067dfcffab57993edb76f6aefe9a0dbdcaa08bdbe302a9babb9aa7feb45754","schema":"livegraph.boolean-propagation.v1","spec_hash":"c04221e8f0a046d870547432c0314fe17b91c6f7d295e0fa39c5c4ab066bf845","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-85e4ae3b135874f91fb0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v01\n- v01 --refutes--> v00\n- v00 --refutes--> v02\n- v02 --supports--> v05\n- v05 --supports--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v07\n- v05 --refutes--> v03\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":"7f75d33c595be70c07a9e96e180816e7518d2bb9175b4c45290a0a91264154a4","schema":"livegraph.boolean-propagation.v1","spec_hash":"85e4ae3b135874f91fb0b6bcea6340c023fa06b6bba8b8b4a4794b374f65f5d1","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":5,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-819b0e56dd1301a16243","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --refutes--> v01\n- v01 --supports--> v02\n- v02 --supports--> v03\n- v03 --refutes--> v00\n- v00 --supports--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v05\n- v07 --refutes--> v06\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":"e0eb93ddeaa043f73af8ea48a873d814ecbc91b5c81ebc144bcab058f982f114","schema":"livegraph.boolean-propagation.v1","spec_hash":"819b0e56dd1301a1624340070ed3d53fa0f229f85c76fdbd2e2b1fe95c0b851d","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-a456c8ab94d139b758d9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --refutes--> v03\n- v03 --supports--> v06\n- v06 --refutes--> v01\n- v01 --refutes--> v07\n- v07 --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v05\n- v06 --supports--> v04\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":"668f698da3b3aa4e06f290f9b22b56e50204dded044282112fdb4a1e4a817086","schema":"livegraph.boolean-propagation.v1","spec_hash":"a456c8ab94d139b758d91f2e571d0799ae29fa35d60369d9e9f21ed7eef78b7b","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-81065db8adea56bc54b0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --supports--> v04\n- v04 --supports--> v02\n- v02 --refutes--> v00\n- v00 --refutes--> v06\n- v06 --refutes--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v07\n- v06 --refutes--> v01\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":"d642d6d666db25d2df8bd8829c518e47dca5f2abb6de4d9895bc5286dd89823b","schema":"livegraph.boolean-propagation.v1","spec_hash":"81065db8adea56bc54b0e8154cb4a82b00ea7744233488605171d21c9ea3f395","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-602894802821163229aa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --refutes--> v01\n- v01 --supports--> v04\n- v04 --refutes--> v00\n- v00 --supports--> v05\n- v05 --supports--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v06\n- v01 --supports--> v03\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":"6e1aa8f53be82376708ed5bd4b3374c9092e97fb2fa15a99cdebc57770e3d078","schema":"livegraph.boolean-propagation.v1","spec_hash":"602894802821163229aa8d7c4556bf4accddb55b4868138b968f99cbd5a723e8","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-2fa16846a7bd218720d0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --supports--> v03\n- v03 --supports--> v06\n- v06 --supports--> v07\n- v07 --supports--> v02\n- v02 --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v04\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":"2f7e62e918754aec3d5bbe0fc0be67f1f8d87b88430c2798c6611d1770e22525","schema":"livegraph.boolean-propagation.v1","spec_hash":"2fa16846a7bd218720d05339b1b1aad7bb702e490cf1315f29846f910dffa5cb","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-8bfe01b09fba2b4075af","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --supports--> v01\n- v01 --supports--> v06\n- v06 --supports--> v00\n- v00 --refutes--> v02\n- v02 --supports--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v03\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":"e828dc60959fe8bc8a45c3e8dc06e5d32ff256f4030778b7889f5263ab0e1514","schema":"livegraph.boolean-propagation.v1","spec_hash":"8bfe01b09fba2b4075af904aef81ec331bc6c25e0d3775aee0ab0281cd798070","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-73f6cbb8b44eb571d472","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --supports--> v02\n- v02 --refutes--> v04\n- v04 --refutes--> v01\n- v01 --refutes--> v06\n- v06 --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v00\n- v04 --supports--> 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":"8665f4792b3972cb09b1eb77c0a641e2e92d9b10737d5cdd862f5ae6f8b56d65","schema":"livegraph.boolean-propagation.v1","spec_hash":"73f6cbb8b44eb571d472111c03cbefdca05337ec322d616dcb2674eed51758a0","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-24bb53dcebed6a1e1eec","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --supports--> v05\n- v05 --supports--> v07\n- v07 --refutes--> v00\n- v00 --supports--> v06\n- v06 --refutes--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v01\n- v07 --refutes--> 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":"6ebac8dc1fd28dd0b6a46062a1b974a7a8816237e200574e09baf9d5968cd640","schema":"livegraph.boolean-propagation.v1","spec_hash":"24bb53dcebed6a1e1eec6b952591857e9db8846689ba2a308a69c72ca6961114","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-3045b181eca3d5641fba","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --refutes--> v07\n- v07 --supports--> v02\n- v02 --refutes--> v06\n- v06 --supports--> v00\n- v00 --refutes--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v01\n- v06 --supports--> 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":"a55df87845cc88b751ed84913ae3aa5f37900bc2403a5aa48234ad89d009bba9","schema":"livegraph.boolean-propagation.v1","spec_hash":"3045b181eca3d5641fbaaed8d1fd9398a611ef0679ec62352c24a51073f92bb3","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-6d66dac49d7813bdf572","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --refutes--> v07\n- v07 --refutes--> v01\n- v01 --supports--> v00\n- v00 --supports--> v04\n- v04 --supports--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v06\n- v07 --refutes--> 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":"ce172e9bda2edaa5416d20fcc527cfc3e66ff5abee7c874c4fc19ee80d884639","schema":"livegraph.boolean-propagation.v1","spec_hash":"6d66dac49d7813bdf57241a0bb20f8308e984b75719baaee3b4e18d796dadeae","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-78093ee066493c521d63","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --supports--> v00\n- v00 --supports--> v04\n- v04 --supports--> v02\n- v02 --refutes--> v06\n- v06 --supports--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v07\n- v06 --refutes--> v01\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":"5ccfdacae9cb0f8e57e18645fd96fd86a4c268689c86bee1dd82d3e3acb1761b","schema":"livegraph.boolean-propagation.v1","spec_hash":"78093ee066493c521d6331a97f3ff2ad7a36b3025ec59a63a353ed5b2be44bd6","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-9655c6c31a1fbea924c0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v03\n- v03 --supports--> v01\n- v01 --refutes--> v04\n- v04 --refutes--> v05\n- v05 --supports--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v02\n- v04 --supports--> 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":"28c1242598aff789867d9fdb91c078568092abff8e943e816180507f9592091d","schema":"livegraph.boolean-propagation.v1","spec_hash":"9655c6c31a1fbea924c06ab58b78577c8ea5edffc7aecd6f4f53341ca498c900","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-18898e7f4b6fc3108c39","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --supports--> v01\n- v01 --supports--> v03\n- v03 --supports--> v00\n- v00 --supports--> v07\n- v07 --refutes--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v06\n- v04 --supports--> v05\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":"f166be7c9fec19b91a2bc8341a9a70e10d5fcf5798cc1152b5dd82d33bc33d1a","schema":"livegraph.boolean-propagation.v1","spec_hash":"18898e7f4b6fc3108c399813bb66ccb4c5319d9a7ddc9856521e23814aecaf1c","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-24bf294744d0e4196014","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v01\n- v01 --supports--> v02\n- v02 --supports--> v05\n- v05 --refutes--> v03\n- v03 --supports--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v06\n- v02 --supports--> 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":"50f4b892a4930e7753b339dd42c06c59a42503362816d06438633d03bc420fcd","schema":"livegraph.boolean-propagation.v1","spec_hash":"24bf294744d0e4196014f903e1de02e4a8b9bec519f6ca343fd8a153b583a141","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-e3adf3b87ce3d67916e6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --refutes--> v05\n- v05 --refutes--> v03\n- v03 --refutes--> v04\n- v04 --supports--> v06\n- v06 --refutes--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v07\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":"86556268465583de6a7032f9ec64a61eb13ae35ac7a8dacdb6018de2661dda24","schema":"livegraph.boolean-propagation.v1","spec_hash":"e3adf3b87ce3d67916e687567b5c957075d75891c4515c2be1f73e5411f5c4ce","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-a1912e0b3739618c8962","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --refutes--> v03\n- v03 --supports--> v01\n- v01 --refutes--> v02\n- v02 --refutes--> v05\n- v05 --supports--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v06\n- v07 --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":"6dddb24c6e2b2cad46e4d45e8d0bc8dbfb9e57d049e460a0f8d8f86aa51e3ff7","schema":"livegraph.boolean-propagation.v1","spec_hash":"a1912e0b3739618c89620c5648f2fcfa20f1419a874d04d364ee61514fc76859","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"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\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-1c8865a706d844d3c11b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v04\n- v04 --supports--> v01\n- v01 --supports--> v07\n- v07 --refutes--> v00\n- v00 --refutes--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --supports--> v05\n- v07 --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":"57a5df833b227ba4325188f5d904990e8b85bacd1148430f604fe7b014b8728d","schema":"livegraph.boolean-propagation.v1","spec_hash":"1c8865a706d844d3c11b272eedc843d34227c1bdbefe5efd16b5cf39f5a3c030","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"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\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-1d42d6de3fb2a961e5d6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --supports--> v04\n- v04 --supports--> v07\n- v07 --refutes--> v03\n- v03 --supports--> v05\n- v05 --refutes--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v06\n- v04 --refutes--> v00\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":"fcfd61782184bf7471f316e5aff593f9e1d821b16025e3a157dc1195b09ba0c7","schema":"livegraph.boolean-propagation.v1","spec_hash":"1d42d6de3fb2a961e5d626a10ebb7c3e75016ae978da7232dfeac90535643322","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-e4f7f5cbfb241382b012","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v02\n- v02 --supports--> v03\n- v03 --supports--> v06\n- v06 --supports--> v07\n- v07 --refutes--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v05\n- v07 --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":"b024b139687e6d09d73379ae9c8025acc1b7f2a7000409f0a181d3c4382b43c7","schema":"livegraph.boolean-propagation.v1","spec_hash":"e4f7f5cbfb241382b0122eab3e822e8c780b8073a42d99ee3b37d90946b28cff","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-25c2cd6ce1140826a1b3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --supports--> v04\n- v04 --supports--> v02\n- v02 --refutes--> v05\n- v05 --supports--> v06\n- v06 --refutes--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --supports--> v00\n- v06 --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":"a567e1e043c9ff59889fcea0aa0789730d52b8858a0c067f3517ce6176bcf78a","schema":"livegraph.boolean-propagation.v1","spec_hash":"25c2cd6ce1140826a1b3bda5b20a2acc8361f73ae717eff2b6efb5a465d4c900","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-d5ac2120a20cfe99bc7e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --refutes--> v04\n- v04 --refutes--> v00\n- v00 --refutes--> v05\n- v05 --supports--> v03\n- v03 --refutes--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v06\n- v05 --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":"5b992d2d4869cd38a384d68f102b90860ade521b5e89e558828897722889bd41","schema":"livegraph.boolean-propagation.v1","spec_hash":"d5ac2120a20cfe99bc7ece92768824907951d8c5450a710dbd156422ea6c5a35","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-a4962a413251f18d5051","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --refutes--> v06\n- v06 --refutes--> v01\n- v01 --refutes--> v05\n- v05 --supports--> v00\n- v00 --supports--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v02\n- v07 --supports--> v03\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":"86cfb82c6129d67834757bb426a385c241a6f73875b86e74a959ed5b63a35da2","schema":"livegraph.boolean-propagation.v1","spec_hash":"a4962a413251f18d5051baa8a570ce29825d4938cbabacfebe7324c62d0d8c0b","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-3c97e1508acf77ca8800","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --supports--> v05\n- v05 --refutes--> v00\n- v00 --refutes--> v03\n- v03 --supports--> v02\n- v02 --supports--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v07\n- v02 --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":"8e0e1f8f825c1d768ba54be9208f18b5b872ff94c833d1b248207222f1190151","schema":"livegraph.boolean-propagation.v1","spec_hash":"3c97e1508acf77ca8800d673234579f8399fd8438f60c6473919271b0dbfc94a","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":5,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-42bd49653757a519eaf0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --refutes--> v03\n- v03 --refutes--> v05\n- v05 --refutes--> v02\n- v02 --supports--> v06\n- v06 --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v01\n- v07 --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":"72ec590a2d0edde765110aa4ed73817b1bde16c9b42aea4f6f749635b5fd0068","schema":"livegraph.boolean-propagation.v1","spec_hash":"42bd49653757a519eaf04a16e4cb0bb7e2abfcae66ca1d86326bb1724facdb64","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-a26dd451ddf92d790aa7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --supports--> v05\n- v05 --supports--> v00\n- v00 --supports--> v04\n- v04 --refutes--> v02\n- v02 --supports--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v01\n- v02 --refutes--> 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":"fc9bb98e641026b30880be21ec9407765d45db556dc594d08193332d5f4b7604","schema":"livegraph.boolean-propagation.v1","spec_hash":"a26dd451ddf92d790aa776f3a4a083e97221710682cb3d930b6fb99cc2f7c07c","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-5b4cb567d167e6d9529f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --refutes--> v03\n- v03 --supports--> v06\n- v06 --supports--> v04\n- v04 --refutes--> v00\n- v00 --supports--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v01\n- v04 --refutes--> v02\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":"6f0e892e826c2abc6005e9ce32e18e0697289693265d7dcff787537955668305","schema":"livegraph.boolean-propagation.v1","spec_hash":"5b4cb567d167e6d9529f99d975201237b592a4c1e4f0a1fa147da25e1ec18bf3","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"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\":\"refutes\",\"src\":\"v04\"},{\"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\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-8eaaf170e7e801b36f2b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --supports--> v07\n- v07 --supports--> v01\n- v01 --refutes--> v00\n- v00 --supports--> v05\n- v05 --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v07 --refutes--> v04\n- v07 --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":"661afb8ae96c7de5c6cd605902cf4de6fbdcbf87cd2ddb5f5ff65cfda8a28f2d","schema":"livegraph.boolean-propagation.v1","spec_hash":"8eaaf170e7e801b36f2b74a0090c574358cbe23ea9eaa3969fdaad656f02ffbc","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-d92c07fadf00f05a23ef","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --supports--> v04\n- v04 --supports--> v06\n- v06 --refutes--> v00\n- v00 --supports--> v05\n- v05 --supports--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v03\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":"7803d5ba4a338be5d5337b7ea8331eca7c8c13da4cdee907287b07b030ffe35f","schema":"livegraph.boolean-propagation.v1","spec_hash":"d92c07fadf00f05a23efdb8ffd9d1816afedf0e05b6dbcf64f5a7f61b909c9f0","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-f2d99b89a5d2f3ce0cb1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --supports--> v04\n- v04 --supports--> v01\n- v01 --refutes--> v07\n- v07 --supports--> v02\n- v02 --supports--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v00\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":"7e27a80a140e94bff248bcfecda7e011c334e8e3a629972eba5b8fdd6e1647e9","schema":"livegraph.boolean-propagation.v1","spec_hash":"f2d99b89a5d2f3ce0cb10c07e770a005f9addff0c6e4edc50153dc794630ae92","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-42af4df78e95fbc27c05","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --refutes--> v03\n- v03 --supports--> v06\n- v06 --refutes--> v02\n- v02 --supports--> v00\n- v00 --refutes--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v04\n- v06 --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":"c35d6fa0e96cc253af4edcc9e748a649656757609c60b0f264d5d7fe96109dcf","schema":"livegraph.boolean-propagation.v1","spec_hash":"42af4df78e95fbc27c05dc81c5f147895782f223a294f15274137c6081e07878","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-cfaef636bfe5c2306150","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --refutes--> v03\n- v03 --supports--> v04\n- v04 --refutes--> v01\n- v01 --refutes--> v06\n- v06 --supports--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v07\n- v04 --supports--> v02\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":"b90f8a99cb7de084995c1c8e7ff862f1f15ca497e34790ee7ac68fb6317c52f9","schema":"livegraph.boolean-propagation.v1","spec_hash":"cfaef636bfe5c2306150fa52c39e6de8a9035e384f5ac5b8b868c61e075ffe52","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-cb9c33370850fb56e33b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --supports--> v01\n- v01 --refutes--> v05\n- v05 --refutes--> v07\n- v07 --supports--> v03\n- v03 --supports--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v04\n- v06 --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":"44694830c924510505e22bed55905b861bea46a9c416763cf39101fc36e8a79e","schema":"livegraph.boolean-propagation.v1","spec_hash":"cb9c33370850fb56e33beb2de8bc16807c47f72602a10a849083b7ddfadcf450","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-e3e68c11ee9458762431","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --supports--> v04\n- v04 --refutes--> v02\n- v02 --supports--> v03\n- v03 --supports--> v00\n- v00 --refutes--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v07\n- v04 --refutes--> v06\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":"2a9fbe6c0eab0e9aa22c4f72edba8367120146e4a6a4d33843e8ffba8ba57bd4","schema":"livegraph.boolean-propagation.v1","spec_hash":"e3e68c11ee9458762431aad4c75ca94d9fb43cf2ad5285b53dd48feba2cb6bb1","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-395208953f0ca37d866a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --supports--> v04\n- v04 --refutes--> v05\n- v05 --refutes--> v03\n- v03 --supports--> v02\n- v02 --supports--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v01\n- v07 --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":"24de6a570d2035e3fb746093a78266d2586f46bd0498c35d4cd15e228fcabba9","schema":"livegraph.boolean-propagation.v1","spec_hash":"395208953f0ca37d866a14e25f7462339281506c1a568851c3ddf51c79919a81","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"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":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-df3c162085b0f86ab1d0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v04\n- v04 --supports--> v06\n- v06 --refutes--> v05\n- v05 --refutes--> v02\n- v02 --refutes--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v03\n- v06 --refutes--> v07\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":"a277ed6c127f6d3829729f58f820839bb93509f1652dc8979b5733a14e7df7ce","schema":"livegraph.boolean-propagation.v1","spec_hash":"df3c162085b0f86ab1d016095f000f936eccfebb5855aaf980111c98ee515b4a","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-310da3626416d072ce9d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --supports--> v03\n- v03 --refutes--> v00\n- v00 --supports--> v01\n- v01 --supports--> v02\n- v02 --supports--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v05\n- v02 --supports--> 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":"c2d5ad95ff2f40b8878d57d9c90cede20d55481043d5c8421d6f6502dee9a2ff","schema":"livegraph.boolean-propagation.v1","spec_hash":"310da3626416d072ce9d49472818c3773bc1ec6785800f72f05a05085c341320","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"}],\"hop\":5,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-9c01eca534235ef02fa3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --refutes--> v04\n- v04 --supports--> v06\n- v06 --supports--> v05\n- v05 --supports--> v03\n- v03 --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v02\n- v06 --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":"9ddd24d9e47d2f3ac324e068ccba1b41a10c04922f9aad97e3776b1ddec394f5","schema":"livegraph.boolean-propagation.v1","spec_hash":"9c01eca534235ef02fa3848c52a6b5431aebf609043aab1123b90e5b1589388e","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-6f1a1dcce6a3322d8ea1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --refutes--> v02\n- v02 --refutes--> v03\n- v03 --refutes--> v07\n- v07 --refutes--> v06\n- v06 --supports--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v05\n- v02 --refutes--> v00\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":"186ccfd7c0b51e45f5a25e3d9551b92537d4e44276e359e8517143d04f5443b2","schema":"livegraph.boolean-propagation.v1","spec_hash":"6f1a1dcce6a3322d8ea19cf1ec7a0a2b85b4daa81ce24c33e6ceca335f48c7f3","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-4be574bcc0b1a63b41d6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --refutes--> v06\n- v06 --supports--> v04\n- v04 --refutes--> v05\n- v05 --supports--> v00\n- v00 --refutes--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v03\n- v01 --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":"fdd1ee69d499beffd9a2b05c07733eaf29118798043b6fdd998ac4e97e6b0489","schema":"livegraph.boolean-propagation.v1","spec_hash":"4be574bcc0b1a63b41d68fcc75ddb914ac693509029dbab6a5385f84c4c22a80","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-bae113fcf32ecc903481","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --refutes--> v05\n- v05 --refutes--> v00\n- v00 --refutes--> v01\n- v01 --supports--> v02\n- v02 --refutes--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v04\n- v05 --refutes--> v06\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":"1631482946ce5ca6fa0007ec70b196745063744d3026867740097d865de2efea","schema":"livegraph.boolean-propagation.v1","spec_hash":"bae113fcf32ecc90348106fd46334f6a9daac66aa6ac8cd47e4f51572b17abfe","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-f627e942b1d8c482ce5c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --supports--> v00\n- v00 --supports--> v07\n- v07 --supports--> v01\n- v01 --supports--> v04\n- v04 --refutes--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v03\n- v07 --refutes--> v02\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":"dc28fe10ce90bc1fab5f773c5bdd487b88999c9f74c7e08839f54bbea18997d7","schema":"livegraph.boolean-propagation.v1","spec_hash":"f627e942b1d8c482ce5c69cddf02acdf1277f74450c3a6b24e3aceca08bb361b","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-fdacfa67f253b3f5cf13","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --supports--> v04\n- v04 --supports--> v00\n- v00 --refutes--> v07\n- v07 --refutes--> v05\n- v05 --refutes--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v03\n- v02 --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":"f9b71a837a07fcafa00a5beb63c10b927f010294e062fa84eb6394b5cd95a9ff","schema":"livegraph.boolean-propagation.v1","spec_hash":"fdacfa67f253b3f5cf13354bdef15393deed89375271a73bab09d93cc1c4cfe9","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-a29c68bac870429737c1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v05\n- v05 --supports--> v06\n- v06 --supports--> v02\n- v02 --refutes--> v03\n- v03 --supports--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v01\n- v05 --supports--> v04\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":"7917b614e9ae2d7ef3544d6dae65a770dbd805a85956f3ab1e4b047bb3952ca7","schema":"livegraph.boolean-propagation.v1","spec_hash":"a29c68bac870429737c129122cfc8d4dfd63368aa2376bbf6ed8c661e66966c1","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-a9c54f0b9bae91c60624","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --supports--> v00\n- v00 --refutes--> v01\n- v01 --refutes--> v02\n- v02 --supports--> v07\n- v07 --supports--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v05\n- v01 --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":"148f170c012f9af770c2841a1d719ad8f8f749d2e29ca6c59f168090eff84fac","schema":"livegraph.boolean-propagation.v1","spec_hash":"a9c54f0b9bae91c60624e6446620b4c73dd46b3bfc8e1caae5dbd1da144f6e82","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-22ee4fbdaf3730a60d72","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --refutes--> v04\n- v04 --refutes--> v05\n- v05 --supports--> v02\n- v02 --refutes--> v00\n- v00 --refutes--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v01\n- v00 --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":"ada5fdcbe463840be4f7d6034b0b17c2235fa30bc0f838bdd792a70e84135c6a","schema":"livegraph.boolean-propagation.v1","spec_hash":"22ee4fbdaf3730a60d72a0e5a95a955fce4789673d164b6a4e23d98952f83f44","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-8b857dc15e23c864b53c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --supports--> v02\n- v02 --refutes--> v07\n- v07 --refutes--> v03\n- v03 --refutes--> v01\n- v01 --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v06\n- v07 --refutes--> 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":"f96979eb928561cef21ceab0dd8425fb1e94f8506fdafb63dc3a6a86e86bcc3c","schema":"livegraph.boolean-propagation.v1","spec_hash":"8b857dc15e23c864b53caeaf295ae8136667293e101f935fd7241084b17a9fbd","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-3c31365ca4003c362f45","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --refutes--> v03\n- v03 --supports--> v00\n- v00 --supports--> v01\n- v01 --supports--> v02\n- v02 --refutes--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v07\n- v03 --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":"6a395cd63e35abf643ccc24fe34b08834ff827268413964b79824878a88392d4","schema":"livegraph.boolean-propagation.v1","spec_hash":"3c31365ca4003c362f4588257693d53c24942a9c423ccc5d9f4e5b697c0c86dc","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":5,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-e9ae95ef1dfc7bbaa308","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --supports--> v07\n- v07 --refutes--> v00\n- v00 --refutes--> v02\n- v02 --refutes--> v05\n- v05 --supports--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v04\n- v06 --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":"ff80c181b04118c5b774b6b8a86df94bb30275f3eb8580fd999c914734ee66cc","schema":"livegraph.boolean-propagation.v1","spec_hash":"e9ae95ef1dfc7bbaa3086beb6b31ba0e4c7ea53248d042a320de8eeebbdb6884","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"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\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-13ce9f6cacdb911861b4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --refutes--> v04\n- v04 --supports--> v03\n- v03 --supports--> v01\n- v01 --refutes--> v06\n- v06 --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v07\n- v04 --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":"8959f481dd2f565bc6a88908781dfe3152cd5b49c9ee6d23ca1a0270bba1933b","schema":"livegraph.boolean-propagation.v1","spec_hash":"13ce9f6cacdb911861b4201370aa373d6688f7489cb694bb5db37aae70716a49","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-4b5a383635a517d62b83","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v05\n- v05 --refutes--> v07\n- v07 --supports--> v01\n- v01 --supports--> v04\n- v04 --supports--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> 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":"577a472ad5ffddb719c7d2c27b5177059a542716ea64df564f1840559f157bb5","schema":"livegraph.boolean-propagation.v1","spec_hash":"4b5a383635a517d62b833210b38bd81b6b26301b618c173ced9547ee7b9e42a6","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-615baafa1a61f311ad20","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --supports--> v05\n- v05 --refutes--> v04\n- v04 --refutes--> v01\n- v01 --supports--> v07\n- v07 --refutes--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v03\n- v07 --supports--> 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":"6e331cb3e792e00e3d6dfba60b43e5b6a357f1ec4a9ae9bd4731bf59548818f4","schema":"livegraph.boolean-propagation.v1","spec_hash":"615baafa1a61f311ad200a6c017f072b02e757f9431bda9666fab625d29978c5","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"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\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-b141eeb98f24abecb94c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v02\n- v02 --refutes--> v05\n- v05 --refutes--> v07\n- v07 --supports--> v03\n- v03 --supports--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v00\n- v05 --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":"25b25bf5929fb98f006062a88f35d36f8463935947ec66d97c86437a14e5d695","schema":"livegraph.boolean-propagation.v1","spec_hash":"b141eeb98f24abecb94c135648c875f57281bfec6397f78773094b8b5c5b10c9","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-d8fd20eb01f9f22777bc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --supports--> v04\n- v04 --supports--> v05\n- v05 --supports--> v02\n- v02 --refutes--> v01\n- v01 --supports--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v03\n- v02 --refutes--> v06\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":"af7a3655db9e293925031fd41212663eaf050aa45d9d8e50ec6b8886a5a735d9","schema":"livegraph.boolean-propagation.v1","spec_hash":"d8fd20eb01f9f22777bc8f3033d652979651a09e417e58f85f441a4bb9f56bcf","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-b3a443f687dfed783041","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --refutes--> v02\n- v02 --supports--> v01\n- v01 --refutes--> v06\n- v06 --refutes--> v07\n- v07 --refutes--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v03\n- v07 --supports--> 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":"c1b3f0237ca167562ef4d4ae8858e31bf49d1dd4f229642ee2866d6801a9b12b","schema":"livegraph.boolean-propagation.v1","spec_hash":"b3a443f687dfed78304123b395172a1d1c9679e88bf930b5798e228740b5e127","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-1e9e340806507e4b8131","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --supports--> v00\n- v00 --supports--> v02\n- v02 --refutes--> v07\n- v07 --supports--> v05\n- v05 --supports--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v03\n- v02 --refutes--> 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":"955eb60925c075ec12c2c3393cb5c9979a239f11e9b351271a948f3bb1bd32cd","schema":"livegraph.boolean-propagation.v1","spec_hash":"1e9e340806507e4b8131648c16f303b83c82ddc2673fb82aa24bbc106660fd65","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-05bbe01648417b8face4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --refutes--> v02\n- v02 --refutes--> v05\n- v05 --refutes--> v03\n- v03 --refutes--> v04\n- v04 --refutes--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v07\n- v04 --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":"f075338117418963c94e54fb045c09f06e07a2d3e3cb624d732c662b2ceaa438","schema":"livegraph.boolean-propagation.v1","spec_hash":"05bbe01648417b8face4cf539923412a64b82f29e27371644eb0ba6efdebe3d0","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"}],\"hop\":5,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-ebca5fe4dab8fb7034d7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --refutes--> v02\n- v02 --refutes--> v06\n- v06 --supports--> v05\n- v05 --refutes--> v01\n- v01 --supports--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v07\n- v05 --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":"30291d703f8a26d7de86751bb7ee2eb2e5c72b80dad0e690bdf50c652d1f1d87","schema":"livegraph.boolean-propagation.v1","spec_hash":"ebca5fe4dab8fb7034d794c7cde10dd2c7fdd1b73a9d1a4e988e7c2a86692af7","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-d279d88d2086a86b81d3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --supports--> v07\n- v07 --refutes--> v06\n- v06 --supports--> v05\n- v05 --refutes--> v00\n- v00 --supports--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v02\n- v07 --supports--> v01\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":"6224df03e4cd0dfdcec71c803777c3da86737b0897a8cf13af31220e71c3e185","schema":"livegraph.boolean-propagation.v1","spec_hash":"d279d88d2086a86b81d3d5d12efaf34578d3dde31d9c4a56f4f328b3a6e6d5b9","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-51d381ffc7aec865e104","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --refutes--> v05\n- v05 --supports--> v04\n- v04 --supports--> v06\n- v06 --supports--> v07\n- v07 --supports--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v03\n- v07 --refutes--> v01\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":"d4cdc1d5a1793fb7086dc6f0e18e062429438d6a938139f1a14158e966cbcd0f","schema":"livegraph.boolean-propagation.v1","spec_hash":"51d381ffc7aec865e1047a23e99c716e59cbb2a1ed9d7a21dc4fec7fbb4d17af","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":5,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":5,"item_id":"lggrpo1-8d797b04c232b2cf14d1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --refutes--> v01\n- v01 --refutes--> v04\n- v04 --supports--> v06\n- v06 --supports--> v02\n- v02 --refutes--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v00\n- v06 --refutes--> v07\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":"c85e3440929d3e98f66c5c5d9a182dea3245fdc76133e588576b8ff7b63a4f5a","schema":"livegraph.boolean-propagation.v1","spec_hash":"8d797b04c232b2cf14d15c550ebf27a91ad7394508814063440955c0847b4b31","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":5,"item_id":"lggrpo1-848d93c8b6fffb026267","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --refutes--> v03\n- v03 --refutes--> v05\n- v05 --supports--> v02\n- v02 --refutes--> v06\n- v06 --supports--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v07\n- v05 --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":"43c9bf94a8345f7fbd181099b36a321a44c7e44beeccc3bbe218bf097d018a28","schema":"livegraph.boolean-propagation.v1","spec_hash":"848d93c8b6fffb0262678ae129be3016e8edaec7eacc083ecfe0e55987964092","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"}],\"hop\":5,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-a3059c245273495cfd3f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v09\n- v09 --supports--> v02\n- v02 --refutes--> v07\n- v07 --refutes--> v01\n- v01 --supports--> v08\n- v08 --supports--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v08 --refutes--> v03\n- v08 --refutes--> v05\n- v08 --supports--> v04\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":"8d5b1fb801aa7d66477765da9931811b32220d86a970a573c1aa8a468e36fab6","schema":"livegraph.boolean-propagation.v1","spec_hash":"a3059c245273495cfd3f426392f3bdf70d7ed53ba46899055454154e8e7ab011","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-7ef4e8453372eb3c56af","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --supports--> v01\n- v01 --refutes--> v04\n- v04 --supports--> v00\n- v00 --supports--> v03\n- v03 --supports--> v07\n- v07 --refutes--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v02\n- v01 --supports--> v06\n- v01 --supports--> v08\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":"fb75027999e5c7189547691eb81d7e26ef5d409f33680242c9c088174744b3a8","schema":"livegraph.boolean-propagation.v1","spec_hash":"7ef4e8453372eb3c56af8be35dfd3bc0162b9a1a96e747bac04dbdcc2211e3a0","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":6,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-3727b239244e41befa20","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --refutes--> v09\n- v09 --supports--> v01\n- v01 --refutes--> v00\n- v00 --refutes--> v05\n- v05 --refutes--> v06\n- v06 --refutes--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v02\n- v07 --supports--> v08\n- v09 --supports--> v03\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":"ab7ecb7be7519864c0e65ddde75f5246b37b3e8403516518b3ad8dbd6635e730","schema":"livegraph.boolean-propagation.v1","spec_hash":"3727b239244e41befa2087584a299b0e4ca8c7d26da7b584c92f366159f407c5","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-af47eaf886a56c42f564","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --supports--> v03\n- v03 --refutes--> v04\n- v04 --refutes--> v08\n- v08 --supports--> v05\n- v05 --supports--> v06\n- v06 --refutes--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v00\n- v05 --refutes--> v09\n- v06 --supports--> 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":"4251efeece4fc3b35555e5eed8e7d57f40bafca6fb9f599ee717cb3e633c4e5f","schema":"livegraph.boolean-propagation.v1","spec_hash":"af47eaf886a56c42f56410fb0791d392bfe1da45fe46b3c888bbf14183553ef6","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-da7fb83b2fd710d99720","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --refutes--> v06\n- v06 --refutes--> v03\n- v03 --refutes--> v07\n- v07 --refutes--> v02\n- v02 --refutes--> v08\n- v08 --supports--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v04\n- v03 --supports--> v00\n- v06 --refutes--> v01\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":"84699cc64486347ab846d067fe44b47cfc3aeaaf36b850a38fd09c2805e1080d","schema":"livegraph.boolean-propagation.v1","spec_hash":"da7fb83b2fd710d997201f294888cd24a7bcf0896700211dd5604284261c01a4","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-ec9d9ba6d7b27fd2bfe8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --supports--> v06\n- v06 --refutes--> v08\n- v08 --supports--> v01\n- v01 --refutes--> v05\n- v05 --refutes--> v04\n- v04 --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v02\n- v04 --refutes--> v07\n- v06 --refutes--> v03\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":"79860088941621b8f71e01178425a752ce4bbed9b5835e6e55f2fe15eb87f3ad","schema":"livegraph.boolean-propagation.v1","spec_hash":"ec9d9ba6d7b27fd2bfe8aaccd5941804a2e3cea4285763bbc7cca17687ddd112","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-b3de18e3f74fd5fa02ca","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --refutes--> v05\n- v05 --supports--> v09\n- v09 --supports--> v04\n- v04 --supports--> v08\n- v08 --supports--> v01\n- v01 --supports--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v02\n- v04 --supports--> v06\n- v08 --supports--> v00\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":"183fc84d0c78bf15437804d1cfd96048508119ac03787c4b40f4830c8bc02abf","schema":"livegraph.boolean-propagation.v1","spec_hash":"b3de18e3f74fd5fa02caa6c1ba20f7462b706b8ce1b3aa1f63bc1c746e719c87","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-be879eeb5764e8c805de","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --supports--> v00\n- v00 --supports--> v01\n- v01 --refutes--> v07\n- v07 --refutes--> v05\n- v05 --supports--> v06\n- v06 --supports--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v02\n- v06 --refutes--> v03\n- v07 --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":"9ecacf5a8a6775b3b381463685607f2c66f2394ae6b16e18ae61591f00a4d1d2","schema":"livegraph.boolean-propagation.v1","spec_hash":"be879eeb5764e8c805de0abec1db2fbb623e42bcae80ac8019b151b7851b9804","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-5ef83a2800f475a15257","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v01\n- v01 --supports--> v08\n- v08 --refutes--> v02\n- v02 --refutes--> v04\n- v04 --supports--> v06\n- v06 --refutes--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v05\n- v04 --refutes--> v03\n- v04 --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":"f653476a3db43902b213aa88833a168d32f403594164aecfde75320f5de3ead5","schema":"livegraph.boolean-propagation.v1","spec_hash":"5ef83a2800f475a152579d6da08b5e5bf37a2b7762ff9f504c38f70fd98c62ca","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-67c2e5d4f71238382fc9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --refutes--> v00\n- v00 --supports--> v08\n- v08 --refutes--> v05\n- v05 --supports--> v01\n- v01 --refutes--> v04\n- v04 --supports--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v06\n- v04 --refutes--> v09\n- v04 --supports--> v03\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":"003a680ccbccc6689e86e4de62d1895e1146db60127578b90ce1f1fc04808a64","schema":"livegraph.boolean-propagation.v1","spec_hash":"67c2e5d4f71238382fc9513bf4785f89e9e284f523e152405bf15d30fc9eb1fa","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-3eddf31bb64148a09c3e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --refutes--> v04\n- v04 --supports--> v03\n- v03 --refutes--> v05\n- v05 --supports--> v07\n- v07 --supports--> v06\n- v06 --refutes--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v02\n- v05 --refutes--> v00\n- v07 --refutes--> v08\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":"77a03e34aaee91d1771aec6f8541218a0f38f14fe80d5292c02446a17e1d3274","schema":"livegraph.boolean-propagation.v1","spec_hash":"3eddf31bb64148a09c3e5516a0f6605c3d48d0a5e6f07b1d432f2ae83907482e","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":6,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-83008244ae1554d2a422","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --supports--> v05\n- v05 --supports--> v08\n- v08 --refutes--> v00\n- v00 --supports--> v09\n- v09 --supports--> v02\n- v02 --supports--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v01\n- v09 --refutes--> v06\n- v09 --refutes--> 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":"494e26768bd690fc86a6ddf93098932badfc8a455b5cd6316252b83a7792d78d","schema":"livegraph.boolean-propagation.v1","spec_hash":"83008244ae1554d2a42229ae7d4f97d0283fa654b0794789b4125cf9e6002328","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-4ed97de6a2bf7cbf5a29","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v03\n- v03 --supports--> v08\n- v08 --supports--> v06\n- v06 --supports--> v02\n- v02 --supports--> v05\n- v05 --supports--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v09\n- v06 --supports--> v00\n- v06 --supports--> v01\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":"c817b029cf4d96b0c809ac1c130ae0b4e05077795bcf1a6a77db4d8a84c90082","schema":"livegraph.boolean-propagation.v1","spec_hash":"4ed97de6a2bf7cbf5a29fa12cf68cb36281c194b17abe505ed74a679b974e626","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-e0271f2e31e5bff3b870","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --supports--> v01\n- v01 --refutes--> v09\n- v09 --supports--> v05\n- v05 --refutes--> v06\n- v06 --supports--> v00\n- v00 --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v09 --refutes--> v02\n- v09 --refutes--> v04\n- v09 --supports--> v07\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":"9bc3c7b2c10cc3ca6fe66e4e385b844c87f580581c045a5387d665ad3d9aadd0","schema":"livegraph.boolean-propagation.v1","spec_hash":"e0271f2e31e5bff3b870a961885fc2fdee135e14d6feff7fbb542762887a0dbc","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-aff54d6c5c445eaafafa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --refutes--> v07\n- v07 --supports--> v06\n- v06 --supports--> v00\n- v00 --refutes--> v05\n- v05 --refutes--> v09\n- v09 --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v08\n- v05 --refutes--> v04\n- v09 --refutes--> 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":"3fe9c37a6df1052064e946f232f5bcfacc781fc3a1c794fdba229a1f1c87a5d7","schema":"livegraph.boolean-propagation.v1","spec_hash":"aff54d6c5c445eaafafa55ffa6ca23fa0a24b2ec656444ba4fbc06bcf5e836dd","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-1352ddc9e8bfaba596e1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --supports--> v02\n- v02 --refutes--> v06\n- v06 --refutes--> v08\n- v08 --refutes--> v00\n- v00 --refutes--> v09\n- v09 --refutes--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v04\n- v05 --supports--> v01\n- v06 --refutes--> 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":"76d177cef93cf244dbf71d81441f529aa89248ce8c77568e0b63d23675292df1","schema":"livegraph.boolean-propagation.v1","spec_hash":"1352ddc9e8bfaba596e1d01b3fb7db54a2eca8b68bd02c47240b17ff54b690ac","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-cbc30ce6b11cd67e2189","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v05\n- v05 --supports--> v02\n- v02 --supports--> v03\n- v03 --supports--> v04\n- v04 --supports--> v06\n- v06 --refutes--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v08\n- v03 --refutes--> v01\n- v06 --refutes--> v09\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":"d06cf33687a66362eb19837af62c3a359349ce4bf19a8d8ee447bcc98d1004de","schema":"livegraph.boolean-propagation.v1","spec_hash":"cbc30ce6b11cd67e218944692a77e08d4221085addda279e8a32e62404cc3573","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"}],\"hop\":6,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-5a8e15762dd626b3b6bc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --refutes--> v01\n- v01 --refutes--> v05\n- v05 --supports--> v09\n- v09 --refutes--> v04\n- v04 --supports--> v07\n- v07 --refutes--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v06\n- v07 --refutes--> v00\n- v07 --refutes--> 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":"f82cc25456e027ecee7ace1e0ab3b77e95f81d668389b25ee97bd43ed35d2721","schema":"livegraph.boolean-propagation.v1","spec_hash":"5a8e15762dd626b3b6bcaa8155291fe2848f6c814b6fabdc7fa08a61887cc209","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-61a4ae28c4a48f2bf79e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v01\n- v01 --refutes--> v08\n- v08 --supports--> v06\n- v06 --refutes--> v02\n- v02 --supports--> v05\n- v05 --supports--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v03\n- v00 --refutes--> v07\n- v01 --refutes--> v09\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":"bf8c8be28649337192d2dadb85da55d660446fed470d0cc36885a30613055513","schema":"livegraph.boolean-propagation.v1","spec_hash":"61a4ae28c4a48f2bf79e357826640474ef936bc98fdb7ea14210444925e25846","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-a9cbfc36f5aae15afab4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --refutes--> v01\n- v01 --refutes--> v04\n- v04 --refutes--> v00\n- v00 --supports--> v09\n- v09 --refutes--> v07\n- v07 --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v06\n- v02 --supports--> v08\n- v04 --supports--> 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":"fe946ac22be706eb9c8dd5df668ff62395a5bd86f0910fa5f2ffc49bc1ec85f5","schema":"livegraph.boolean-propagation.v1","spec_hash":"a9cbfc36f5aae15afab4348111401c2c84cb5ef3f128b1f1eb2954290ffdabf9","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-9c6dad0457bd1711a841","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --supports--> v04\n- v04 --supports--> v02\n- v02 --supports--> v06\n- v06 --supports--> v03\n- v03 --supports--> v07\n- v07 --supports--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v01\n- v06 --refutes--> v09\n- v07 --supports--> 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":"c78137e8de4b032f8332e031b3b5b52bf57b2d6c5059289eb5467d9b09c3b027","schema":"livegraph.boolean-propagation.v1","spec_hash":"9c6dad0457bd1711a841e13bedc1714e77e76a4ec2fc91421ef0a03dd0506b76","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":6,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-48ffc0ed1e9460a2f07d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --refutes--> v05\n- v05 --supports--> v04\n- v04 --supports--> v06\n- v06 --supports--> v00\n- v00 --refutes--> v07\n- v07 --supports--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v01\n- v05 --refutes--> v02\n- v05 --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":"f851b8ac56c6f77da76ef20f81947b58ac7d18e0d6b12d50858365797fd4d27b","schema":"livegraph.boolean-propagation.v1","spec_hash":"48ffc0ed1e9460a2f07dc671f6fd6a6ab4fdd63f07ffac1a6f2c907965ac6a2a","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-20bdd4b1d24faad698f1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --supports--> v02\n- v02 --supports--> v09\n- v09 --supports--> v07\n- v07 --refutes--> v00\n- v00 --refutes--> v03\n- v03 --refutes--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v08\n- v03 --supports--> v01\n- v04 --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":"dbf1a693dfd3a145b4da734d7f21894ea40ae9a5b5f553b7d592229c84d7f2d5","schema":"livegraph.boolean-propagation.v1","spec_hash":"20bdd4b1d24faad698f1fe89613cc451ba58f13778394758edb1889305dc4e98","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"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\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-7067252a7d194025e9a8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --supports--> v02\n- v02 --supports--> v01\n- v01 --supports--> v08\n- v08 --refutes--> v06\n- v06 --supports--> v05\n- v05 --refutes--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v03\n- v02 --supports--> v07\n- v06 --supports--> v00\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":"661df4ff695529ee7fba41044b34eca398e56ecda92210f2432bcf54c60dfeeb","schema":"livegraph.boolean-propagation.v1","spec_hash":"7067252a7d194025e9a858974ed3100769d189c29866cd635ebef31fd2010d7e","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-6169b763331adcd95ef8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --refutes--> v00\n- v00 --refutes--> v08\n- v08 --supports--> v03\n- v03 --refutes--> v07\n- v07 --supports--> v02\n- v02 --refutes--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v09\n- v05 --supports--> v01\n- v05 --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":"b1b6e40ecf9f8b76825523007cb1b918359a41e37a8425521881248b8568a259","schema":"livegraph.boolean-propagation.v1","spec_hash":"6169b763331adcd95ef84e6d529b5a7ffd843ce6f7a138649fa27d6c9913a49a","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-68d91e49173fff212831","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --supports--> v08\n- v08 --supports--> v03\n- v03 --supports--> v05\n- v05 --refutes--> v00\n- v00 --supports--> v02\n- v02 --refutes--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v06\n- v02 --refutes--> v09\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":"62a1002457b61022844bb3a51896056294b8ce33d50447902dd4388b770b7aeb","schema":"livegraph.boolean-propagation.v1","spec_hash":"68d91e49173fff212831aa63a89988169cdecba08e00c23a1dca81ebb1b65090","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-08dad124651d3923b179","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v08\n- v08 --supports--> v00\n- v00 --refutes--> v09\n- v09 --refutes--> v03\n- v03 --refutes--> v04\n- v04 --refutes--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v09 --refutes--> v02\n- v09 --refutes--> v07\n- v09 --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":"0243663229bf86c7bf8aeca0bdddfd43eeef8df35c61e0af4f646e5160f689b9","schema":"livegraph.boolean-propagation.v1","spec_hash":"08dad124651d3923b179b452fd0285676cca20625caba095300e2d9f5e3ff07a","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-73d4d95f2d2307b94a57","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --supports--> v04\n- v04 --refutes--> v05\n- v05 --refutes--> v01\n- v01 --refutes--> v07\n- v07 --supports--> v08\n- v08 --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v06\n- v05 --refutes--> v02\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":"3a723134d6799e94eb8039d420f09e3def353cd9fc12a9085dc30befde3b830b","schema":"livegraph.boolean-propagation.v1","spec_hash":"73d4d95f2d2307b94a5736e9199ba6f4185c455eecb47fedaea21b73805b69e6","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-ed5a73c633408a7c153b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v06\n- v06 --refutes--> v02\n- v02 --supports--> v01\n- v01 --supports--> v08\n- v08 --supports--> v04\n- v04 --supports--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v03\n- v01 --supports--> v07\n- v06 --supports--> v09\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":"598412a7ac2da6ea4babc0aa023b957260da4669426aff50c5fe2df1b90abdd7","schema":"livegraph.boolean-propagation.v1","spec_hash":"ed5a73c633408a7c153b402c20744ed9177bb6e6d02cc7a297903e8456c5a054","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-efd45fd45ae19dc41821","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --refutes--> v08\n- v08 --refutes--> v03\n- v03 --supports--> v01\n- v01 --supports--> v07\n- v07 --refutes--> v09\n- v09 --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v05\n- v03 --supports--> v04\n- v09 --refutes--> v06\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":"e2bc14ed701c91536c8b4d8e6fc3ce0f06e1cd7ec87b4a9ae73798a9eafff272","schema":"livegraph.boolean-propagation.v1","spec_hash":"efd45fd45ae19dc41821a77ce8bcf6c2a5d9528f535271019c3697bdebe5cd16","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-68e79c4b7786fc2876cf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v04\n- v04 --supports--> v05\n- v05 --supports--> v01\n- v01 --refutes--> v02\n- v02 --refutes--> v03\n- v03 --refutes--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v09\n- v01 --refutes--> v06\n- v04 --supports--> v07\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":"cea5d01356e4cbfa9a21f02dcc48dc33b1152c7822c862bbcd15e7dbc6d6aee0","schema":"livegraph.boolean-propagation.v1","spec_hash":"68e79c4b7786fc2876cf0146194830e32145c8714fa24cc1546ab86afd84bcbc","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"}],\"hop\":6,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-cd3951c0fe4b0edaf69d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --supports--> v07\n- v07 --supports--> v02\n- v02 --supports--> v05\n- v05 --supports--> v03\n- v03 --refutes--> v08\n- v08 --supports--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v00\n- v05 --refutes--> v09\n- v08 --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":"e4126f3b4d6efa5c810d42da1d992efe0913e6a8596c690f5a9a98d1ade8386d","schema":"livegraph.boolean-propagation.v1","spec_hash":"cd3951c0fe4b0edaf69dcfe0ac4505630f0e88e794fa15e32de268d4cccf488d","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-6ea29e3c4112634f41fc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --refutes--> v02\n- v02 --supports--> v06\n- v06 --refutes--> v09\n- v09 --refutes--> v07\n- v07 --supports--> v03\n- v03 --refutes--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v05\n- v06 --supports--> v01\n- v09 --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":"a642f22812e47975dff1fb8a687e68c5e4500d3164fbe6d7849565c2d31a5352","schema":"livegraph.boolean-propagation.v1","spec_hash":"6ea29e3c4112634f41fc874ecc91dcd6351f345cdb5a2d067fe423ad16b8f8bf","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-f4341dbfcf057b894b53","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --supports--> v07\n- v07 --refutes--> v04\n- v04 --supports--> v00\n- v00 --supports--> v01\n- v01 --supports--> v06\n- v06 --refutes--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v09\n- v06 --supports--> v03\n- v07 --refutes--> 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":"d3f05c833978226bc9efb1c142b9f86e781c0d6351ed8ce9565212f6a1023183","schema":"livegraph.boolean-propagation.v1","spec_hash":"f4341dbfcf057b894b53002053380775226a8d248717a02a509e35ada8c9cd53","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":6,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-812f404b8d959645dbf9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --supports--> v05\n- v05 --supports--> v03\n- v03 --supports--> v00\n- v00 --refutes--> v01\n- v01 --supports--> v02\n- v02 --supports--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v07\n- v03 --refutes--> v06\n- v03 --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":"96f1edfb34c75fbf35217ab92a34af6794704de305de2311f48855a9be845d13","schema":"livegraph.boolean-propagation.v1","spec_hash":"812f404b8d959645dbf9c1be009b9082eb4b393f4d6a8fd915944e291946893b","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-61453931a7f1d7069523","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --refutes--> v04\n- v04 --refutes--> v03\n- v03 --supports--> v05\n- v05 --refutes--> v08\n- v08 --supports--> v07\n- v07 --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v09\n- v07 --supports--> v02\n- v07 --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":"2ad3ac2f3861195b98742b834615baa8272cde2ce7fea6d460d298fe2b79df13","schema":"livegraph.boolean-propagation.v1","spec_hash":"61453931a7f1d7069523f4b96dbfb1cff986bd865b4399f97b6d2d361f158804","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-cb6f9e6fbf6ce3af9b0f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --supports--> v06\n- v06 --supports--> v07\n- v07 --refutes--> v02\n- v02 --refutes--> v03\n- v03 --refutes--> v01\n- v01 --supports--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v04\n- v03 --supports--> v08\n- v06 --supports--> v00\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":"58fd84cbdf4660a9a304a733f39892b5a322beef254e2016c71855d51d6e6199","schema":"livegraph.boolean-propagation.v1","spec_hash":"cb6f9e6fbf6ce3af9b0f261007165c0962823ac24f4ceeddbc6b6b04f2f236c1","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-f451e528f01657e697b0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --refutes--> v03\n- v03 --supports--> v02\n- v02 --refutes--> v01\n- v01 --supports--> v09\n- v09 --refutes--> v08\n- v08 --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v04\n- v02 --supports--> v06\n- v09 --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":"4b7b376ad3dc9724e3f735eb3443cc24b72f914fddaab2f06f18da1b662168a8","schema":"livegraph.boolean-propagation.v1","spec_hash":"f451e528f01657e697b05a38c29b2048d7b7812e076ec912100a1bf214beeead","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-9a0dda148e5ef39db9ac","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --supports--> v04\n- v04 --supports--> v07\n- v07 --supports--> v08\n- v08 --supports--> v05\n- v05 --supports--> v03\n- v03 --refutes--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v00\n- v07 --supports--> v09\n- v08 --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":"e2fb218fcb2505296a748d78915a19f718afd520eac14fcbe4dd5373f3737a86","schema":"livegraph.boolean-propagation.v1","spec_hash":"9a0dda148e5ef39db9ac08b583f0c325b13fc57d1537ca830c8c8d99c1a86a24","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-b1697224ea80fb91878a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --refutes--> v07\n- v07 --supports--> v08\n- v08 --supports--> v03\n- v03 --supports--> v06\n- v06 --supports--> v02\n- v02 --refutes--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v01\n- v06 --supports--> v05\n- v07 --supports--> v04\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":"2a5badfdb6d711bd3fde79350f110beed832235d8d719549652285d0570c7d04","schema":"livegraph.boolean-propagation.v1","spec_hash":"b1697224ea80fb91878add912e5d3f46a872cd968ebd520715c5bd71f8304dfe","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-e6019c43823a9f3a0e5a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --supports--> v09\n- v09 --supports--> v05\n- v05 --refutes--> v03\n- v03 --refutes--> v01\n- v01 --refutes--> v02\n- v02 --supports--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v00\n- v01 --refutes--> v08\n- v02 --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":"99cdf7faede008a20032e708a8049c0748d8354f26052b571313abfb5feac797","schema":"livegraph.boolean-propagation.v1","spec_hash":"e6019c43823a9f3a0e5af60f53d7042ffa27589a90ee792c73609e3c3e80d06b","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-cef0814bc3ea038ee397","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --supports--> v08\n- v08 --supports--> v03\n- v03 --refutes--> v07\n- v07 --refutes--> v06\n- v06 --refutes--> v01\n- v01 --supports--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v05\n- v06 --refutes--> v00\n- v07 --refutes--> 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":"5baae9b9733aabb24d5c1949c2a2682ae1cab5c3c53b2e3edda4edefd0b485ae","schema":"livegraph.boolean-propagation.v1","spec_hash":"cef0814bc3ea038ee3971af4200a2c5b22dd20d764582a4a244a34a2a6743bc2","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-49aaf387e0a255f5e718","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v03\n- v03 --refutes--> v07\n- v07 --supports--> v02\n- v02 --supports--> v01\n- v01 --refutes--> v09\n- v09 --supports--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v08\n- v03 --supports--> v05\n- v06 --supports--> v04\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":"bd26d7d2e8a90be8c5c6b22614e95d91ffb64cd2092f93cae27e822fd94c0ccb","schema":"livegraph.boolean-propagation.v1","spec_hash":"49aaf387e0a255f5e7181528802c7d1cb6f358cc6fda15805df2fe621dc18d87","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-9ec4c1081399bd243b4c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --refutes--> v08\n- v08 --refutes--> v03\n- v03 --supports--> v07\n- v07 --refutes--> v01\n- v01 --refutes--> v00\n- v00 --refutes--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v02\n- v03 --refutes--> v09\n- v08 --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":"77530bf585e15cc6906eb1a04a8c5ba59c88af75c5da727bb3c9ab3e7051b6ab","schema":"livegraph.boolean-propagation.v1","spec_hash":"9ec4c1081399bd243b4cdcb9f6e270ad4eb023abface57f77cdf4c030eabc733","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-bc6370e414ce6b9062ab","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --supports--> v03\n- v03 --refutes--> v00\n- v00 --supports--> v08\n- v08 --supports--> v05\n- v05 --supports--> v09\n- v09 --supports--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v08 --refutes--> v01\n- v08 --refutes--> v02\n- v09 --supports--> 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":"2b41f3cb19b0042207e168c1411d83a9f15019083848353d53e6fa1acace6cc8","schema":"livegraph.boolean-propagation.v1","spec_hash":"bc6370e414ce6b9062abd2136339af607139e39c8aab91f087163155da0086ed","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-46c73a6f8d758135daad","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --refutes--> v04\n- v04 --supports--> v03\n- v03 --refutes--> v01\n- v01 --refutes--> v06\n- v06 --supports--> v00\n- v00 --supports--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v02\n- v03 --supports--> v05\n- v04 --refutes--> v09\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":"37798efac6197026cd69659d60366b71b847e15bdffa534072080d773efaa0df","schema":"livegraph.boolean-propagation.v1","spec_hash":"46c73a6f8d758135daad62e126f99a2d3c08894eca0621170f91d39375c5d759","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"}],\"hop\":6,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-3a1d79cd467381c4ae43","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --supports--> v03\n- v03 --supports--> v08\n- v08 --refutes--> v00\n- v00 --supports--> v07\n- v07 --supports--> v01\n- v01 --refutes--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v02\n- v07 --supports--> v05\n- v08 --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":"ea4c8c1c8fd1b11e5ca1d9ba111de404ac855e350da40dcd7cfdd4fbb7a0fc98","schema":"livegraph.boolean-propagation.v1","spec_hash":"3a1d79cd467381c4ae43af5001bce26d53d1f05651849e9df7bd175ba739ec1b","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-4697ee7258127cb92d9c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --supports--> v03\n- v03 --refutes--> v06\n- v06 --refutes--> v00\n- v00 --supports--> v09\n- v09 --refutes--> v02\n- v02 --refutes--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v01\n- v03 --refutes--> v07\n- v08 --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":"2b1598156f9beea28c93c7f5a994b97062d4d8382a3c34c8f3203f320ab2a6d4","schema":"livegraph.boolean-propagation.v1","spec_hash":"4697ee7258127cb92d9c5f6ccb058eabb3127cd2e8f1a3d5287d2b3e58ad4227","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-6e9cec7d2091076f2a75","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v07\n- v07 --refutes--> v01\n- v01 --refutes--> v08\n- v08 --refutes--> v00\n- v00 --supports--> v05\n- v05 --refutes--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v09\n- v05 --supports--> v04\n- v07 --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":"d28922fec53274e8ffc88a71caee3f0371eb3612e60687db5cd6d171f33d8a41","schema":"livegraph.boolean-propagation.v1","spec_hash":"6e9cec7d2091076f2a7532020af14f60143831239068c72e4d00119fed4b5ea5","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-450dbc175f58dbeccb94","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --supports--> v07\n- v07 --supports--> v00\n- v00 --supports--> v04\n- v04 --supports--> v02\n- v02 --refutes--> v06\n- v06 --refutes--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v01\n- v04 --refutes--> v03\n- v04 --supports--> v08\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":"3f8243dc091c002c103f90337e489106bdcc1c28e9747661481f985ba7364c34","schema":"livegraph.boolean-propagation.v1","spec_hash":"450dbc175f58dbeccb94aea8d4579595fd3bc6b9a7da56d624cd93fe4e2f3768","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-0677290fdaeded74cd1b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --refutes--> v06\n- v06 --refutes--> v07\n- v07 --refutes--> v03\n- v03 --supports--> v08\n- v08 --refutes--> v01\n- v01 --refutes--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v07 --refutes--> v04\n- v07 --supports--> v05\n- v09 --supports--> v00\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":"4ab5600be513603edc3a8828d3481e481512afcc7b185d546391b8a8eb4c9ad7","schema":"livegraph.boolean-propagation.v1","spec_hash":"0677290fdaeded74cd1b68555ac92f1965ec6e683679ec2957a000be1ce92ec3","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-242320221c4803ce9f8d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --supports--> v09\n- v09 --supports--> v03\n- v03 --supports--> v00\n- v00 --supports--> v08\n- v08 --supports--> v02\n- v02 --supports--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v06\n- v02 --supports--> v01\n- v09 --refutes--> v05\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":"1bf5e57cd21677f56b2c3b95b23267b58c5abb5c756c0863f8a857b5dea4ebd4","schema":"livegraph.boolean-propagation.v1","spec_hash":"242320221c4803ce9f8d725b507b45e8f8976200b230047a80ad226b9f289f33","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-0faf55390bb6df91a564","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --supports--> v09\n- v09 --supports--> v06\n- v06 --supports--> v07\n- v07 --refutes--> v04\n- v04 --supports--> v01\n- v01 --supports--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v08\n- v09 --refutes--> v00\n- v09 --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":"c283c9e66c505895eb4b87f872c2092556472cdf84b1c3bf7373fc0f624b9e6c","schema":"livegraph.boolean-propagation.v1","spec_hash":"0faf55390bb6df91a5648dbe9cf7510759e8ccea70760f9a9219a0f7bfc4126b","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-42602ffa1cd6bad313ea","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --supports--> v07\n- v07 --supports--> v00\n- v00 --refutes--> v05\n- v05 --supports--> v08\n- v08 --supports--> v02\n- v02 --supports--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v06\n- v02 --supports--> v03\n- v08 --refutes--> v04\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":"413704f28ff6f642c37cdcfa9c6b6a825bff190758a044c930538782b15ce97a","schema":"livegraph.boolean-propagation.v1","spec_hash":"42602ffa1cd6bad313ea8b4c1d7385c6e3c3b727942f8265f71c7e54e6228cf2","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-ea695674bb5f5b892c48","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --refutes--> v00\n- v00 --supports--> v04\n- v04 --supports--> v07\n- v07 --supports--> v09\n- v09 --supports--> v08\n- v08 --supports--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v07 --supports--> v05\n- v08 --refutes--> v02\n- v09 --refutes--> v03\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":"d5b76dcbb273cbe35bb6765ceb5fa841a6025eda1435d45a8e783d4c30fdfadd","schema":"livegraph.boolean-propagation.v1","spec_hash":"ea695674bb5f5b892c48d2854f2d03d608cfc2e4e65c535548c4bf30f7d4a6dc","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-61d876d711d54334635e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --refutes--> v02\n- v02 --supports--> v04\n- v04 --supports--> v00\n- v00 --supports--> v05\n- v05 --supports--> v01\n- v01 --supports--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v07\n- v02 --supports--> v08\n- v04 --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":"2b0972c9754cbd56f01c418297e8e168534ed46f9165d200ef18e8480ef143cd","schema":"livegraph.boolean-propagation.v1","spec_hash":"61d876d711d54334635e0fb9723d6c3dc0083e2e042480f9ea73b9577b2254b4","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-051edbbb93eebd474cfc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --supports--> v05\n- v05 --supports--> v06\n- v06 --supports--> v03\n- v03 --supports--> v08\n- v08 --refutes--> v04\n- v04 --refutes--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --supports--> v07\n- v08 --refutes--> v00\n- v08 --refutes--> 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":"ba60ee135f3c86100f19ab7bdb423508618295ab6c8a183099d461234ac4077b","schema":"livegraph.boolean-propagation.v1","spec_hash":"051edbbb93eebd474cfcf5491f83688943cf77c9908c7b60d2447ee0024185cc","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-d55a718c443a681d27e0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --supports--> v00\n- v00 --supports--> v08\n- v08 --refutes--> v09\n- v09 --refutes--> v07\n- v07 --supports--> v04\n- v04 --supports--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v05\n- v04 --supports--> v02\n- v07 --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":"0e3d57d2b9ae8f90f3f4f6fa1447c2a7f2e1b34cce60e217e934dd4b3665a3ad","schema":"livegraph.boolean-propagation.v1","spec_hash":"d55a718c443a681d27e0e670fa005d5b149aa39f6bf2eb163a5b2683ecf09a9d","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-5052ea8b66845406ef10","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --refutes--> v06\n- v06 --refutes--> v03\n- v03 --refutes--> v01\n- v01 --refutes--> v07\n- v07 --supports--> v00\n- v00 --supports--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v08\n- v01 --refutes--> v02\n- v09 --refutes--> 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":"10ee5031dbfe22654354f456ca0e606a0d09b03bf2eda6f76f8fac0e5d22f62a","schema":"livegraph.boolean-propagation.v1","spec_hash":"5052ea8b66845406ef10e2570009f1688cac006a2d949c66da386692a702e933","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-3f1674381ddb70e56bfe","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --refutes--> v08\n- v08 --supports--> v02\n- v02 --refutes--> v00\n- v00 --refutes--> v04\n- v04 --supports--> v01\n- v01 --supports--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v07\n- v01 --refutes--> v03\n- v04 --refutes--> v05\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":"ae1ddac4682ce3016b3a747a3c4c525fc458ce105db0f6bc1872cbd2e9342075","schema":"livegraph.boolean-propagation.v1","spec_hash":"3f1674381ddb70e56bfe374c82e2f82c933952ab8cb6ad644f8c5b6ecbe9cbef","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-19a8a4054b0d52508495","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --supports--> v00\n- v00 --supports--> v04\n- v04 --refutes--> v01\n- v01 --supports--> v07\n- v07 --refutes--> v08\n- v08 --supports--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v02\n- v00 --refutes--> v03\n- v08 --refutes--> v05\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":"b93d26339c05b15c6bc5b59ceb9188a61abcabeb8963a02eccd831dc8c1e013b","schema":"livegraph.boolean-propagation.v1","spec_hash":"19a8a4054b0d52508495eb0f1ca7fa0495bc3fa333fc8bfe94059b8111c21ae2","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":6,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-beb0ded8dcbb2574e7bd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --refutes--> v04\n- v04 --refutes--> v09\n- v09 --refutes--> v07\n- v07 --refutes--> v05\n- v05 --supports--> v00\n- v00 --supports--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v06\n- v05 --refutes--> v03\n- v07 --refutes--> 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":"8aab4baeab0456be7d19e3ff867fcc43f136ea399428d14bfd483d52ca506c50","schema":"livegraph.boolean-propagation.v1","spec_hash":"beb0ded8dcbb2574e7bd02197eb5a1387367a0106e13560dbcb5eb7f11546856","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":6,"item_id":"lggrpo1-f9b075723cca56c32cd1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --supports--> v06\n- v06 --supports--> v07\n- v07 --refutes--> v03\n- v03 --supports--> v09\n- v09 --refutes--> v01\n- v01 --refutes--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v00\n- v05 --refutes--> v08\n- v07 --refutes--> 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":"5f66e7a90cbc075213690c0e1f2c0e0b4e550ae4ec1b68f56358f7aa655a2a4f","schema":"livegraph.boolean-propagation.v1","spec_hash":"f9b075723cca56c32cd19973a4fa7b1f70d100f209f1dac8750ccd74542f3cbf","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":6,"item_id":"lggrpo1-cd29da9d8ec456ad3320","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --supports--> v06\n- v06 --refutes--> v00\n- v00 --refutes--> v08\n- v08 --supports--> v09\n- v09 --supports--> v05\n- v05 --refutes--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v04\n- v06 --supports--> v02\n- v09 --refutes--> 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":"deba559d612fd3d6b13fd1120098fa089ad738d5b2ef8aa1777fe7f70a581d7c","schema":"livegraph.boolean-propagation.v1","spec_hash":"cd29da9d8ec456ad33201f579dd5921f752d2b4cf68cb39f06fb2d5f60ce4306","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":6,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-bf419708e701475f8e6e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --supports--> v04\n- v04 --refutes--> v08\n- v08 --refutes--> v01\n- v01 --supports--> v09\n- v09 --refutes--> v05\n- v05 --refutes--> v03\n- v03 --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v07\n- v06 --supports--> v02\n- v08 --refutes--> v0a\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":"0511cf4ebfe02ea22c1dbee3fe3f640c893d4a308fb02550d2a065e343aa175f","schema":"livegraph.boolean-propagation.v1","spec_hash":"bf419708e701475f8e6e9eff5e7f48d518df03a17d382a46b77217308be015d0","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-e531353d14f2bc3ff837","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --refutes--> v05\n- v05 --refutes--> v0a\n- v0a --refutes--> v07\n- v07 --refutes--> v08\n- v08 --supports--> v03\n- v03 --refutes--> v02\n- v02 --supports--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v06\n- v05 --supports--> v00\n- v08 --supports--> v04\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":"f8ddca82dda1c60e059fb34db145fca82952e8153cc3fb412b1cc0a13a5fd599","schema":"livegraph.boolean-propagation.v1","spec_hash":"e531353d14f2bc3ff8379863d48138c83519eece553a43405b7970080cbcba2a","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-ee9ccbbc78829041a877","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --supports--> v01\n- v01 --supports--> v00\n- v00 --supports--> v04\n- v04 --supports--> v06\n- v06 --supports--> v07\n- v07 --refutes--> v05\n- v05 --refutes--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v0a\n- v06 --refutes--> v02\n- v07 --supports--> v03\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":"57ae6144fc4252ae2c76b772ef770ff00ae5dd4a2828d1d061322e023ed961d1","schema":"livegraph.boolean-propagation.v1","spec_hash":"ee9ccbbc78829041a8778d8efcd15751ff1488ec9b5623f7a3c4ef2a3b8cdc48","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":7,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-66d62c8aa78bde39f62f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --refutes--> v05\n- v05 --refutes--> v01\n- v01 --refutes--> v04\n- v04 --supports--> v0a\n- v0a --supports--> v03\n- v03 --supports--> v02\n- v02 --supports--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v06\n- v09 --supports--> v08\n- v0a --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":"369c14d2d549761d679ae548b58eccea10013ea6a9aab24d12a13c74ac3958e3","schema":"livegraph.boolean-propagation.v1","spec_hash":"66d62c8aa78bde39f62f50472dc39be389a55a9d74e4d24d21602bdd7aba9d41","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-9e0942a23a4030ad6ac3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --supports--> v04\n- v04 --refutes--> v00\n- v00 --refutes--> v07\n- v07 --refutes--> v05\n- v05 --refutes--> v09\n- v09 --refutes--> v08\n- v08 --refutes--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v02\n- v05 --supports--> v03\n- v08 --refutes--> v0a\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":"940c025b39bdf17a8f2e94ff8e2832ebd583db37f8f2100229f1a00f5998ef6a","schema":"livegraph.boolean-propagation.v1","spec_hash":"9e0942a23a4030ad6ac36f04d2a3511dbd5784191f1ffcd520bf9dc95550fbe2","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-718755adf47bd17d7bb5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --refutes--> v08\n- v08 --refutes--> v0a\n- v0a --refutes--> v07\n- v07 --refutes--> v09\n- v09 --supports--> v06\n- v06 --supports--> v01\n- v01 --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v07 --refutes--> v05\n- v08 --supports--> v04\n- v09 --refutes--> v00\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":"c505ec6d534f55ee90607c200cd33691b55de9cbfbbc1264192144617fb3ad09","schema":"livegraph.boolean-propagation.v1","spec_hash":"718755adf47bd17d7bb5ec578a6adbb1dcfa799cf304b49c7e92510e712b1637","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-85e968ca519618dab99e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --supports--> v04\n- v04 --supports--> v09\n- v09 --refutes--> v02\n- v02 --supports--> v01\n- v01 --refutes--> v08\n- v08 --supports--> v05\n- v05 --supports--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v00\n- v04 --refutes--> v06\n- v04 --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":"fe0818b1ae508b94ac1af793caa0f1d9002e7a4995f45047823a0d6688f358fa","schema":"livegraph.boolean-propagation.v1","spec_hash":"85e968ca519618dab99eea3097bfa9201321ba41d55987a47e95868426cc923a","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-5132738f873e07d1ea13","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --supports--> v09\n- v09 --supports--> v03\n- v03 --refutes--> v06\n- v06 --refutes--> v0a\n- v0a --supports--> v08\n- v08 --refutes--> v07\n- v07 --refutes--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v08 --supports--> v01\n- v09 --supports--> v05\n- v0a --supports--> 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":"2767256f82442f751bf4d8add5743cddd9be95bae3ef9f9f469e6ee76d2956d4","schema":"livegraph.boolean-propagation.v1","spec_hash":"5132738f873e07d1ea137a6f2ed5d97b6ad12109018ceceeea8b6908dbb22797","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-7b3955914f36a6cd572a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --supports--> v05\n- v05 --refutes--> v09\n- v09 --supports--> v08\n- v08 --supports--> v03\n- v03 --supports--> v04\n- v04 --supports--> v06\n- v06 --refutes--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v00\n- v05 --supports--> v0a\n- v08 --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":"626637a6b7d85a5f361050b8d48fe9fb921c273c7a4129646812fc447ad245dc","schema":"livegraph.boolean-propagation.v1","spec_hash":"7b3955914f36a6cd572a3d53919dae8c76d55e3cdccd5178845d26bb095ab3dd","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-14a9d37107744084b83a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v00\n- v00 --supports--> v0a\n- v0a --supports--> v08\n- v08 --refutes--> v05\n- v05 --refutes--> v07\n- v07 --supports--> v09\n- v09 --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v02\n- v04 --supports--> v01\n- v09 --refutes--> v06\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":"bfee8910b554d59eb6e40fc1d295d3eacd506e74662d2e86ecc4f74110d61e39","schema":"livegraph.boolean-propagation.v1","spec_hash":"14a9d37107744084b83aab2d77fdaf9e8f7eba86f34e5a823997b1b462eb8970","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-b1621db33782c4604aaa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --supports--> v01\n- v01 --refutes--> v03\n- v03 --supports--> v08\n- v08 --supports--> v02\n- v02 --refutes--> v00\n- v00 --supports--> v09\n- v09 --supports--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v07\n- v03 --supports--> v06\n- v09 --supports--> v05\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":"799ffe6ecc1521cec84cba5f7eb943c5af7f0527af1911a9b57b23647ae2a158","schema":"livegraph.boolean-propagation.v1","spec_hash":"b1621db33782c4604aaa161a95a9b86776c25d89f4b0a1c6f125fca9df1274d5","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-f63a6a7dab58dc573795","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v01\n- v01 --supports--> v04\n- v04 --refutes--> v09\n- v09 --refutes--> v07\n- v07 --supports--> v03\n- v03 --supports--> v0a\n- v0a --refutes--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v00\n- v04 --refutes--> v02\n- v04 --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":"38c3d80d24e19f6e6febc8e0b9a383e5be7da402cb81e1b5f964e03eb1ee5a3f","schema":"livegraph.boolean-propagation.v1","spec_hash":"f63a6a7dab58dc573795d3faadc4d5187de200602aa385481e84b93dac2f32ca","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-b940909d5100748f5683","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --supports--> v04\n- v04 --supports--> v0a\n- v0a --refutes--> v06\n- v06 --supports--> v09\n- v09 --refutes--> v05\n- v05 --refutes--> v08\n- v08 --supports--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v02\n- v04 --supports--> v00\n- v08 --supports--> 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":"8c37dcaadb108068fc5876f67dfaee0300f2faf949f7e879413cba361e45bcc8","schema":"livegraph.boolean-propagation.v1","spec_hash":"b940909d5100748f568325965c2172d8730c49b52f93482346d60ab1fe85fa6b","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-6cd5369f88a743f153b8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v02\n- v02 --supports--> v09\n- v09 --supports--> v0a\n- v0a --refutes--> v05\n- v05 --supports--> v03\n- v03 --supports--> v01\n- v01 --supports--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v00\n- v0a --supports--> v07\n- v0a --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":"fbd3f6f11bb9044b1aad465caee6fe041bb0010eebac9c55be9fb0ae57fd9427","schema":"livegraph.boolean-propagation.v1","spec_hash":"6cd5369f88a743f153b89c779a76688a3e637215853ea5c5d1aff0d41a7f0c8c","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-5b2338982c049be8e22d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --supports--> v06\n- v06 --supports--> v01\n- v01 --refutes--> v00\n- v00 --refutes--> v09\n- v09 --refutes--> v0a\n- v0a --supports--> v08\n- v08 --supports--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v04\n- v02 --supports--> v03\n- v08 --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":"6ca9b4a7ac329cacba894b9a7cbc59d71c096afcbc0aa7f50dfb135a036be5b5","schema":"livegraph.boolean-propagation.v1","spec_hash":"5b2338982c049be8e22dd848ef1767e458634688f2a2a498c0760a21a2d1b5f1","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-b762d0011749a2d9034b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --supports--> v04\n- v04 --supports--> v00\n- v00 --supports--> v07\n- v07 --supports--> v08\n- v08 --refutes--> v0a\n- v0a --supports--> v01\n- v01 --refutes--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v05\n- v08 --refutes--> v03\n- v09 --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":"ac45b85293194c2f32dbb99bee916a6d8dff8c2c18b962d244945f3606740781","schema":"livegraph.boolean-propagation.v1","spec_hash":"b762d0011749a2d9034b2ca69ecc0458363db30b25e6f5c6328ba066d8492986","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-fb3e23ad45bf4a615166","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --refutes--> v0a\n- v0a --supports--> v09\n- v09 --supports--> v05\n- v05 --supports--> v04\n- v04 --supports--> v07\n- v07 --supports--> v03\n- v03 --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v06\n- v03 --supports--> v08\n- v0a --supports--> v01\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":"14aca39a5a9c80ab521578fd619e78be0971a7d31d00bc475c97d86546a23528","schema":"livegraph.boolean-propagation.v1","spec_hash":"fb3e23ad45bf4a6151664814557051505f3f7eeb9afc9388c88694010b1eed2f","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-33dcdd7c708e1fe8c1b5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --supports--> v03\n- v03 --supports--> v00\n- v00 --supports--> v06\n- v06 --supports--> v08\n- v08 --supports--> v0a\n- v0a --supports--> v01\n- v01 --supports--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v02\n- v01 --supports--> v09\n- v03 --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":"d4c1644e196d2ccbbec339c8eba2563649261debdd091e42b53ba93a52c776a5","schema":"livegraph.boolean-propagation.v1","spec_hash":"33dcdd7c708e1fe8c1b56553c0671aa0e1b89c3f42bdef9ed62e2b43abc2e516","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-b2abfaf5026ad321447b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --refutes--> v0a\n- v0a --supports--> v00\n- v00 --refutes--> v09\n- v09 --supports--> v07\n- v07 --refutes--> v04\n- v04 --supports--> v06\n- v06 --supports--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v01\n- v00 --supports--> v08\n- v04 --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":"7f9436e0ab04242ae0b8a8c55b8e1421b16d9ee837958b5143301a49855c08f0","schema":"livegraph.boolean-propagation.v1","spec_hash":"b2abfaf5026ad321447b8b2f16dd297d44845884d6e3649c99bd9006318a1a3c","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-10b9907e490907467fac","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --refutes--> v06\n- v06 --refutes--> v03\n- v03 --refutes--> v05\n- v05 --supports--> v00\n- v00 --refutes--> v08\n- v08 --refutes--> v04\n- v04 --refutes--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v01\n- v06 --refutes--> v0a\n- v08 --refutes--> 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":"e695d4b299209786469f86a17980a9c039fa7429c5cae4e48cffa37e9d5dce60","schema":"livegraph.boolean-propagation.v1","spec_hash":"10b9907e490907467fac607c525750f90ea3cce2db88f881e21735fdc1516de9","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-e581ee878b729eb7e8e7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v04\n- v04 --supports--> v02\n- v02 --refutes--> v08\n- v08 --refutes--> v06\n- v06 --supports--> v05\n- v05 --refutes--> v09\n- v09 --refutes--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v03\n- v06 --refutes--> v07\n- v06 --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":"b86b0b958774978a01950bfce22152f8e1d0ef4db2bfaecd90943fb85fc3170f","schema":"livegraph.boolean-propagation.v1","spec_hash":"e581ee878b729eb7e8e79f67d4b9eb08f962de1649f0477c3c58c9743568e010","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-e6db0983743bf9bd0767","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v01\n- v01 --refutes--> v03\n- v03 --supports--> v0a\n- v0a --supports--> v07\n- v07 --refutes--> v02\n- v02 --supports--> v00\n- v00 --refutes--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v09\n- v01 --supports--> v04\n- v02 --supports--> 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":"23bc2854e13943c9136febab5873a641126da9a388b3c02371d3fadbcf6c9681","schema":"livegraph.boolean-propagation.v1","spec_hash":"e6db0983743bf9bd076754c97e6c40320ab493042ee4700320e652d2418849db","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-772ab449f222724bdce0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --refutes--> v07\n- v07 --refutes--> v06\n- v06 --supports--> v08\n- v08 --supports--> v02\n- v02 --refutes--> v0a\n- v0a --supports--> v03\n- v03 --refutes--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v04\n- v06 --supports--> v09\n- v07 --refutes--> 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":"ca7ef327f5c87e73e8976da41f16ecb873322b47c4fa51b9cd4075be7faee6bd","schema":"livegraph.boolean-propagation.v1","spec_hash":"772ab449f222724bdce03271cf0257a79fc31d88ea27b7479b5217216bc5d73d","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-0e7e800e90dd6d4a2261","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --refutes--> v06\n- v06 --refutes--> v08\n- v08 --supports--> v05\n- v05 --supports--> v04\n- v04 --supports--> v07\n- v07 --supports--> v09\n- v09 --supports--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v08 --supports--> v02\n- v09 --supports--> v03\n- v09 --supports--> v0a\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":"e3c4861aba89117508569fbb9e59b498a4252fd58d6bab03d1e2b717603c843b","schema":"livegraph.boolean-propagation.v1","spec_hash":"0e7e800e90dd6d4a22617063dfc9efc1a6c05207be5db00633ef97a19158dea6","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-37ebc1969ace834910f6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --refutes--> v07\n- v07 --supports--> v06\n- v06 --supports--> v00\n- v00 --supports--> v09\n- v09 --supports--> v08\n- v08 --refutes--> v01\n- v01 --refutes--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v05\n- v01 --refutes--> v03\n- v01 --supports--> v0a\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":"97ce3b694aeef517a104b8f43e83d2822da167108b81fa77fa45c1e582217072","schema":"livegraph.boolean-propagation.v1","spec_hash":"37ebc1969ace834910f69d538f313be013f223ea4a09d4cc8d35ca3333a3c04c","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-0e5a2f50f7455b51a9a5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --supports--> v04\n- v04 --refutes--> v0a\n- v0a --supports--> v01\n- v01 --supports--> v07\n- v07 --refutes--> v05\n- v05 --supports--> v03\n- v03 --refutes--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v00\n- v05 --refutes--> v08\n- v05 --supports--> v09\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":"8a6fc8ac78b330b7b508613499e9f74500c0f90facd9d3bb3331c375bb100593","schema":"livegraph.boolean-propagation.v1","spec_hash":"0e5a2f50f7455b51a9a55ccdf1c796650bdae51d80ee60d51a52d70340e35ced","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-aa87e20011403b01f776","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --supports--> v05\n- v05 --refutes--> v00\n- v00 --refutes--> v04\n- v04 --supports--> v03\n- v03 --refutes--> v0a\n- v0a --refutes--> v02\n- v02 --supports--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v07\n- v03 --refutes--> v08\n- v04 --refutes--> v09\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":"b357440b05627215efa0a79315803b5df0bee5b8b570daba70cfa8d6e8dc5590","schema":"livegraph.boolean-propagation.v1","spec_hash":"aa87e20011403b01f77644bca76f07bf3fa5aabd1428f64c480656f699f741eb","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-dbee2d5472b0e4665860","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --refutes--> v00\n- v00 --refutes--> v06\n- v06 --supports--> v09\n- v09 --refutes--> v05\n- v05 --supports--> v01\n- v01 --refutes--> v03\n- v03 --refutes--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v02\n- v03 --supports--> v04\n- v03 --supports--> v08\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":"c2f730c63d4e66c34930533888e5c30ea93fa9346968c2a6f81f7a47da7d3a11","schema":"livegraph.boolean-propagation.v1","spec_hash":"dbee2d5472b0e46658603c2d3de792e59d3bf248db6c5be3dfd9df68772c9d5e","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-caefa8e8272dbce93e48","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --refutes--> v08\n- v08 --supports--> v01\n- v01 --supports--> v07\n- v07 --supports--> v0a\n- v0a --supports--> v05\n- v05 --supports--> v00\n- v00 --supports--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v06\n- v09 --supports--> v03\n- v0a --refutes--> v02\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":"ea5298251bf33ed4bc273c829dcca3e4d05bcf4ae9dfd88bf817765caa140f75","schema":"livegraph.boolean-propagation.v1","spec_hash":"caefa8e8272dbce93e48c3b0c410a60148cc4bad4fb33d7bb725c9fbd3d5b0b1","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-9b81a25bafbe89a78572","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --supports--> v04\n- v04 --refutes--> v09\n- v09 --refutes--> v01\n- v01 --supports--> v00\n- v00 --supports--> v02\n- v02 --supports--> v06\n- v06 --supports--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v07\n- v06 --supports--> v0a\n- v09 --supports--> v08\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":"f77202b24a6cdfb4395aa772ab50b5bf601db7f28a9142633f535152dce35994","schema":"livegraph.boolean-propagation.v1","spec_hash":"9b81a25bafbe89a78572dc9a2e277551c42c0f012d194537b9c3ccc94ab2915d","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-c1f1f8a384d771f648d7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --supports--> v00\n- v00 --refutes--> v07\n- v07 --refutes--> v01\n- v01 --refutes--> v09\n- v09 --refutes--> v06\n- v06 --refutes--> v03\n- v03 --refutes--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v04\n- v01 --refutes--> v05\n- v02 --refutes--> v0a\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":"f9c40523ab1b3c693fec437146d17b9e9b8e91477efef4ac3de3c383ea622efa","schema":"livegraph.boolean-propagation.v1","spec_hash":"c1f1f8a384d771f648d79e546d6ef5d05d24e19f043112d60485b2e7c668a031","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-2e49afbdfcf123929e9c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --supports--> v07\n- v07 --supports--> v01\n- v01 --supports--> v04\n- v04 --supports--> v06\n- v06 --refutes--> v08\n- v08 --supports--> v03\n- v03 --supports--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v07 --refutes--> v00\n- v09 --refutes--> v02\n- v09 --supports--> v05\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":"0a8a110e302314823efba68359e1c3bda4baa62bd0723201c375bbf3dbdb2c04","schema":"livegraph.boolean-propagation.v1","spec_hash":"2e49afbdfcf123929e9c5af54d0b512fa777383affb210fbd835a2ee10934fbf","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-1bfe5423a08c6e646250","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --supports--> v08\n- v08 --refutes--> v07\n- v07 --refutes--> v00\n- v00 --supports--> v03\n- v03 --refutes--> v0a\n- v0a --refutes--> v09\n- v09 --refutes--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v01\n- v02 --refutes--> v04\n- v02 --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":"46c1a989645573122e0696735ebe72bec7eb48411dbf9a7d7e77b15769828259","schema":"livegraph.boolean-propagation.v1","spec_hash":"1bfe5423a08c6e646250adc88451d82e7522548586870a8df53debf72080a6fc","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-f4b1c5fd078bcdb7072f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --refutes--> v08\n- v08 --supports--> v04\n- v04 --refutes--> v06\n- v06 --supports--> v01\n- v01 --supports--> v07\n- v07 --supports--> v09\n- v09 --refutes--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v03\n- v08 --supports--> v00\n- v08 --supports--> v05\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":"83caa91b17c623074babbd3c80d4312ea00e8b25ecf267f2c9748a0df28e8955","schema":"livegraph.boolean-propagation.v1","spec_hash":"f4b1c5fd078bcdb7072f005a8102111cd97505b8bb21722866ed6cabe2c7831e","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-7712b14ffa974aadf3f0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --supports--> v08\n- v08 --refutes--> v02\n- v02 --supports--> v01\n- v01 --refutes--> v00\n- v00 --supports--> v06\n- v06 --refutes--> v09\n- v09 --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v07\n- v00 --supports--> v05\n- v08 --refutes--> v04\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":"955cf66fd93c244034567ac9853a93ce144535a2e80cc68bf75fb6535238ca9c","schema":"livegraph.boolean-propagation.v1","spec_hash":"7712b14ffa974aadf3f0da4d57a567bdf75723331eace5c89358e531991f6c64","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-2f4d310ee3aeb1e98c7b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --refutes--> v00\n- v00 --supports--> v03\n- v03 --supports--> v09\n- v09 --supports--> v08\n- v08 --supports--> v06\n- v06 --refutes--> v04\n- v04 --supports--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v0a\n- v08 --supports--> v01\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":"ea78aa813424ed4fd8d40a672a98c9fe8b6022b7d53869e9c8fcca2d2f1f9167","schema":"livegraph.boolean-propagation.v1","spec_hash":"2f4d310ee3aeb1e98c7b3395cf228f32a746e4d3f4da0f240095628a76aca21c","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-244647ec6fb4eaeab359","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --refutes--> v03\n- v03 --refutes--> v08\n- v08 --supports--> v0a\n- v0a --refutes--> v02\n- v02 --supports--> v09\n- v09 --refutes--> v04\n- v04 --refutes--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v00\n- v07 --refutes--> v01\n- v08 --refutes--> 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":"a305d027048513642c08289e407b62e914cbbe0e31c023ecf67f2298653ec250","schema":"livegraph.boolean-propagation.v1","spec_hash":"244647ec6fb4eaeab3598fae14685204b5b47e8c7749faf3d0ab2f0b669dd31f","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-d652ebe2d90c8f7184d7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v0a\n- v0a --supports--> v08\n- v08 --supports--> v04\n- v04 --refutes--> v03\n- v03 --refutes--> v09\n- v09 --supports--> v06\n- v06 --supports--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v07\n- v04 --supports--> v01\n- v06 --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":"3cd3fe651913f5bece025c70027c7299c4d4a956dec543af1aabc61aee8c1e67","schema":"livegraph.boolean-propagation.v1","spec_hash":"d652ebe2d90c8f7184d7466a447ec1828f03fce0eaa1e8f6d12711a1fc106484","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-04f7f5cc899affd6f958","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --refutes--> v03\n- v03 --supports--> v08\n- v08 --supports--> v06\n- v06 --supports--> v02\n- v02 --supports--> v07\n- v07 --refutes--> v00\n- v00 --supports--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v09\n- v07 --refutes--> v01\n- v07 --supports--> 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":"6c60b50d67ce8b6b9a436ef87611543f323903c5b4e5963afc5af1b203ae1414","schema":"livegraph.boolean-propagation.v1","spec_hash":"04f7f5cc899affd6f95812007ecb4261036bc038a65052880d88843bcd1caf5e","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"}],\"hop\":7,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-122eb91fb4874543fa61","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --supports--> v01\n- v01 --supports--> v02\n- v02 --refutes--> v03\n- v03 --refutes--> v04\n- v04 --supports--> v05\n- v05 --refutes--> v07\n- v07 --refutes--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v0a\n- v07 --refutes--> v08\n- v09 --refutes--> v00\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":"004593570c7004cac0d378015629b069ac3d886310d516f67ac844ffc0949ad2","schema":"livegraph.boolean-propagation.v1","spec_hash":"122eb91fb4874543fa61f338663ff71c59eac9cdf9b41c7676f7b2da32945580","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-1c6bdae0e68d6286c62c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v00\n- v00 --refutes--> v0a\n- v0a --refutes--> v05\n- v05 --supports--> v06\n- v06 --refutes--> v01\n- v01 --refutes--> v08\n- v08 --supports--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v07\n- v05 --supports--> v09\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":"4cef8b834037e69af2c75889eaae47bc6075f88d9c27f1add84cf8ba96127253","schema":"livegraph.boolean-propagation.v1","spec_hash":"1c6bdae0e68d6286c62ca22fe0397d9e56d50f9b96d0a65f2c7b72fc18a88633","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-7bd04a0c44d919447b79","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --supports--> v01\n- v01 --supports--> v05\n- v05 --supports--> v09\n- v09 --supports--> v07\n- v07 --supports--> v0a\n- v0a --supports--> v04\n- v04 --refutes--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v06\n- v09 --supports--> v00\n- v0a --refutes--> v08\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":"716d8ebacf23ba3d8c34d8b13f797e571e5c29941b3c3a061a067c0c1e355503","schema":"livegraph.boolean-propagation.v1","spec_hash":"7bd04a0c44d919447b7916c2f4aab795149d1ffc79743dab3f5a5bb5aecf63c7","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-655bb07f905a67c4ca9c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --refutes--> v02\n- v02 --refutes--> v06\n- v06 --refutes--> v01\n- v01 --supports--> v05\n- v05 --refutes--> v07\n- v07 --refutes--> v09\n- v09 --supports--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0a\n- v02 --refutes--> v04\n- v09 --supports--> v03\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":"dbfc983f7e99b73a8baa973fb6520fa5eb795e00b7a6e2538957a6fc820b271a","schema":"livegraph.boolean-propagation.v1","spec_hash":"655bb07f905a67c4ca9ca4245c7fe1c0a6f7cdd520dfe79f15f6623062139d01","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"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\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-d36c6ff0d8d1a051458e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --refutes--> v04\n- v04 --supports--> v06\n- v06 --refutes--> v08\n- v08 --refutes--> v00\n- v00 --refutes--> v03\n- v03 --supports--> v09\n- v09 --refutes--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v02\n- v04 --supports--> v05\n- v06 --refutes--> v0a\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":"802bca4027b1e5c786aec9d8d62cc255e5187143c9fef7752dcf2e8b570606fd","schema":"livegraph.boolean-propagation.v1","spec_hash":"d36c6ff0d8d1a051458e75db0210c5c793b350ac6cff1a9d15265f3b1d1f3bec","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-6a0f4883ab8ad3b0bb5e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --supports--> v02\n- v02 --refutes--> v04\n- v04 --refutes--> v08\n- v08 --refutes--> v09\n- v09 --refutes--> v03\n- v03 --refutes--> v00\n- v00 --supports--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v06\n- v03 --refutes--> v05\n- v03 --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":"c7163da50ccccc4c370dc2c9b454ad01bd5a247c840d360750b604005048b349","schema":"livegraph.boolean-propagation.v1","spec_hash":"6a0f4883ab8ad3b0bb5eb7c9014ff2e609fb8900ad7fe6414461370b0fcbd82f","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-74cac16c8b672efcd21e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --supports--> v01\n- v01 --refutes--> v0a\n- v0a --refutes--> v07\n- v07 --supports--> v03\n- v03 --supports--> v06\n- v06 --supports--> v05\n- v05 --refutes--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v04\n- v05 --refutes--> v00\n- v07 --refutes--> v02\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":"005d97356e273a155279afec3f3450734ea0d69d7489a33ee41cb0f593992fd2","schema":"livegraph.boolean-propagation.v1","spec_hash":"74cac16c8b672efcd21e9cc4a4f014200f664a9c45edeeda65760ea40f95b15a","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-63913859d2f3186c5c1f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --refutes--> v01\n- v01 --refutes--> v03\n- v03 --supports--> v06\n- v06 --supports--> v07\n- v07 --supports--> v05\n- v05 --refutes--> v0a\n- v0a --supports--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v09\n- v07 --supports--> v00\n- v08 --refutes--> v04\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":"b6e35e22a9697de90508e97e275bc996719645c49fd90d9919ed53df425d7925","schema":"livegraph.boolean-propagation.v1","spec_hash":"63913859d2f3186c5c1ff224cd99022f9d8c82d9f6ed2023eb7e52322baf1f96","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-f89ebb73acc6667a786f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --supports--> v05\n- v05 --supports--> v06\n- v06 --supports--> v09\n- v09 --supports--> v01\n- v01 --refutes--> v00\n- v00 --supports--> v04\n- v04 --supports--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v02\n- v03 --refutes--> v08\n- v06 --supports--> v0a\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":"c50add2f23c334b8719ebbe34908694a6b2e045c0d2e047d1d6e2f0c5e5fae1d","schema":"livegraph.boolean-propagation.v1","spec_hash":"f89ebb73acc6667a786f06ac58fc1e027380326a1c5195fc24aba4619872a89c","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-90ed8e7f0459b5cd67e2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v00\n- v00 --supports--> v01\n- v01 --refutes--> v07\n- v07 --supports--> v06\n- v06 --supports--> v05\n- v05 --supports--> v03\n- v03 --refutes--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v09\n- v03 --supports--> v02\n- v05 --refutes--> v0a\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":"778682df27a23ddc7959895e21f8f92bcbcd4e2a2ce6f849349f8e6b393327bc","schema":"livegraph.boolean-propagation.v1","spec_hash":"90ed8e7f0459b5cd67e21573b9151b27054b091e3e3fc162561a8198ab030f94","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-f507701c4e82a8f1a160","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --refutes--> v03\n- v03 --refutes--> v09\n- v09 --supports--> v08\n- v08 --supports--> v02\n- v02 --refutes--> v05\n- v05 --supports--> v0a\n- v0a --refutes--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v08 --supports--> v06\n- v09 --supports--> v00\n- v09 --supports--> 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":"24970269f1d89ddad18ce1a5355f6f631dd6bc5c2b2cfc794f4a302ce30078bc","schema":"livegraph.boolean-propagation.v1","spec_hash":"f507701c4e82a8f1a16019251eb473811fab73a14ea73321e7e4957abadc8c11","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-ee8b92ef4319988236ad","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --supports--> v09\n- v09 --supports--> v02\n- v02 --refutes--> v06\n- v06 --supports--> v07\n- v07 --supports--> v04\n- v04 --supports--> v03\n- v03 --refutes--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v00\n- v06 --refutes--> v08\n- v07 --supports--> v0a\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":"6489d5a6a0e928141c99ec4c9ec8209c628d9c345a6397f01bc097a0eca5fae6","schema":"livegraph.boolean-propagation.v1","spec_hash":"ee8b92ef4319988236ad65e00ade7b0731d1315545694333835573d0c3eb7cc3","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-4f9fad082d0df46127d6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v01\n- v01 --supports--> v08\n- v08 --refutes--> v03\n- v03 --refutes--> v09\n- v09 --supports--> v07\n- v07 --supports--> v05\n- v05 --refutes--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v02\n- v06 --supports--> v04\n- v07 --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":"e420bb21b490108c793cddfffb82921e1ed86ce0647dccea1084c699bfd770a6","schema":"livegraph.boolean-propagation.v1","spec_hash":"4f9fad082d0df46127d6f29e8d4ea49e74c42aaa93e7f93b29c15a991c7a8d3c","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-65619d80fcc6f9747c38","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --supports--> v02\n- v02 --refutes--> v06\n- v06 --supports--> v04\n- v04 --supports--> v01\n- v01 --refutes--> v0a\n- v0a --refutes--> v00\n- v00 --supports--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v09\n- v07 --supports--> v08\n- v0a --refutes--> 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":"992e1baee580168f902436f6d8e5e3cc01ceb3a1905b39322c796616a1a0e9b8","schema":"livegraph.boolean-propagation.v1","spec_hash":"65619d80fcc6f9747c3885a4f2002110c6ce592d257da3a3733faa4da27514d7","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-50f9cb978f350403f78c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --refutes--> v05\n- v05 --refutes--> v00\n- v00 --supports--> v04\n- v04 --refutes--> v06\n- v06 --refutes--> v07\n- v07 --refutes--> v02\n- v02 --supports--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v03\n- v09 --supports--> v01\n- v09 --supports--> v0a\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":"6d238db4e7a9f505efe661127dfd9bf9208c3e2633ed858459f10f145d3e7b54","schema":"livegraph.boolean-propagation.v1","spec_hash":"50f9cb978f350403f78c03033fd0430af540b5629d5a907dfbb4cede74b7d1eb","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-f300d21d13e434095852","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v05\n- v05 --refutes--> v0a\n- v0a --supports--> v00\n- v00 --supports--> v02\n- v02 --refutes--> v07\n- v07 --supports--> v03\n- v03 --refutes--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v01\n- v04 --supports--> v09\n- v07 --refutes--> 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":"e639e311ee79b8d3b4dd3bfb363276965e32f659e49609d2217d41f2ca69e01d","schema":"livegraph.boolean-propagation.v1","spec_hash":"f300d21d13e434095852eb6ed6c52e66131f89079e2fd5110efcc56695ec480f","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-882094eedd7906910fd2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --refutes--> v06\n- v06 --refutes--> v05\n- v05 --supports--> v09\n- v09 --supports--> v0a\n- v0a --supports--> v00\n- v00 --refutes--> v03\n- v03 --supports--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v02\n- v06 --refutes--> v07\n- v0a --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":"93768dabcc4a05d58d731f7afd4c346f6581a36824bce10336005481a38fc5c5","schema":"livegraph.boolean-propagation.v1","spec_hash":"882094eedd7906910fd2f4c6a6342ea221a3972a3f55816a5886a32a7cad4f89","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-9408b99a37ee5eb1b22f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --refutes--> v05\n- v05 --supports--> v0a\n- v0a --supports--> v06\n- v06 --refutes--> v02\n- v02 --supports--> v08\n- v08 --supports--> v09\n- v09 --supports--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --supports--> v07\n- v0a --refutes--> v04\n- v0a --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":"79e0cb1f0bb141c64617f8f9814f39ea72a5d74536dd2583e27996b37c2f0a67","schema":"livegraph.boolean-propagation.v1","spec_hash":"9408b99a37ee5eb1b22f7d3e34d6f4c9041182f7e0e0ac28d0330f9ab221348b","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"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\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-77745079e4747db2fc93","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --refutes--> v09\n- v09 --refutes--> v00\n- v00 --refutes--> v0a\n- v0a --refutes--> v01\n- v01 --refutes--> v07\n- v07 --refutes--> v04\n- v04 --refutes--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v08\n- v04 --refutes--> v02\n- v09 --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":"4b77bfeb9dc64b1f40b3bc0a622a33d8876db5414dfa872a345898812d5598d9","schema":"livegraph.boolean-propagation.v1","spec_hash":"77745079e4747db2fc9351cba775a3703a649615207b14949449ca0ed9c207b4","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-a2b4138d200225b66b5b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --refutes--> v08\n- v08 --supports--> v07\n- v07 --refutes--> v00\n- v00 --supports--> v09\n- v09 --supports--> v04\n- v04 --refutes--> v02\n- v02 --supports--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v0a\n- v03 --refutes--> v05\n- v07 --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":"e64ccb942f478cf6715103c4f29095b3b2d915c76683820fc95b24b2638a0546","schema":"livegraph.boolean-propagation.v1","spec_hash":"a2b4138d200225b66b5b3a98967a60faa696e95a2249308d6eea1c0bb8fc1284","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-7318d16852abef06efcd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --supports--> v05\n- v05 --refutes--> v07\n- v07 --supports--> v02\n- v02 --supports--> v03\n- v03 --refutes--> v06\n- v06 --refutes--> v09\n- v09 --supports--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v04\n- v08 --refutes--> v01\n- v09 --refutes--> v00\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":"dd5ec1a821243019bb7411974b33d316d38b687a51dc6e995f57a1075c3a698b","schema":"livegraph.boolean-propagation.v1","spec_hash":"7318d16852abef06efcd19670ee579ae801a0335f38bf5ac85516066296eb266","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-3bf4298d7b8d5c01e5df","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --refutes--> v05\n- v05 --supports--> v03\n- v03 --refutes--> v00\n- v00 --supports--> v09\n- v09 --refutes--> v04\n- v04 --refutes--> v0a\n- v0a --supports--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v02\n- v05 --supports--> v06\n- v05 --supports--> 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":"502abb1adcf526cbb9fc7f6a8fcc351644774764312354bfbe26a4ebf935eae3","schema":"livegraph.boolean-propagation.v1","spec_hash":"3bf4298d7b8d5c01e5dfe9a72b6ca18d0ceec9033ff59b9c09e3609713c3a449","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-e955087bacb4afa76a96","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v08\n- v08 --supports--> v09\n- v09 --supports--> v06\n- v06 --supports--> v00\n- v00 --refutes--> v0a\n- v0a --supports--> v05\n- v05 --supports--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v07\n- v04 --refutes--> v03\n- v06 --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":"89029228435fd1389b9065b4ae0e0cb0f0bcd0eadfaee946fb4ddd6cc8df539d","schema":"livegraph.boolean-propagation.v1","spec_hash":"e955087bacb4afa76a96799043748d1a4e6000184e29f1382071d4341eeac7cf","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":7,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":7,"item_id":"lggrpo1-f70014e3e452b4e23e4f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --refutes--> v06\n- v06 --supports--> v09\n- v09 --refutes--> v04\n- v04 --refutes--> v07\n- v07 --refutes--> v03\n- v03 --refutes--> v02\n- v02 --supports--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v00\n- v07 --supports--> v05\n- v07 --supports--> 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":"ab19e73dabf66641a1fafece15000d21d5f549ab84b07273f3c928fcb6732ad4","schema":"livegraph.boolean-propagation.v1","spec_hash":"f70014e3e452b4e23e4f8d82a473c58d57a82f1cde8d46c5939578608efe81a0","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":7,"item_id":"lggrpo1-20e1f67496272a4ba8c5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --supports--> v03\n- v03 --supports--> v09\n- v09 --supports--> v06\n- v06 --supports--> v02\n- v02 --refutes--> v07\n- v07 --refutes--> v05\n- v05 --supports--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --supports--> v00\n- v06 --supports--> v0a\n- v07 --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":"e3ff3279b77ae3b1f96a4fac79b37209e2ce04c2225b4b14eee6f4d7dcb8fd92","schema":"livegraph.boolean-propagation.v1","spec_hash":"20e1f67496272a4ba8c5936476e83064efe67c3b4fc8e49e67a28e36a9afced0","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"}],\"hop\":7,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-b23edf488bc7e0365916","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v07\n- v07 --refutes--> v0a\n- v0a --refutes--> v05\n- v05 --refutes--> v06\n- v06 --supports--> v0c\n- v0c --supports--> v02\n- v02 --refutes--> v09\n- v09 --refutes--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v0b\n- v07 --refutes--> v03\n- v0a --refutes--> v01\n- v0a --refutes--> v08\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":"c97bba05632bf3e4b87b86a8bfd15c674c121a0557b56dd6f4eaf5565829edbb","schema":"livegraph.boolean-propagation.v1","spec_hash":"b23edf488bc7e0365916aa0bdf758891b8eec04bf2d93d45aff68cd514a48b7f","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-893dde37dfddac65ce67","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --supports--> v09\n- v09 --refutes--> v06\n- v06 --supports--> v05\n- v05 --refutes--> v07\n- v07 --refutes--> v08\n- v08 --refutes--> v0a\n- v0a --supports--> v0b\n- v0b --refutes--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v00\n- v05 --supports--> v04\n- v0a --supports--> v03\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":"03f4a529cab1b78e4375a9e1c4b96a9ac978ffcd564bb2266ad6b81023924e25","schema":"livegraph.boolean-propagation.v1","spec_hash":"893dde37dfddac65ce67a308b14d90838557281abca43be3a86e283071335221","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-d2ad3377cf8539749dd6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --refutes--> v06\n- v06 --refutes--> v05\n- v05 --refutes--> v08\n- v08 --refutes--> v04\n- v04 --supports--> v07\n- v07 --supports--> v0b\n- v0b --refutes--> v0a\n- v0a --supports--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v00\n- v05 --refutes--> v03\n- v07 --refutes--> v01\n- v08 --supports--> v0c\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":"c13b1d8ffecf7e30a2dc9e7dfa8b6c08d8918018e95ecaa33fcfc0dfd0ad37d4","schema":"livegraph.boolean-propagation.v1","spec_hash":"d2ad3377cf8539749dd61fe0a2af2f0ad4adcc5f3c31cce52b555b295fb33601","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-57f3e269fe67aa1c2d40","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --supports--> v04\n- v04 --supports--> v06\n- v06 --refutes--> v08\n- v08 --supports--> v02\n- v02 --refutes--> v03\n- v03 --supports--> v09\n- v09 --supports--> v0c\n- v0c --refutes--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v0b\n- v0a --refutes--> v00\n- v0c --refutes--> v05\n- v0c --supports--> v01\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":"893b75e2cb2e80e81ce106215e104893ae9173bdc499c4d0fb7ecc3ea97848c7","schema":"livegraph.boolean-propagation.v1","spec_hash":"57f3e269fe67aa1c2d401a0f7b9c9bccae896835a70f5ef15434474e4a93b76a","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-ceee71168184e4e0bfa1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --supports--> v00\n- v00 --refutes--> v03\n- v03 --refutes--> v0b\n- v0b --refutes--> v08\n- v08 --supports--> v0c\n- v0c --supports--> v0a\n- v0a --supports--> v04\n- v04 --refutes--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v01\n- v02 --supports--> v07\n- v04 --supports--> v05\n- v0b --supports--> v06\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":"aeac9cd922e476c6376ec44b59a168c73a7b8d67a5fa814b03195730fddbbcc9","schema":"livegraph.boolean-propagation.v1","spec_hash":"ceee71168184e4e0bfa150738446259add927ad8cdd7db3e24a9cd7c4d97752f","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-d4ddf521046dc70500a5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v07\n- v07 --refutes--> v08\n- v08 --supports--> v05\n- v05 --supports--> v0a\n- v0a --refutes--> v04\n- v04 --refutes--> v00\n- v00 --refutes--> v0c\n- v0c --refutes--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v02\n- v04 --supports--> v03\n- v0a --supports--> v09\n- v0c --refutes--> v01\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":"334a1f1ed2f271498a18e5ff12866bfe1deb7754ecb4b48d3faad97ea3a21e37","schema":"livegraph.boolean-propagation.v1","spec_hash":"d4ddf521046dc70500a54c0c6205fc761adcfff98314b116ba4d5409f660c67b","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-0cfb0a3d9f7336b7e83f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0b --refutes--> v06\n- v06 --refutes--> v00\n- v00 --supports--> v0a\n- v0a --supports--> v0c\n- v0c --refutes--> v07\n- v07 --refutes--> v09\n- v09 --refutes--> v03\n- v03 --supports--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v02\n- v07 --refutes--> v01\n- v0a --refutes--> v08\n- v0b --supports--> v05\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":"6fa10f00055eb652ac43d58caa516b2cd1a38787fb490fd0d11819d332eead82","schema":"livegraph.boolean-propagation.v1","spec_hash":"0cfb0a3d9f7336b7e83f50a57640569c9fc9a5af836b30e8a7276b7654518f0d","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-ece22aba083e31ab1916","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --refutes--> v05\n- v05 --supports--> v06\n- v06 --supports--> v09\n- v09 --supports--> v0a\n- v0a --refutes--> v03\n- v03 --supports--> v07\n- v07 --refutes--> v0c\n- v0c --supports--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v0b\n- v09 --refutes--> v08\n- v09 --supports--> v00\n- v0a --supports--> 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":"5051c89852dc97c4d963f3f697127480fe6ff01b3ea11f37f3593eb7c7156e9f","schema":"livegraph.boolean-propagation.v1","spec_hash":"ece22aba083e31ab19166166beb31a20b9c3a662d2c4ad64220535e4c249d4ed","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-ae4c9556425a8b76e62d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --refutes--> v06\n- v06 --refutes--> v00\n- v00 --refutes--> v08\n- v08 --refutes--> v07\n- v07 --supports--> v03\n- v03 --refutes--> v09\n- v09 --supports--> v0b\n- v0b --refutes--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v01\n- v00 --refutes--> v02\n- v03 --supports--> v0a\n- v07 --refutes--> v05\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":"3a5dedff7a4d44d253b92dbad67186ffe90ad12015a7be2341712ec4d157b696","schema":"livegraph.boolean-propagation.v1","spec_hash":"ae4c9556425a8b76e62d8a2ed448b7ba19a3d4271674aaf91fdeaf47504ad5b7","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-3298bd59ca0a7cb75270","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --refutes--> v0c\n- v0c --refutes--> v01\n- v01 --supports--> v06\n- v06 --refutes--> v04\n- v04 --refutes--> v09\n- v09 --supports--> v02\n- v02 --refutes--> v00\n- v00 --refutes--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v03\n- v06 --supports--> v0b\n- v0c --refutes--> v07\n- v0c --supports--> v0a\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":"a4a4e3bef1d00f1467bf9f679111865dd53ca2d6bf16273ef8e7aaa40ce3e228","schema":"livegraph.boolean-propagation.v1","spec_hash":"3298bd59ca0a7cb75270107fabb0374f92624d9b72504263d5bbcb91358ae092","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-52bdbff9537a19b4498e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0b --refutes--> v03\n- v03 --refutes--> v05\n- v05 --refutes--> v0a\n- v0a --supports--> v07\n- v07 --supports--> v00\n- v00 --refutes--> v02\n- v02 --supports--> v04\n- v04 --refutes--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v0c\n- v03 --supports--> v08\n- v05 --supports--> v01\n- v0a --refutes--> v09\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":"376f72dced3e422efe29eacb369e4daadd76e9de75137e324e8c82ea97660023","schema":"livegraph.boolean-propagation.v1","spec_hash":"52bdbff9537a19b4498ecc485b8933d5f73ecc0ef04d6f1b84091d25f7d9f68f","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-d39fa528366429b48f89","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --refutes--> v00\n- v00 --supports--> v0a\n- v0a --supports--> v07\n- v07 --supports--> v06\n- v06 --supports--> v02\n- v02 --supports--> v09\n- v09 --supports--> v05\n- v05 --supports--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v08\n- v02 --supports--> v0c\n- v03 --refutes--> v04\n- v07 --supports--> v01\n\n[VERIFIED ROOT]\nv03 = 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":"6a03820fe46d29f005ea6eccd52a07a020da1d48d34b3abbf2f9dcc5b6b5b873","schema":"livegraph.boolean-propagation.v1","spec_hash":"d39fa528366429b48f89d77bda290e9e09d6842be651e2422ebad44e61f8e241","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":8,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-25906ea5039cf6ae27f0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v00\n- v00 --supports--> v0b\n- v0b --supports--> v09\n- v09 --supports--> v08\n- v08 --refutes--> v04\n- v04 --refutes--> v0a\n- v0a --refutes--> v0c\n- v0c --supports--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v02\n- v09 --refutes--> v03\n- v09 --supports--> v01\n- v0a --supports--> v05\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":"7423bfaf6dc7e325c0cf6f4ad72f3da66398c2e5d68dfcbcd49b311d47aa7e98","schema":"livegraph.boolean-propagation.v1","spec_hash":"25906ea5039cf6ae27f062d18a9df8172a7f73be812107a8f1ec0409bd60a678","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-bda66e862c46159e785a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v03\n- v03 --supports--> v0a\n- v0a --refutes--> v01\n- v01 --refutes--> v08\n- v08 --refutes--> v09\n- v09 --supports--> v07\n- v07 --refutes--> v04\n- v04 --supports--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v02\n- v06 --supports--> v0b\n- v06 --supports--> v0c\n- v0a --supports--> v00\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":"a5940864d083ab777148c4bba6c37efd2e5e6741aa8c5c04977cc6a323532c3a","schema":"livegraph.boolean-propagation.v1","spec_hash":"bda66e862c46159e785abd0269f9f729488a241455a7ce5c9e83ffcf814adcc2","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":8,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-4f38f28e3f9855b83ee8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --supports--> v09\n- v09 --supports--> v03\n- v03 --refutes--> v06\n- v06 --refutes--> v0b\n- v0b --refutes--> v00\n- v00 --refutes--> v01\n- v01 --supports--> v05\n- v05 --refutes--> v0c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v09 --refutes--> v04\n- v09 --refutes--> v0a\n- v0b --supports--> v02\n- v0b --supports--> 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":"df35a206b71837499071007e8d604203a2ba56298414bc0081ceb619434a47bb","schema":"livegraph.boolean-propagation.v1","spec_hash":"4f38f28e3f9855b83ee8df3694afe90578c3ed6c6db076393bd5292910cb437d","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-7ff287147d7f09989783","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --supports--> v01\n- v01 --refutes--> v03\n- v03 --supports--> v06\n- v06 --refutes--> v00\n- v00 --refutes--> v0a\n- v0a --refutes--> v02\n- v02 --refutes--> v09\n- v09 --supports--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v04\n- v00 --refutes--> v0b\n- v02 --supports--> v08\n- v03 --refutes--> 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":"77cea749e3d75c91d2de020724e2ed7436c5b1fae8eca41beeb3806ba0650ac9","schema":"livegraph.boolean-propagation.v1","spec_hash":"7ff287147d7f0998978355810c4358bbcc94912bc8cd349f10eba858fa943425","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"}],\"hop\":8,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-65b4a1d63ee7cd3b087d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --refutes--> v06\n- v06 --refutes--> v0c\n- v0c --refutes--> v00\n- v00 --refutes--> v0b\n- v0b --supports--> v03\n- v03 --refutes--> v07\n- v07 --refutes--> v01\n- v01 --refutes--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v09\n- v00 --supports--> v02\n- v01 --supports--> v08\n- v0b --supports--> v05\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":"1cac415438ad82246a47565e7867efa81b12590ed7c2e95c62e75b16ab3e28cf","schema":"livegraph.boolean-propagation.v1","spec_hash":"65b4a1d63ee7cd3b087dcb889e44509df3f7cf897af7319ae97acc105a0fac75","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-9ce6517393a61f93944c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --refutes--> v09\n- v09 --supports--> v04\n- v04 --supports--> v01\n- v01 --supports--> v07\n- v07 --refutes--> v02\n- v02 --supports--> v03\n- v03 --refutes--> v08\n- v08 --refutes--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v05\n- v03 --supports--> v06\n- v09 --supports--> v00\n- v0a --supports--> v0c\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":"ead79f6124df32b1d493e808cb97e0ff790365e6677e5922dbc94547695b3153","schema":"livegraph.boolean-propagation.v1","spec_hash":"9ce6517393a61f93944c581d2c2337ef9131d1541535744d6bcd28941a0f1b1f","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":8,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-1edccad80f96df1bbd63","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --refutes--> v06\n- v06 --supports--> v04\n- v04 --supports--> v01\n- v01 --supports--> v00\n- v00 --supports--> v05\n- v05 --supports--> v07\n- v07 --supports--> v0c\n- v0c --supports--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v0a\n- v02 --supports--> v08\n- v05 --supports--> v03\n- v05 --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":"d2e182b401b4c9b6c2e1a5838434edc243187944ccd06b02c729e22b8f6928f1","schema":"livegraph.boolean-propagation.v1","spec_hash":"1edccad80f96df1bbd6361a0c303370316050d87d2cd72f321fe4183d866019f","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-6ee3212c17019aa72656","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --supports--> v04\n- v04 --refutes--> v01\n- v01 --refutes--> v08\n- v08 --refutes--> v0b\n- v0b --supports--> v05\n- v05 --refutes--> v03\n- v03 --supports--> v00\n- v00 --supports--> v0c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v07\n- v05 --refutes--> v02\n- v0b --refutes--> v0a\n- v0b --supports--> 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":"487558bd1725ed104f56f67881b1a0e8469be853739c2834c378b6db9d830e99","schema":"livegraph.boolean-propagation.v1","spec_hash":"6ee3212c17019aa726562bc2a0b4e655479a3caa2c4237ee007ea6577b3c620c","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-dd7007005054fd03f951","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --supports--> v0c\n- v0c --supports--> v09\n- v09 --supports--> v03\n- v03 --supports--> v0b\n- v0b --supports--> v01\n- v01 --refutes--> v02\n- v02 --supports--> v0a\n- v0a --refutes--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v00\n- v06 --supports--> v05\n- v09 --supports--> v07\n- v0c --supports--> v04\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":"72e7252e63e943cb5173709043691f6fa73137fa42eb856c6a6a3bf444b4f94f","schema":"livegraph.boolean-propagation.v1","spec_hash":"dd7007005054fd03f951d7e0fd9d911b5113226a7a14b912cfe94ac01cf87d8c","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-45776c277fe2dbb714bc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --refutes--> v08\n- v08 --refutes--> v06\n- v06 --refutes--> v09\n- v09 --supports--> v05\n- v05 --supports--> v0a\n- v0a --supports--> v01\n- v01 --refutes--> v0c\n- v0c --refutes--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v0b\n- v09 --refutes--> v03\n- v0a --refutes--> v04\n- v0c --refutes--> 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":"82e78923ed96294dcf0c7a5730f5c4c4b87a77acb93e5002ab703f7f613ba07d","schema":"livegraph.boolean-propagation.v1","spec_hash":"45776c277fe2dbb714bc30309c899faa1b06f65ced86433ea05a375e2aa0af18","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-4859d65a237fed065dc3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --supports--> v0c\n- v0c --supports--> v07\n- v07 --refutes--> v06\n- v06 --refutes--> v01\n- v01 --supports--> v09\n- v09 --refutes--> v08\n- v08 --supports--> v0a\n- v0a --supports--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --supports--> v00\n- v08 --supports--> v02\n- v09 --refutes--> v05\n- v0a --supports--> v04\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":"fe51aa74edd055c4cba84d15002037ad5ab8cda4920201d13d3b67ba4e81ad97","schema":"livegraph.boolean-propagation.v1","spec_hash":"4859d65a237fed065dc3bcf1bfe34cb8eacb8b0b30f145b097813a738512a0e3","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-79ba0b18259b4aa1fc11","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v01\n- v01 --supports--> v05\n- v05 --supports--> v0a\n- v0a --supports--> v02\n- v02 --supports--> v06\n- v06 --supports--> v07\n- v07 --supports--> v0c\n- v0c --refutes--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v04\n- v00 --refutes--> v0b\n- v05 --supports--> v03\n- v05 --supports--> 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":"869937436741334c5d2bd2a153de89525719ed37c4029f2e63e4ee38fda82f5c","schema":"livegraph.boolean-propagation.v1","spec_hash":"79ba0b18259b4aa1fc11c6315b3856786794a49937805fcad38c825f9fd61586","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-8cc84a72b4dbd8bd771c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --supports--> v01\n- v01 --supports--> v04\n- v04 --refutes--> v0b\n- v0b --supports--> v08\n- v08 --refutes--> v05\n- v05 --supports--> v0a\n- v0a --refutes--> v0c\n- v0c --supports--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v03\n- v08 --refutes--> v09\n- v0c --refutes--> v00\n- v0c --refutes--> v02\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":"82c09821793176c28bd9a3168a797fa5f38b0847796b4e8079ac14a7b2d207af","schema":"livegraph.boolean-propagation.v1","spec_hash":"8cc84a72b4dbd8bd771c54d2e407b2976ba405be95327f8e0dac1c993ffdca5d","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-fad04161d9130b39a5dc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --refutes--> v08\n- v08 --supports--> v05\n- v05 --refutes--> v07\n- v07 --supports--> v09\n- v09 --refutes--> v02\n- v02 --supports--> v03\n- v03 --supports--> v0b\n- v0b --refutes--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v04\n- v00 --supports--> v06\n- v02 --supports--> v0a\n- v09 --refutes--> v0c\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":"afab239825f84be663a38eedaf6031f164a8ec051f536c1fd93dbe6dea9d04d9","schema":"livegraph.boolean-propagation.v1","spec_hash":"fad04161d9130b39a5dc8656a4eb2a999b13ce1ceee4d93477c085ce64bd4f73","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-70ca8368a0b8f013d1e2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --supports--> v0a\n- v0a --supports--> v08\n- v08 --supports--> v0b\n- v0b --refutes--> v00\n- v00 --refutes--> v09\n- v09 --refutes--> v01\n- v01 --supports--> v02\n- v02 --refutes--> v0c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v04\n- v07 --supports--> v05\n- v0b --refutes--> v03\n- v0b --supports--> v06\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":"d6b42f4301c82a23bf39e111c00a1a460a2d7ee771881b35e367e050025f93dc","schema":"livegraph.boolean-propagation.v1","spec_hash":"70ca8368a0b8f013d1e2d121a89cc0f47e86cd695d2a43bd4ca796f924728e27","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-75a85955e864cd49f6ec","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --supports--> v05\n- v05 --supports--> v0c\n- v0c --refutes--> v08\n- v08 --refutes--> v06\n- v06 --refutes--> v09\n- v09 --refutes--> v07\n- v07 --supports--> v00\n- v00 --supports--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v07 --supports--> v0b\n- v09 --supports--> v03\n- v0a --refutes--> v01\n- v0a --refutes--> v02\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":"d0caa216095969f9582e58731ee2f532655a0dbe6ec4fa0d9b4f36226521fa70","schema":"livegraph.boolean-propagation.v1","spec_hash":"75a85955e864cd49f6ec6690fd1465cb6d41fd85532646a254cb2561a9d51158","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-d53da848a2203b4f5036","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --supports--> v0b\n- v0b --refutes--> v0c\n- v0c --supports--> v04\n- v04 --supports--> v01\n- v01 --supports--> v07\n- v07 --refutes--> v0a\n- v0a --refutes--> v08\n- v08 --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v05\n- v08 --refutes--> v00\n- v08 --supports--> v02\n- v0a --supports--> v09\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":"22d6d6afa5ff2f62120a1d1f2ac77163026586adf499793bd56233cdb4e533a5","schema":"livegraph.boolean-propagation.v1","spec_hash":"d53da848a2203b4f5036eb75a36434eb1b1ca427eabb795672739f56301ceee6","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-2a04dd740eea7670fdfc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v03\n- v03 --supports--> v07\n- v07 --refutes--> v05\n- v05 --refutes--> v09\n- v09 --refutes--> v0c\n- v0c --supports--> v02\n- v02 --supports--> v04\n- v04 --refutes--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v06\n- v00 --supports--> v08\n- v02 --refutes--> v0b\n- v02 --supports--> v01\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":"2bad09e372e2d2b9aac952997a5005aaeadf910b8cfa92df333d833a84776aa3","schema":"livegraph.boolean-propagation.v1","spec_hash":"2a04dd740eea7670fdfc6109546d13165905d2092ddcf82d8d1714bef6c4c23b","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-d817adfd1e5424f40f9c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v01\n- v01 --supports--> v06\n- v06 --supports--> v07\n- v07 --refutes--> v0a\n- v0a --refutes--> v0b\n- v0b --supports--> v05\n- v05 --refutes--> v0c\n- v0c --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v09\n- v06 --refutes--> v02\n- v0a --refutes--> v03\n- v0a --refutes--> v08\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":"5931042ad3cab8a0cf265a3ed135a510d8ed85f1df5b1f20b0a6996f7ec1b93c","schema":"livegraph.boolean-propagation.v1","spec_hash":"d817adfd1e5424f40f9cce6ff36ed29a2dbc0f74c348e12efaec1fcac6850fad","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-c1c70783ddb156fc6fe7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --refutes--> v08\n- v08 --refutes--> v06\n- v06 --refutes--> v01\n- v01 --refutes--> v04\n- v04 --refutes--> v09\n- v09 --refutes--> v07\n- v07 --supports--> v0b\n- v0b --supports--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v0c\n- v01 --supports--> v03\n- v07 --refutes--> v02\n- v08 --supports--> v00\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":"5a88a99d9ade0fb1a53f090f2e4fede4ca04e52e5f992bada5621c0668da65c8","schema":"livegraph.boolean-propagation.v1","spec_hash":"c1c70783ddb156fc6fe73b2ef064957e7e66a8038b80a3f3aa737680b1028c45","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-b4b99ea96469d44bbf6c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --supports--> v03\n- v03 --refutes--> v06\n- v06 --supports--> v01\n- v01 --supports--> v0a\n- v0a --refutes--> v05\n- v05 --refutes--> v0b\n- v0b --supports--> v00\n- v00 --supports--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v02\n- v05 --supports--> v08\n- v0b --refutes--> v07\n- v0b --supports--> v09\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":"682ad0d39386d040241b50a0ac85b0da19ae4eb9d4a203eb411eaff7961da5d3","schema":"livegraph.boolean-propagation.v1","spec_hash":"b4b99ea96469d44bbf6c38f43469b7a9b1ac5c77fadd9b0e76fc0c87af02cc6c","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-649ebcf1bbd0bee12d96","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --supports--> v02\n- v02 --refutes--> v09\n- v09 --supports--> v0c\n- v0c --refutes--> v03\n- v03 --refutes--> v0a\n- v0a --refutes--> v0b\n- v0b --supports--> v00\n- v00 --refutes--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v04\n- v09 --supports--> v08\n- v0a --supports--> v07\n- v0b --supports--> v06\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":"78644d86b97a32a659b884e10508e30849bffe7e0adb32eb5405de1a542422c7","schema":"livegraph.boolean-propagation.v1","spec_hash":"649ebcf1bbd0bee12d9610f30669f3533572db4de293e05731654c29588bdb1c","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-f9ea41263ae1a7500264","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --supports--> v0b\n- v0b --supports--> v05\n- v05 --supports--> v01\n- v01 --refutes--> v00\n- v00 --supports--> v03\n- v03 --supports--> v06\n- v06 --supports--> v0a\n- v0a --refutes--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v07\n- v02 --refutes--> v08\n- v03 --refutes--> v04\n- v0a --refutes--> v0c\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":"5c84ec4c1d7c99e87c09918d50e567368d3e3015370d48be117019280bfe4f44","schema":"livegraph.boolean-propagation.v1","spec_hash":"f9ea41263ae1a7500264a3d8d9554877f8b707c586569e95d9b6195e95c69eaa","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-1e06b4d84fea6e7e9221","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v0a\n- v0a --supports--> v05\n- v05 --refutes--> v00\n- v00 --supports--> v02\n- v02 --supports--> v01\n- v01 --supports--> v04\n- v04 --supports--> v0c\n- v0c --refutes--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v03\n- v02 --refutes--> v09\n- v05 --supports--> v07\n- v0a --refutes--> v0b\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":"783f078d5826d7b19419f92b49b2a96da0fad9802a5c5904feafbc1622838b6a","schema":"livegraph.boolean-propagation.v1","spec_hash":"1e06b4d84fea6e7e92210938168a4c037007f062b484f2e2ab441473432c5a53","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"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\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-80bd49cb199e59bbd726","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --refutes--> v05\n- v05 --supports--> v08\n- v08 --supports--> v09\n- v09 --supports--> v0b\n- v0b --refutes--> v06\n- v06 --refutes--> v07\n- v07 --supports--> v0a\n- v0a --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v04\n- v07 --supports--> v0c\n- v09 --refutes--> v02\n- v0a --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":"42d92045331f29c99e75aa761c780d8c2f05815f998f07a2e1b77bafc392f09f","schema":"livegraph.boolean-propagation.v1","spec_hash":"80bd49cb199e59bbd72606ec958aa490ad953c3bb08788a9b18e4f08fdd45563","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-0015fb3e5608d0de1166","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --supports--> v08\n- v08 --supports--> v0c\n- v0c --supports--> v0b\n- v0b --supports--> v00\n- v00 --refutes--> v01\n- v01 --refutes--> v0a\n- v0a --refutes--> v09\n- v09 --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v07\n- v09 --supports--> v05\n- v0b --refutes--> v02\n- v0b --refutes--> 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":"fede557193235241416ef7cb394e6aef69899bb4023784c628f63d3c250c67e7","schema":"livegraph.boolean-propagation.v1","spec_hash":"0015fb3e5608d0de116617b2c3eb77fccc32fb907468fa1db4c47fb1ee7e09dc","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-d77e34a2b85195013037","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0b --refutes--> v01\n- v01 --supports--> v00\n- v00 --supports--> v06\n- v06 --supports--> v08\n- v08 --supports--> v0c\n- v0c --supports--> v05\n- v05 --refutes--> v04\n- v04 --supports--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v07\n- v04 --refutes--> v09\n- v05 --refutes--> v0a\n- v08 --supports--> v02\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":"4130a80d3cce1bdd3399d8d7da865c57a731f80ea449e86d3ca8e64ea2d8da86","schema":"livegraph.boolean-propagation.v1","spec_hash":"d77e34a2b85195013037ecd0b68e38501abca24fb4666e1e3239398292a2feba","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-2cf5af9a161faee31ee5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --refutes--> v0a\n- v0a --refutes--> v00\n- v00 --refutes--> v04\n- v04 --refutes--> v05\n- v05 --refutes--> v02\n- v02 --refutes--> v08\n- v08 --supports--> v01\n- v01 --supports--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v07\n- v08 --supports--> v03\n- v08 --supports--> v0b\n- v0a --supports--> v09\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":"7269acd6a5c0a42ced84abf9dd8248d18ae85c798196c52efefdc44ca54a5420","schema":"livegraph.boolean-propagation.v1","spec_hash":"2cf5af9a161faee31ee5e830788b05c83a5cbf1c943b8aae2c91a2ef23d43e1c","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-d1029370d8ed155374ac","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0b --refutes--> v02\n- v02 --refutes--> v07\n- v07 --supports--> v01\n- v01 --refutes--> v00\n- v00 --supports--> v08\n- v08 --refutes--> v05\n- v05 --refutes--> v09\n- v09 --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v0c\n- v01 --refutes--> v04\n- v05 --refutes--> v0a\n- v05 --supports--> v06\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":"208a8aec5e98ed4ff22df6986fb8194a499689f18f98337e01a24d5ffd908ddd","schema":"livegraph.boolean-propagation.v1","spec_hash":"d1029370d8ed155374ac980c86089721f648abc0af830a1b8e9da6ed8d263ad3","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-7352f82d2d9f316485f9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --refutes--> v03\n- v03 --refutes--> v05\n- v05 --supports--> v0b\n- v0b --supports--> v01\n- v01 --refutes--> v08\n- v08 --refutes--> v06\n- v06 --supports--> v09\n- v09 --refutes--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v08 --refutes--> v07\n- v08 --supports--> v04\n- v09 --refutes--> v00\n- v0a --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":"a06671765eb8b6ac6a4c2c98ff87c7004871f066736017fe56631bd1afb8cc37","schema":"livegraph.boolean-propagation.v1","spec_hash":"7352f82d2d9f316485f961607703884745f6f26e47798d4cbc468fcecf7e7462","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-0203e3f224acfd16ce28","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --supports--> v04\n- v04 --supports--> v01\n- v01 --supports--> v06\n- v06 --supports--> v0c\n- v0c --supports--> v07\n- v07 --refutes--> v03\n- v03 --refutes--> v00\n- v00 --refutes--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v02\n- v07 --refutes--> v05\n- v07 --refutes--> v0a\n- v0c --refutes--> v0b\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":"2d340e9584a92a15d2658c1a8a7636342f642102455ec083a11e2295eb6f9742","schema":"livegraph.boolean-propagation.v1","spec_hash":"0203e3f224acfd16ce28491a54ef2d9830abab5c4190203ec18b07361c7d47a9","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-8634fe38270219085a9c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --refutes--> v0c\n- v0c --supports--> v05\n- v05 --supports--> v06\n- v06 --refutes--> v0a\n- v0a --supports--> v09\n- v09 --supports--> v03\n- v03 --refutes--> v00\n- v00 --supports--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v01\n- v06 --supports--> v08\n- v06 --supports--> v0b\n- v07 --supports--> 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":"c376f0059bd79b1f927700bfa2e815df3a5441ff979f04f0cbfaedb511301c90","schema":"livegraph.boolean-propagation.v1","spec_hash":"8634fe38270219085a9c32fea6d15fc630193c23958c1095bf0086b68b86ed6e","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-72d0c309f7058b0f376b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --supports--> v08\n- v08 --refutes--> v01\n- v01 --refutes--> v03\n- v03 --supports--> v00\n- v00 --refutes--> v07\n- v07 --supports--> v0b\n- v0b --supports--> v06\n- v06 --supports--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v0c\n- v06 --supports--> v0a\n- v08 --supports--> v04\n- v09 --refutes--> 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":"78f8adc757ae7ffa23d87cfeb21ffd47e99b5b99941f518e0aef3dba92c056e0","schema":"livegraph.boolean-propagation.v1","spec_hash":"72d0c309f7058b0f376b88545c24238529da09c9913e0c4025bb4026ac10ea5d","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-f1a30b961e1f3c6dff11","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --refutes--> v02\n- v02 --supports--> v0b\n- v0b --supports--> v05\n- v05 --supports--> v08\n- v08 --refutes--> v06\n- v06 --supports--> v01\n- v01 --supports--> v0a\n- v0a --supports--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v07\n- v06 --supports--> v04\n- v08 --supports--> v03\n- v0b --refutes--> v00\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":"2953f8c1287bcd5fe65b404ef9912a08e99d9325cf2d92fd21b86b31cc9ae1a2","schema":"livegraph.boolean-propagation.v1","spec_hash":"f1a30b961e1f3c6dff11377f20fd2b9c68650d3e6bd1072260a382654ee0e6df","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-80136de43e855a273e16","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v0b\n- v0b --refutes--> v0c\n- v0c --supports--> v02\n- v02 --refutes--> v03\n- v03 --supports--> v07\n- v07 --supports--> v0a\n- v0a --supports--> v00\n- v00 --refutes--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v06\n- v03 --refutes--> v05\n- v07 --supports--> v01\n- v0b --supports--> v08\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":"f261abe8aefa2de1378096cb1bb794fdd0a531876f7d71b715bbb9280013a5fb","schema":"livegraph.boolean-propagation.v1","spec_hash":"80136de43e855a273e16db66cc38f5e2cc7d6e26716da26035b5ffa15a2d468f","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-088e2c42ba70435f7921","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --supports--> v04\n- v04 --supports--> v01\n- v01 --supports--> v05\n- v05 --refutes--> v09\n- v09 --supports--> v0a\n- v0a --refutes--> v0c\n- v0c --refutes--> v06\n- v06 --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v0b\n- v06 --refutes--> v02\n- v09 --supports--> v07\n- v0a --supports--> 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":"1e3ac3378823ebcad4afd8066cc825ab112e4d90f14f37ae1a8ec9b95ba6594c","schema":"livegraph.boolean-propagation.v1","spec_hash":"088e2c42ba70435f79215ea46ff493cbc226c6d5917e90536042b915e1e99aa0","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-4ea4afccac43e7a1841d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --supports--> v00\n- v00 --refutes--> v09\n- v09 --supports--> v07\n- v07 --supports--> v08\n- v08 --refutes--> v0a\n- v0a --supports--> v03\n- v03 --supports--> v04\n- v04 --refutes--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v02\n- v03 --refutes--> v0b\n- v03 --refutes--> v0c\n- v07 --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":"da47b887a1795712bf67d32d0b23d719d5129f510154f4e8311053edecc2e50d","schema":"livegraph.boolean-propagation.v1","spec_hash":"4ea4afccac43e7a1841d740194077c9792b556efd96540dce0bc249edf51458c","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":8,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-eeadf4e75ad797d86c96","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v07\n- v07 --refutes--> v05\n- v05 --refutes--> v03\n- v03 --refutes--> v0a\n- v0a --refutes--> v08\n- v08 --supports--> v00\n- v00 --supports--> v0b\n- v0b --refutes--> v0c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v09\n- v07 --supports--> v01\n- v08 --refutes--> v02\n- v0a --refutes--> v04\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":"01db3fcee1b43b2875f34c824c94d7e75f54d03458fd67f7fa43f8449515bdcc","schema":"livegraph.boolean-propagation.v1","spec_hash":"eeadf4e75ad797d86c9684a02fc0319af1b8af4ca46e14aa4be1953cc5bdb5bf","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-33482ce47ac728994605","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v02\n- v02 --supports--> v05\n- v05 --refutes--> v0c\n- v0c --refutes--> v07\n- v07 --refutes--> v08\n- v08 --supports--> v04\n- v04 --refutes--> v01\n- v01 --refutes--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v03\n- v07 --supports--> v09\n- v07 --supports--> v0a\n- v0c --refutes--> v00\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":"d25b3ad01d14cf20957dc04fde1ef57e6bf03fed49040e4820632419ec324788","schema":"livegraph.boolean-propagation.v1","spec_hash":"33482ce47ac72899460509b6a9b0a15ab09f7f543f40ca258aaba66302d74321","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-c8512068808e789124a9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --refutes--> v03\n- v03 --refutes--> v08\n- v08 --supports--> v00\n- v00 --refutes--> v0a\n- v0a --refutes--> v07\n- v07 --supports--> v0c\n- v0c --refutes--> v01\n- v01 --supports--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v02\n- v08 --supports--> v0b\n- v0a --refutes--> v09\n- v0c --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":"65480f3cf3955c38ce936962382393a78bf6157895c32b698c0e24016fc48583","schema":"livegraph.boolean-propagation.v1","spec_hash":"c8512068808e789124a921153ebed61ba8bdabcb0a10bf98556c5971b298ff71","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-79217d882440efdedaa1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --refutes--> v07\n- v07 --supports--> v0a\n- v0a --supports--> v03\n- v03 --supports--> v00\n- v00 --refutes--> v06\n- v06 --refutes--> v05\n- v05 --refutes--> v01\n- v01 --supports--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v04\n- v01 --supports--> v0b\n- v07 --supports--> v0c\n- v08 --supports--> v02\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":"a31b07d975947ada4747c0c87a5ec73e5732a3c3857e4bd63835a4be686c0695","schema":"livegraph.boolean-propagation.v1","spec_hash":"79217d882440efdedaa193aacc27612f7af7d0083b2eba5c62f8cef3ae678623","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":8,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-4df86b30796bf3d01b47","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0b --supports--> v05\n- v05 --refutes--> v04\n- v04 --refutes--> v03\n- v03 --supports--> v06\n- v06 --supports--> v0c\n- v0c --supports--> v01\n- v01 --supports--> v08\n- v08 --refutes--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v09\n- v04 --supports--> v07\n- v05 --refutes--> v00\n- v0c --refutes--> v02\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":"1adc8e3471ed99e2f2f702c0ec888b284ca5313ef02711f39c4b11e340d2c854","schema":"livegraph.boolean-propagation.v1","spec_hash":"4df86b30796bf3d01b4709f942b6c1cfcbc924c5c0a4a8a56394063e22d33900","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-9a845bad73a16fea0f6b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --refutes--> v09\n- v09 --supports--> v03\n- v03 --supports--> v07\n- v07 --supports--> v08\n- v08 --refutes--> v05\n- v05 --supports--> v01\n- v01 --supports--> v04\n- v04 --refutes--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v0a\n- v03 --refutes--> v0b\n- v05 --supports--> v06\n- v07 --refutes--> v00\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":"180a467610335fecb92f7783d0bd131548e3afeb8e4fea06baa089fca023c116","schema":"livegraph.boolean-propagation.v1","spec_hash":"9a845bad73a16fea0f6b847ab62a57ff91f7918b81a946523a68f5462124d7d6","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-1ed642e7fbfcd39ee913","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --supports--> v05\n- v05 --supports--> v02\n- v02 --refutes--> v01\n- v01 --supports--> v06\n- v06 --refutes--> v07\n- v07 --supports--> v0c\n- v0c --refutes--> v08\n- v08 --refutes--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v04\n- v02 --supports--> v0b\n- v0c --refutes--> v00\n- v0c --supports--> v03\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":"649624cf11bdb5ed560e6edf47bd704f1a472312cd75b7a9d14d4f40d5d96fe7","schema":"livegraph.boolean-propagation.v1","spec_hash":"1ed642e7fbfcd39ee91322f4aec487ca948b7c1bfc0f9541842329e33bb1035e","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-fade9664d7918e47b997","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0b --refutes--> v04\n- v04 --refutes--> v09\n- v09 --refutes--> v08\n- v08 --supports--> v07\n- v07 --refutes--> v01\n- v01 --supports--> v03\n- v03 --refutes--> v0c\n- v0c --supports--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v00\n- v09 --supports--> v06\n- v0b --supports--> v05\n- v0c --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":"ca8fd984a85bd82f5bda0ad4d22cdc0cbf2adb84d35f2e7be88526dc7a6a3ad0","schema":"livegraph.boolean-propagation.v1","spec_hash":"fade9664d7918e47b9970f231c213ee2508e7b2a45270c8aa7e1fa91babb7e69","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-5d9a83ba08f60750f7d9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --refutes--> v00\n- v00 --refutes--> v03\n- v03 --refutes--> v0a\n- v0a --supports--> v0b\n- v0b --supports--> v08\n- v08 --refutes--> v01\n- v01 --supports--> v09\n- v09 --supports--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v07\n- v01 --supports--> v02\n- v03 --supports--> v06\n- v0a --refutes--> v0c\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":"e76b41eb91234e307510f26d7e0998844c6333f6169768d3826f1d4d5d6eb2cb","schema":"livegraph.boolean-propagation.v1","spec_hash":"5d9a83ba08f60750f7d9da7853a710a1881e813852de1327b2a4b051662acf12","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-af0afa76ace2e554d9e2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v09\n- v09 --supports--> v07\n- v07 --supports--> v01\n- v01 --supports--> v00\n- v00 --supports--> v0c\n- v0c --supports--> v02\n- v02 --refutes--> v08\n- v08 --refutes--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v0b\n- v02 --supports--> v05\n- v04 --refutes--> v03\n- v04 --refutes--> v06\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":"bfac76b33eac2013dd4a59b1fd57498649ca7893662aac266e5a3e272aaab6e7","schema":"livegraph.boolean-propagation.v1","spec_hash":"af0afa76ace2e554d9e2162ae0279cdfe8b0c1fe1a840beb75dcf0a7ab41ee2d","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-643a3cf295e4b70c13c5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --refutes--> v09\n- v09 --refutes--> v0b\n- v0b --supports--> v00\n- v00 --refutes--> v01\n- v01 --refutes--> v06\n- v06 --supports--> v02\n- v02 --supports--> v03\n- v03 --supports--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v0a\n- v08 --refutes--> v04\n- v08 --supports--> v07\n- v09 --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":"938908611e4ef9e3622b1035067a95a8ed8668ea1fec627e5e38989939fbeeb1","schema":"livegraph.boolean-propagation.v1","spec_hash":"643a3cf295e4b70c13c53dde4697fb7bdd18356428fefa699aca36ef4a5e43ec","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-233ea7b56fc8a45272bf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0b --refutes--> v0c\n- v0c --supports--> v07\n- v07 --refutes--> v0a\n- v0a --refutes--> v09\n- v09 --refutes--> v01\n- v01 --supports--> v05\n- v05 --refutes--> v03\n- v03 --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v09 --supports--> v02\n- v09 --supports--> v06\n- v0c --refutes--> v04\n- v0c --supports--> v08\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":"9cd7237c0e676f8a97af2c116aa1b6d4c7494279d7e5a5e40872164207780c89","schema":"livegraph.boolean-propagation.v1","spec_hash":"233ea7b56fc8a45272bfd743d1a87a7e7c498f3465d05b8d15893adb42195127","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-022d08500234aecaa282","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --refutes--> v0b\n- v0b --supports--> v02\n- v02 --supports--> v09\n- v09 --refutes--> v0a\n- v0a --supports--> v01\n- v01 --supports--> v04\n- v04 --supports--> v06\n- v06 --refutes--> v0c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v08\n- v05 --refutes--> v03\n- v09 --refutes--> v00\n- v0a --refutes--> v07\n\n[VERIFIED ROOT]\nv05 = 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":"ace2284ac524f7877246beb07f76c33b766841c7641740a27932e3cd233d6c79","schema":"livegraph.boolean-propagation.v1","spec_hash":"022d08500234aecaa28224bbc3a3c8d895b427e6110be84cbc50f3e89ef574a2","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":8,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":8,"item_id":"lggrpo1-217f3e501f8ff901d136","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --refutes--> v02\n- v02 --supports--> v00\n- v00 --supports--> v04\n- v04 --supports--> v03\n- v03 --refutes--> v07\n- v07 --supports--> v06\n- v06 --supports--> v0c\n- v0c --refutes--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v05\n- v00 --supports--> v09\n- v04 --refutes--> v0a\n- v07 --refutes--> v0b\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":"0abfb01bad3cb4c9b987762c27911856e63ebeb25557fe38ea49958809f93ca3","schema":"livegraph.boolean-propagation.v1","spec_hash":"217f3e501f8ff901d136a28067176bcafe7d223cc35f61be07bca525522d7c95","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":8,"item_id":"lggrpo1-b272257ac73478f8a8a1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --supports--> v01\n- v01 --refutes--> v0c\n- v0c --refutes--> v05\n- v05 --supports--> v09\n- v09 --refutes--> v0b\n- v0b --refutes--> v07\n- v07 --refutes--> v06\n- v06 --supports--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v03\n- v06 --supports--> v02\n- v07 --supports--> v04\n- v0a --supports--> v00\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":"82d45161fffaace416847635cb9a37f6ca537f9b5035995ed87b6583f57a0804","schema":"livegraph.boolean-propagation.v1","spec_hash":"b272257ac73478f8a8a132eed6e2db5cc4b9cbfbd324b4d18ffdb62030ef51c7","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":8,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-3d005ae36d90184f4033","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --supports--> v07\n- v07 --refutes--> v01\n- v01 --refutes--> v08\n- v08 --refutes--> v0c\n- v0c --supports--> v0d\n- v0d --refutes--> v05\n- v05 --supports--> v02\n- v02 --refutes--> v0b\n- v0b --supports--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v00\n- v05 --supports--> v06\n- v08 --refutes--> v04\n- v0c --refutes--> v03\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":"596977a7c49f7e5a2468270e1cdddd28e27a154e99013461a8f5c8a694f7aced","schema":"livegraph.boolean-propagation.v1","spec_hash":"3d005ae36d90184f4033ff70b3834844cb326753024be7ba3f2821ca157f0b4e","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-aa5caf9df28fece335ff","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --refutes--> v0b\n- v0b --supports--> v03\n- v03 --supports--> v0c\n- v0c --supports--> v0d\n- v0d --refutes--> v08\n- v08 --refutes--> v02\n- v02 --supports--> v07\n- v07 --refutes--> v04\n- v04 --refutes--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v0b --refutes--> v00\n- v0b --refutes--> v06\n- v0c --supports--> v05\n- v0d --supports--> v0a\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":"273375a8dded6498aae20ee1df0b5d846c26e4846f247e6cf500c98c92346187","schema":"livegraph.boolean-propagation.v1","spec_hash":"aa5caf9df28fece335ff960b1354ad27834b73d49a5978bc7bed71aeee18e2a9","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-b9e323a81ca9b68179c6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v0d\n- v0d --supports--> v01\n- v01 --supports--> v00\n- v00 --supports--> v04\n- v04 --supports--> v02\n- v02 --refutes--> v0b\n- v0b --refutes--> v0a\n- v0a --supports--> v07\n- v07 --supports--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v08\n- v02 --refutes--> v0c\n- v07 --supports--> v09\n- v0a --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":"0f958b00c18e6bad11ad9502b84c3c5035558560d93e862805119b1fae85dda6","schema":"livegraph.boolean-propagation.v1","spec_hash":"b9e323a81ca9b68179c6c3a7b082709843468b9a39c5313175146bc51b842891","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-7c5920d3ddaaadafd442","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --refutes--> v0d\n- v0d --supports--> v0b\n- v0b --refutes--> v0a\n- v0a --supports--> v0c\n- v0c --supports--> v07\n- v07 --supports--> v09\n- v09 --refutes--> v05\n- v05 --refutes--> v04\n- v04 --supports--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v0b --supports--> v01\n- v0c --refutes--> v06\n- v0c --supports--> v03\n- v0d --supports--> v02\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":"f764475c86a0dc15d138164bd02a90ca4c7498ab6f869c1485e583e50eb90990","schema":"livegraph.boolean-propagation.v1","spec_hash":"7c5920d3ddaaadafd44228086e9bd21e11a7cf59a0f2f7f0fe402fcfd2b33916","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-c76c5aef7f73cff5f233","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --refutes--> v06\n- v06 --refutes--> v05\n- v05 --refutes--> v07\n- v07 --supports--> v03\n- v03 --refutes--> v04\n- v04 --refutes--> v0a\n- v0a --supports--> v01\n- v01 --supports--> v02\n- v02 --refutes--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v0d\n- v05 --refutes--> v0b\n- v06 --refutes--> v00\n- v07 --supports--> v0c\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":"8232f21ece5adad36619f25639e1e7b04f69638ca3a65f2aaf9355d6c343de6c","schema":"livegraph.boolean-propagation.v1","spec_hash":"c76c5aef7f73cff5f2333c877b7add79a869459900a4314fbcbb3103c07272e6","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":9,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-9c0c46908d6cc7fbd495","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0d --refutes--> v04\n- v04 --supports--> v03\n- v03 --supports--> v09\n- v09 --supports--> v00\n- v00 --supports--> v07\n- v07 --refutes--> v01\n- v01 --refutes--> v0a\n- v0a --refutes--> v08\n- v08 --supports--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v02\n- v07 --refutes--> v0b\n- v08 --refutes--> v0c\n- v08 --supports--> v06\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":"b9d725dbd6530f16adb5e174929a2846420d395b4a20e8bfe3d9319fc6309605","schema":"livegraph.boolean-propagation.v1","spec_hash":"9c0c46908d6cc7fbd495bf0316cae6c18eab18a43e50cca2af488444bc4cb9fa","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-7c6737ea56e41c90baa4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --refutes--> v09\n- v09 --supports--> v02\n- v02 --supports--> v03\n- v03 --refutes--> v07\n- v07 --supports--> v06\n- v06 --supports--> v08\n- v08 --refutes--> v0b\n- v0b --supports--> v05\n- v05 --supports--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v0c\n- v02 --supports--> v0d\n- v09 --refutes--> v00\n- v0b --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":"44ff7d8e0d541e6deb30dd053418f74e4efc64a3fbf472aaf29dc0eb634c0de8","schema":"livegraph.boolean-propagation.v1","spec_hash":"7c6737ea56e41c90baa47ab980d543ce0c80471ffd175c765fa3cae89e04db91","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":9,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-dce51a91c0eb9808fd19","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v0a\n- v0a --refutes--> v01\n- v01 --supports--> v02\n- v02 --refutes--> v04\n- v04 --refutes--> v0c\n- v0c --refutes--> v07\n- v07 --supports--> v0d\n- v0d --refutes--> v09\n- v09 --refutes--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v0b\n- v0a --refutes--> v03\n- v0a --refutes--> v06\n- v0d --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":"73b5787bb4cea57e8fb7a5fa59c992141c0768d1cdb8edaea6fa5aad67a5ed76","schema":"livegraph.boolean-propagation.v1","spec_hash":"dce51a91c0eb9808fd1941138108e9a19b8a26b250bb03671fb0ff44c2a4a3b6","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-2316179990173ae7dea0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v0d\n- v0d --refutes--> v07\n- v07 --refutes--> v05\n- v05 --supports--> v02\n- v02 --refutes--> v0b\n- v0b --refutes--> v00\n- v00 --supports--> v03\n- v03 --supports--> v0c\n- v0c --supports--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v09\n- v04 --supports--> v08\n- v04 --supports--> v0a\n- v0d --refutes--> v01\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":"d0422f6e3b0e20cf6502d49b124ee40d5e258af7674d0ba9f2a13786b85e9441","schema":"livegraph.boolean-propagation.v1","spec_hash":"2316179990173ae7dea06a0177093e7e88734bb28acd2df89d8f0f2aa6670bec","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-74623583595aa9d6c789","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --supports--> v01\n- v01 --refutes--> v0c\n- v0c --refutes--> v0a\n- v0a --supports--> v02\n- v02 --supports--> v0d\n- v0d --refutes--> v0b\n- v0b --refutes--> v00\n- v00 --supports--> v06\n- v06 --refutes--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v07\n- v06 --refutes--> v04\n- v0c --supports--> v03\n- v0d --refutes--> v05\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":"c7d60921be9c7d9822dd4a3aee2647de30b4b2c1209f64d95c851a2bd245dae0","schema":"livegraph.boolean-propagation.v1","spec_hash":"74623583595aa9d6c7892069528d199f80cb5fbd38919efe43a2280bf70b6ec5","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-67c92ed3c24db3c9757d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --supports--> v0d\n- v0d --supports--> v07\n- v07 --refutes--> v0a\n- v0a --refutes--> v0b\n- v0b --supports--> v03\n- v03 --supports--> v01\n- v01 --refutes--> v00\n- v00 --supports--> v08\n- v08 --supports--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v06\n- v02 --supports--> v0c\n- v03 --supports--> v05\n- v07 --refutes--> 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":"4dbd252c9aba5f59bdfa1c68dde21361e1b37c00ccd5c22b98c4714cbae9ac94","schema":"livegraph.boolean-propagation.v1","spec_hash":"67c92ed3c24db3c9757d1c7c37f9bc3478bcccc80e9dac34ae0f72e67687993d","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-e90c4bb3c19401847e05","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --refutes--> v02\n- v02 --supports--> v0b\n- v0b --refutes--> v06\n- v06 --supports--> v00\n- v00 --supports--> v04\n- v04 --supports--> v0a\n- v0a --supports--> v07\n- v07 --refutes--> v09\n- v09 --supports--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v08\n- v09 --refutes--> v0d\n- v0a --refutes--> v01\n- v0a --supports--> v03\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":"a0cddb87cd9b89bd0070c90a4b020e4c8deda87e0eb70732c3ff2cbeaf9991ee","schema":"livegraph.boolean-propagation.v1","spec_hash":"e90c4bb3c19401847e0569a736fbb4b12efa7430697a5e1437d876615042f5da","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-7f1a1f776790318976ac","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --refutes--> v07\n- v07 --supports--> v0b\n- v0b --refutes--> v01\n- v01 --supports--> v03\n- v03 --refutes--> v06\n- v06 --supports--> v08\n- v08 --supports--> v0c\n- v0c --refutes--> v0d\n- v0d --supports--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v08 --refutes--> v00\n- v0a --supports--> v09\n- v0b --supports--> v05\n- v0d --supports--> 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":"79e8b05326af3b7590eb57f46a86e07c23d2f1e871499a9014dbc696dbdf9efd","schema":"livegraph.boolean-propagation.v1","spec_hash":"7f1a1f776790318976ac7ae97fe395dba588953fff5ea0467981ff82c5036e11","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-00cedb0428d18548a8ee","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --refutes--> v05\n- v05 --refutes--> v09\n- v09 --supports--> v0a\n- v0a --supports--> v00\n- v00 --refutes--> v06\n- v06 --refutes--> v0b\n- v0b --supports--> v01\n- v01 --supports--> v0d\n- v0d --refutes--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v03\n- v06 --refutes--> v04\n- v0a --supports--> v02\n- v0a --supports--> v08\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":"3dea37124d93342270b5008d44c548e062f00a4ecf19497b577643ecb92ee562","schema":"livegraph.boolean-propagation.v1","spec_hash":"00cedb0428d18548a8eed30361e6806ac0c45030889419513db16666313e88a9","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-8dcb4bedfad8b5c55d85","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --refutes--> v06\n- v06 --supports--> v00\n- v00 --supports--> v0d\n- v0d --supports--> v0a\n- v0a --refutes--> v01\n- v01 --refutes--> v05\n- v05 --refutes--> v04\n- v04 --refutes--> v02\n- v02 --supports--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v0b\n- v0a --refutes--> v09\n- v0c --supports--> v08\n- v0d --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":"1ac6b4a4d80fa994adf7d72d404271cbaa67a917874b0a4b3572bf690776c7d7","schema":"livegraph.boolean-propagation.v1","spec_hash":"8dcb4bedfad8b5c55d859fff12e1eeaed2d726a756ed8cf3da251babfa256d40","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"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\"}],\"hop\":9,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-7dde3af1a2e51090a203","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0d --supports--> v0c\n- v0c --refutes--> v05\n- v05 --refutes--> v07\n- v07 --refutes--> v09\n- v09 --refutes--> v01\n- v01 --supports--> v03\n- v03 --refutes--> v06\n- v06 --supports--> v0a\n- v0a --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v02\n- v05 --supports--> v0b\n- v07 --supports--> v08\n- v0a --supports--> v04\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":"5fd07cde8bd68ecd6fca7ea5703667873d3f942270c9d78d45f23a0f63e76a5e","schema":"livegraph.boolean-propagation.v1","spec_hash":"7dde3af1a2e51090a203f99bb5dc0ef996d6f9a5e618aa31db726fcc3eced563","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-274ed83fd7e1c72dae7a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --refutes--> v07\n- v07 --refutes--> v0b\n- v0b --supports--> v00\n- v00 --refutes--> v0c\n- v0c --supports--> v05\n- v05 --refutes--> v03\n- v03 --refutes--> v04\n- v04 --supports--> v09\n- v09 --refutes--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v08\n- v07 --refutes--> v06\n- v09 --supports--> v01\n- v0b --refutes--> v0d\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":"8db64e7ad954cff6642a73030325e02bba638fceac6c68778547225420d2cd17","schema":"livegraph.boolean-propagation.v1","spec_hash":"274ed83fd7e1c72dae7a0ec246c1db2f2e27acccf73502a8910892f1dd79f993","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-fd42a4f28d8ac6daf7f4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --supports--> v00\n- v00 --supports--> v0b\n- v0b --supports--> v0d\n- v0d --supports--> v06\n- v06 --refutes--> v01\n- v01 --supports--> v02\n- v02 --supports--> v09\n- v09 --refutes--> v05\n- v05 --refutes--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v03\n- v06 --refutes--> v0c\n- v09 --refutes--> v04\n- v0b --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":"c3d19ce396d74cd3c3949f34b58c5eda92ce83ccff0b3c7367391298bbcef205","schema":"livegraph.boolean-propagation.v1","spec_hash":"fd42a4f28d8ac6daf7f4c888abbf50975a1d8b6d03c4d347105c803030fcea8a","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-aa9d309d220e6c1fe6a5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --supports--> v0c\n- v0c --supports--> v01\n- v01 --refutes--> v0b\n- v0b --supports--> v03\n- v03 --refutes--> v0a\n- v0a --supports--> v08\n- v08 --refutes--> v06\n- v06 --refutes--> v02\n- v02 --refutes--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v04\n- v03 --supports--> v0d\n- v08 --supports--> v00\n- v0b --supports--> v07\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":"9428116771ef69350855b868e7873238e99f298138cfac4b34101c8bd1898ab1","schema":"livegraph.boolean-propagation.v1","spec_hash":"aa9d309d220e6c1fe6a5ba702e0409b183fcececde34cb1dbf2ff00a38847247","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-17521ec295c86527e515","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --refutes--> v04\n- v04 --supports--> v00\n- v00 --supports--> v0a\n- v0a --refutes--> v03\n- v03 --supports--> v09\n- v09 --supports--> v02\n- v02 --refutes--> v07\n- v07 --supports--> v08\n- v08 --refutes--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v0d\n- v04 --refutes--> v0c\n- v07 --refutes--> v06\n- v0a --supports--> v0b\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":"ebcfdb0fd54ac8e8482d85037970c958ecda3f74eea7bc001c504472df86aa57","schema":"livegraph.boolean-propagation.v1","spec_hash":"17521ec295c86527e515383a2570cf1d74571128680895c0b3a654c898ddaa72","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":9,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-359a4c0c1dcde55fed6e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0d --refutes--> v00\n- v00 --supports--> v01\n- v01 --refutes--> v02\n- v02 --supports--> v06\n- v06 --refutes--> v08\n- v08 --refutes--> v07\n- v07 --supports--> v04\n- v04 --refutes--> v09\n- v09 --refutes--> v0c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v03\n- v08 --refutes--> v05\n- v08 --supports--> v0b\n- v09 --supports--> v0a\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":"d2f2662a86ceeacd05345bc7abd1e383466e38a6c4bc18f43013d12d0dbada7b","schema":"livegraph.boolean-propagation.v1","spec_hash":"359a4c0c1dcde55fed6ef07b72cf2725f5f9de31e725484bc11a140f43730f2e","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-736a2e7b20b694a05c43","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --supports--> v04\n- v04 --supports--> v0b\n- v0b --supports--> v06\n- v06 --refutes--> v03\n- v03 --supports--> v01\n- v01 --refutes--> v0d\n- v0d --refutes--> v08\n- v08 --supports--> v07\n- v07 --refutes--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v09\n- v05 --supports--> v00\n- v07 --refutes--> v0c\n- v07 --supports--> 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":"47e6c9ed86324f021eafd4a647bd9f4ec97acc614456e2a6b7177ecbe9fa783b","schema":"livegraph.boolean-propagation.v1","spec_hash":"736a2e7b20b694a05c439aa90861e25b816394773e7e76459b217168cf809869","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-38d50d6bbd1a5f14c122","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v03\n- v03 --supports--> v05\n- v05 --supports--> v04\n- v04 --refutes--> v02\n- v02 --supports--> v0d\n- v0d --supports--> v0c\n- v0c --refutes--> v08\n- v08 --refutes--> v09\n- v09 --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v01\n- v03 --refutes--> v07\n- v03 --refutes--> v0b\n- v06 --refutes--> v0a\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":"cecc5736e4778e80913fdbc3d2ccfdb2cc5edcf9eed288d55008a15b321de40f","schema":"livegraph.boolean-propagation.v1","spec_hash":"38d50d6bbd1a5f14c12254dc9637e58e4ce4bab7aed0805a4bc6e365fb052249","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-792a88f32947f8e54fa6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --supports--> v0b\n- v0b --supports--> v00\n- v00 --supports--> v02\n- v02 --supports--> v06\n- v06 --supports--> v08\n- v08 --refutes--> v04\n- v04 --supports--> v03\n- v03 --refutes--> v05\n- v05 --refutes--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v09\n- v05 --supports--> v0a\n- v06 --refutes--> v0c\n- v06 --supports--> v0d\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":"09d16946dbf42b3bd20a483537411c51d745af2d382b81eb7406e1c01a777b16","schema":"livegraph.boolean-propagation.v1","spec_hash":"792a88f32947f8e54fa6ca8e8320d8ac86626d9d52770343c05e7f2fdf704f60","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":9,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-91a2a144bff9f5b71e0e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --supports--> v09\n- v09 --refutes--> v01\n- v01 --refutes--> v07\n- v07 --refutes--> v02\n- v02 --refutes--> v06\n- v06 --supports--> v0a\n- v0a --supports--> v04\n- v04 --supports--> v05\n- v05 --supports--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v0d\n- v07 --refutes--> v0b\n- v08 --refutes--> v0c\n- v0a --supports--> 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":"2eef3040d34b7fd8038a76d41557e1598a59d7f24c2eb57f654a8f383a8550cf","schema":"livegraph.boolean-propagation.v1","spec_hash":"91a2a144bff9f5b71e0e3cdd7cbf784c6c13b3685cf096039c2309922187cf70","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":9,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-c04fa744857029a816ca","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --refutes--> v05\n- v05 --supports--> v01\n- v01 --supports--> v0c\n- v0c --supports--> v0a\n- v0a --supports--> v06\n- v06 --refutes--> v0d\n- v0d --supports--> v0b\n- v0b --refutes--> v00\n- v00 --supports--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v04\n- v05 --refutes--> v03\n- v05 --supports--> v09\n- v08 --refutes--> v07\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":"568c85cc670aa2a8e135bbfa621a9271b0422060ccb49766363d3fd1bd41619f","schema":"livegraph.boolean-propagation.v1","spec_hash":"c04fa744857029a816ca8d53b6369920b3eac7f35d91eba37184ff66da1827d4","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-4fb7b0a4b1455571185d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --supports--> v03\n- v03 --refutes--> v07\n- v07 --supports--> v09\n- v09 --refutes--> v02\n- v02 --refutes--> v0b\n- v0b --supports--> v01\n- v01 --supports--> v04\n- v04 --refutes--> v0c\n- v0c --supports--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v05\n- v03 --supports--> v08\n- v0a --refutes--> v0d\n- v0c --supports--> v06\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":"0f10b91f41657f4803c65ec0ab67c651234d2068317c7a9b4b021fbc5b95304b","schema":"livegraph.boolean-propagation.v1","spec_hash":"4fb7b0a4b1455571185dd4641c11c04b416ced3c7b292bfa069fb63201bf6b44","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-11a85397968c7f11af88","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --refutes--> v06\n- v06 --refutes--> v09\n- v09 --supports--> v07\n- v07 --refutes--> v0b\n- v0b --supports--> v04\n- v04 --supports--> v00\n- v00 --supports--> v0a\n- v0a --refutes--> v03\n- v03 --refutes--> v0c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v07 --supports--> v05\n- v0a --refutes--> v02\n- v0a --refutes--> v0d\n- v0b --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":"752e25f60c7800f9fd9784b14895adacbd8476c50ed355024eae499b8828c77d","schema":"livegraph.boolean-propagation.v1","spec_hash":"11a85397968c7f11af8815d05afbe5b866499439e7e170466dfa8a5a49e04c0c","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":9,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-b4ebd446902a17605a6f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v0d\n- v0d --supports--> v03\n- v03 --refutes--> v05\n- v05 --refutes--> v02\n- v02 --refutes--> v0c\n- v0c --supports--> v0a\n- v0a --supports--> v08\n- v08 --refutes--> v01\n- v01 --supports--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v06\n- v03 --supports--> v09\n- v08 --supports--> v07\n- v0c --refutes--> v00\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":"5da9f76fb1d936b1a85fc9edc6eb6892c8b40a2a48a6be08ee38fb03ca2837a4","schema":"livegraph.boolean-propagation.v1","spec_hash":"b4ebd446902a17605a6f0ea3fc76339d311468ccd2f38fa334085dfd00f3fb48","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-03a6043031bf6b942af8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --refutes--> v0b\n- v0b --refutes--> v04\n- v04 --refutes--> v02\n- v02 --refutes--> v03\n- v03 --supports--> v0d\n- v0d --refutes--> v00\n- v00 --supports--> v09\n- v09 --refutes--> v0c\n- v0c --refutes--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v08\n- v02 --supports--> v07\n- v0a --supports--> v06\n- v0c --refutes--> v01\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":"adf8bdc72b93c49ee56b927df325eaf086a316e82e0fb5550dfe121b6c1f5c84","schema":"livegraph.boolean-propagation.v1","spec_hash":"03a6043031bf6b942af826892b65d3a7060996f4f6bb358a6adf8f3fdc9859ed","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-37f535d68feb94063de4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --refutes--> v01\n- v01 --refutes--> v00\n- v00 --supports--> v02\n- v02 --refutes--> v0d\n- v0d --refutes--> v0b\n- v0b --refutes--> v0c\n- v0c --supports--> v06\n- v06 --supports--> v09\n- v09 --refutes--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v0a\n- v09 --refutes--> v04\n- v09 --supports--> v05\n- v0c --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":"f605087ac733fe83ef9adf26ef39141382dbf55e49ea3e18d78c0f214d9c6fab","schema":"livegraph.boolean-propagation.v1","spec_hash":"37f535d68feb94063de4bb0914340d7ecce015049d9029b9a4db44d15b94ac12","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-d40c1fc396fb0504f611","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --refutes--> v0a\n- v0a --supports--> v0b\n- v0b --supports--> v06\n- v06 --supports--> v03\n- v03 --supports--> v04\n- v04 --supports--> v09\n- v09 --supports--> v0c\n- v0c --supports--> v05\n- v05 --refutes--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v00\n- v04 --supports--> v07\n- v0a --refutes--> v08\n- v0b --supports--> v0d\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":"b586e004048feb9ac829bf5c1b2ea3d79ddde410311ee3dca0ddf9b683aa7471","schema":"livegraph.boolean-propagation.v1","spec_hash":"d40c1fc396fb0504f611be350a31784c2bcf4352a00af9edf8230571843067b0","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-caf7bc09169a477ec719","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v02\n- v02 --refutes--> v01\n- v01 --supports--> v08\n- v08 --supports--> v0a\n- v0a --refutes--> v09\n- v09 --refutes--> v00\n- v00 --supports--> v0c\n- v0c --supports--> v06\n- v06 --supports--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v07\n- v06 --refutes--> v05\n- v06 --supports--> v0d\n- v08 --supports--> v0b\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":"3cf128df2edc251bcd56cb9c9bf6207490c06da99c626fbf57fef7d458b45b01","schema":"livegraph.boolean-propagation.v1","spec_hash":"caf7bc09169a477ec71982ea1c77c9aa1b11078660b137648c28bca305ab5be6","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-2fcbcf05c34b379f7052","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --supports--> v0d\n- v0d --supports--> v02\n- v02 --refutes--> v00\n- v00 --supports--> v0b\n- v0b --supports--> v04\n- v04 --supports--> v03\n- v03 --supports--> v0c\n- v0c --refutes--> v06\n- v06 --refutes--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v01\n- v06 --refutes--> v05\n- v06 --refutes--> v09\n- v08 --refutes--> v07\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":"2fda963dcf8747c9b7beeb85cf8bc41137d9db648da24dc2f0be9bf06f864cb9","schema":"livegraph.boolean-propagation.v1","spec_hash":"2fcbcf05c34b379f7052914c1f4eda85226e927f69c560b9368584dc790cb50b","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-589ede4d827d464d3ed4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --refutes--> v07\n- v07 --supports--> v05\n- v05 --supports--> v0b\n- v0b --supports--> v02\n- v02 --supports--> v06\n- v06 --supports--> v09\n- v09 --refutes--> v03\n- v03 --refutes--> v0c\n- v0c --supports--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v0a\n- v09 --refutes--> v04\n- v0b --supports--> v0d\n- v0c --supports--> v08\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":"a79dd4b19baa068c1b386299a60528f49d536bd2f1c1b58a21b1f98520be404e","schema":"livegraph.boolean-propagation.v1","spec_hash":"589ede4d827d464d3ed4d63b86edb1b91b0932d88db23df03ae5e6879923fee7","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-6272427889b681114ddd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v04\n- v04 --supports--> v09\n- v09 --supports--> v0c\n- v0c --refutes--> v05\n- v05 --refutes--> v00\n- v00 --refutes--> v08\n- v08 --supports--> v01\n- v01 --supports--> v0b\n- v0b --supports--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v03\n- v04 --refutes--> v02\n- v05 --supports--> v07\n- v0b --supports--> v0d\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":"1f8c8270b6fa75996fafd3cb0d2e2851ab005a7eb8b0cb4c1e424e96c4fbeb9b","schema":"livegraph.boolean-propagation.v1","spec_hash":"6272427889b681114ddd367d5faaa5538a15a25a28e9ca26070888b9c354deb6","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-47848365b9d0ab7a2f4f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v0c\n- v0c --refutes--> v02\n- v02 --supports--> v0d\n- v0d --refutes--> v06\n- v06 --refutes--> v08\n- v08 --supports--> v03\n- v03 --refutes--> v0b\n- v0b --supports--> v07\n- v07 --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v01\n- v04 --refutes--> v05\n- v08 --supports--> v09\n- v0b --supports--> 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":"34d89a4164da73c46f8f486460c41e43e48965e6687f7a9cda81e6e98926dbb6","schema":"livegraph.boolean-propagation.v1","spec_hash":"47848365b9d0ab7a2f4f59b05eeb49222f4a85433f31ae88dd18758f6ec752ec","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-802c42b2b21e8c87bc74","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v08\n- v08 --refutes--> v0c\n- v0c --supports--> v07\n- v07 --supports--> v09\n- v09 --supports--> v0b\n- v0b --supports--> v03\n- v03 --refutes--> v02\n- v02 --refutes--> v01\n- v01 --supports--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0a\n- v01 --refutes--> v06\n- v07 --refutes--> v05\n- v08 --refutes--> v0d\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":"7fb1d5370e95f020cbab7b1d53987b18969f596ad94720a539bc721266d95d10","schema":"livegraph.boolean-propagation.v1","spec_hash":"802c42b2b21e8c87bc742057b5eba7445188a039a5a29632d14495ae7f70d55c","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-a7bffb4ea5ac5d65677f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --supports--> v07\n- v07 --refutes--> v0d\n- v0d --supports--> v00\n- v00 --refutes--> v02\n- v02 --refutes--> v03\n- v03 --refutes--> v09\n- v09 --supports--> v01\n- v01 --supports--> v0b\n- v0b --supports--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v08\n- v03 --supports--> v04\n- v03 --supports--> v0a\n- v0b --refutes--> v05\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":"e124c69b2429128d9cecd59cd821265841e9d92340bccf4261ce68e543cef3eb","schema":"livegraph.boolean-propagation.v1","spec_hash":"a7bffb4ea5ac5d65677ff5e562aa0f0f9f502354009c121d213c0ca4a580c410","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-23726d52afa5c65cb3f6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --refutes--> v0b\n- v0b --supports--> v05\n- v05 --supports--> v01\n- v01 --refutes--> v0a\n- v0a --supports--> v03\n- v03 --refutes--> v08\n- v08 --supports--> v06\n- v06 --supports--> v02\n- v02 --refutes--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v00\n- v02 --supports--> v0c\n- v05 --refutes--> v0d\n- v07 --supports--> 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":"7ff216d08e5c0abbe26e60d3d4a311650059cf1f73199f1d4d390cb48a1463e9","schema":"livegraph.boolean-propagation.v1","spec_hash":"23726d52afa5c65cb3f62b4174b9e4ae4c96a0925973d2b68afffcafbb4e2e45","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":9,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-fbd3914e006cce85e7b6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --supports--> v06\n- v06 --supports--> v04\n- v04 --supports--> v03\n- v03 --refutes--> v09\n- v09 --refutes--> v02\n- v02 --refutes--> v08\n- v08 --refutes--> v01\n- v01 --supports--> v0a\n- v0a --refutes--> v0c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v00\n- v08 --supports--> v0b\n- v09 --refutes--> v0d\n- v0a --supports--> v05\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":"425ee301b4fa8423083dc0604cfb25fbf1ff5cae0200d319d5204c64b5743972","schema":"livegraph.boolean-propagation.v1","spec_hash":"fbd3914e006cce85e7b6ae007292196eb8a937b6f1c40e3325cea7df03fa01d1","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"}],\"hop\":9,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-848655e43385425adffd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --refutes--> v02\n- v02 --supports--> v04\n- v04 --supports--> v0d\n- v0d --refutes--> v0b\n- v0b --refutes--> v0c\n- v0c --supports--> v03\n- v03 --supports--> v05\n- v05 --refutes--> v0a\n- v0a --refutes--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v06\n- v0a --refutes--> v00\n- v0b --supports--> v08\n- v0c --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":"1de11b2f657d158618ed2d1eddcc74d11001da9ba4b84e0a9576622926336c86","schema":"livegraph.boolean-propagation.v1","spec_hash":"848655e43385425adffd665c68ca8773c49e490a6c8a45f81c72cb5332a0965e","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-a249072811c6c24db848","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --supports--> v09\n- v09 --refutes--> v0c\n- v0c --refutes--> v0d\n- v0d --refutes--> v05\n- v05 --supports--> v0b\n- v0b --supports--> v04\n- v04 --refutes--> v02\n- v02 --supports--> v06\n- v06 --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v08\n- v02 --supports--> v07\n- v09 --supports--> v01\n- v0b --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":"7e0f54a3eb996628395c02560628dfac1a72d5f0de798ec140ec03f7519d4dfb","schema":"livegraph.boolean-propagation.v1","spec_hash":"a249072811c6c24db848ad4df0f4f5d88598096a00defee713830f2161abb61d","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-0ba46b6898d58103a943","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --supports--> v0d\n- v0d --refutes--> v00\n- v00 --refutes--> v08\n- v08 --supports--> v0b\n- v0b --refutes--> v07\n- v07 --refutes--> v06\n- v06 --supports--> v0c\n- v0c --supports--> v09\n- v09 --supports--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0a\n- v02 --refutes--> v01\n- v06 --supports--> v03\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":"724e71ee61b78ca9c07a4baa11fa80c26408d9c4ee652d1fb1718a4b2367569b","schema":"livegraph.boolean-propagation.v1","spec_hash":"0ba46b6898d58103a9434a5c3eb02f70178857f6dea3232e9f7200188f0e8280","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-524cac051a519edda8c6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --refutes--> v06\n- v06 --supports--> v00\n- v00 --supports--> v01\n- v01 --refutes--> v02\n- v02 --refutes--> v09\n- v09 --supports--> v08\n- v08 --supports--> v0a\n- v0a --refutes--> v0b\n- v0b --supports--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v04\n- v05 --refutes--> v0c\n- v08 --refutes--> v07\n- v08 --supports--> v03\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":"791ba25e82593c04cd53fd9c3607fee5b1bbf5b2e003beb944d66bc8f82efe72","schema":"livegraph.boolean-propagation.v1","spec_hash":"524cac051a519edda8c65ba7a926a76a5defc719cf21964e60b996298d642335","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":9,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-0b24d67fa757955ee3b8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --refutes--> v06\n- v06 --supports--> v04\n- v04 --supports--> v07\n- v07 --supports--> v01\n- v01 --refutes--> v0b\n- v0b --refutes--> v08\n- v08 --supports--> v0d\n- v0d --refutes--> v0a\n- v0a --refutes--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v09\n- v00 --supports--> v0c\n- v04 --refutes--> v02\n- v07 --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":"bb3c5845830dd172a9cfbf30c131888d4f6f872511e53c807d621acf9a147e49","schema":"livegraph.boolean-propagation.v1","spec_hash":"0b24d67fa757955ee3b8491a8f0548c2712a8656ac0f155c75a8be35d8028a51","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-2aedf095cecea73a8f20","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --refutes--> v03\n- v03 --supports--> v0c\n- v0c --refutes--> v04\n- v04 --refutes--> v0a\n- v0a --supports--> v09\n- v09 --refutes--> v07\n- v07 --supports--> v08\n- v08 --supports--> v0b\n- v0b --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v05\n- v08 --supports--> v06\n- v0a --refutes--> v0d\n- v0c --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":"e64491bc8fc50a1c2cc68f8e5d70ec41bc2f42c09082d2d5082323a08f3ebb04","schema":"livegraph.boolean-propagation.v1","spec_hash":"2aedf095cecea73a8f20b2790e64e064cfb85b1579481cab9058ff1877941377","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-f8dd8f06244972644312","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --refutes--> v03\n- v03 --refutes--> v0d\n- v0d --supports--> v06\n- v06 --supports--> v00\n- v00 --refutes--> v0c\n- v0c --refutes--> v09\n- v09 --supports--> v05\n- v05 --supports--> v07\n- v07 --refutes--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v04\n- v03 --supports--> v01\n- v05 --refutes--> v08\n- v07 --supports--> v0b\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":"d801acbc30850a7d2c27c454f2d65ddda282045126ceb3eefcdcb41f8ea98893","schema":"livegraph.boolean-propagation.v1","spec_hash":"f8dd8f06244972644312c47d11d288d453ad05c834ec9f12b8f78f1b8139116d","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-2aea5c2802d11dd94d6c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --refutes--> v02\n- v02 --refutes--> v0c\n- v0c --supports--> v08\n- v08 --supports--> v04\n- v04 --supports--> v0b\n- v0b --refutes--> v03\n- v03 --refutes--> v01\n- v01 --supports--> v07\n- v07 --supports--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v0a\n- v07 --refutes--> v0d\n- v08 --refutes--> v09\n- v0b --refutes--> 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":"236fd881e1e08f46c1518c46c3939b14a03c44df08bfbf5fca5b8287a6aee0a7","schema":"livegraph.boolean-propagation.v1","spec_hash":"2aea5c2802d11dd94d6c828e087e168a016479b54f18a871925e41e0373c33c8","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-0a523c1ff5d2668d7d38","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --supports--> v04\n- v04 --supports--> v08\n- v08 --supports--> v0c\n- v0c --supports--> v01\n- v01 --supports--> v0a\n- v0a --refutes--> v02\n- v02 --supports--> v06\n- v06 --supports--> v00\n- v00 --supports--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0d\n- v00 --supports--> v07\n- v0a --supports--> v09\n- v0c --refutes--> v0b\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":"0e57aed156485df17116b48fa94cc2fe42dc21e53b368369968c18eeabd9508b","schema":"livegraph.boolean-propagation.v1","spec_hash":"0a523c1ff5d2668d7d384da5949f80138219c1761fbc92f2a55c860532ca6844","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-d8dc797b6635acd1cc69","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --refutes--> v0c\n- v0c --supports--> v04\n- v04 --refutes--> v02\n- v02 --refutes--> v07\n- v07 --supports--> v0b\n- v0b --supports--> v0d\n- v0d --supports--> v05\n- v05 --refutes--> v00\n- v00 --supports--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v03\n- v05 --supports--> v0a\n- v08 --refutes--> v09\n- v0b --supports--> v06\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":"c27044f51b6692c8b4e6f5a06cd1f606b52540ea6eaba382eb0cab99a2587fe6","schema":"livegraph.boolean-propagation.v1","spec_hash":"d8dc797b6635acd1cc694a99d09d43db09539a550b24f3410c8b08569829168f","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-c8338e684819ec55f912","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --supports--> v07\n- v07 --refutes--> v04\n- v04 --refutes--> v09\n- v09 --supports--> v06\n- v06 --refutes--> v0b\n- v0b --refutes--> v01\n- v01 --refutes--> v00\n- v00 --refutes--> v0c\n- v0c --refutes--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v08\n- v04 --supports--> v0a\n- v07 --supports--> v02\n- v0c --refutes--> v05\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":"f674f7898729f830ff774c3ed6a5eb75674f9293274f1f2bd4d5eb71a164f62b","schema":"livegraph.boolean-propagation.v1","spec_hash":"c8338e684819ec55f9122ca8ba381c4d60b32b1d44127d11a90eeddf4310915e","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-e1b34303f9d927fad052","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --refutes--> v01\n- v01 --refutes--> v0b\n- v0b --refutes--> v02\n- v02 --supports--> v09\n- v09 --refutes--> v0a\n- v0a --supports--> v0d\n- v0d --refutes--> v04\n- v04 --refutes--> v00\n- v00 --supports--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v03\n- v08 --refutes--> v0c\n- v09 --refutes--> v07\n- v0d --refutes--> v05\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":"ae06ae1a8e19af2b37606a5a5e1a11f197f83b6498f73de3ff994abc383bd499","schema":"livegraph.boolean-propagation.v1","spec_hash":"e1b34303f9d927fad052a72ae6a9cc0525ba8b1410c0b099b5f7d48f60526904","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-f82a5d001cb32a876289","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --supports--> v0a\n- v0a --refutes--> v0d\n- v0d --refutes--> v08\n- v08 --supports--> v06\n- v06 --refutes--> v05\n- v05 --refutes--> v04\n- v04 --supports--> v01\n- v01 --supports--> v00\n- v00 --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v0c\n- v01 --refutes--> v07\n- v05 --refutes--> v0b\n- v0a --supports--> 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":"307e4b1cecfc57d94acff33523899895012afab286ea206bb815905be723fd2f","schema":"livegraph.boolean-propagation.v1","spec_hash":"f82a5d001cb32a87628939025f57bb914a67cbe55b40438382c21fdd99bc53b9","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-a0ef7b94f37a8eb09d28","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v02\n- v02 --refutes--> v03\n- v03 --supports--> v0c\n- v0c --supports--> v07\n- v07 --refutes--> v0b\n- v0b --supports--> v08\n- v08 --refutes--> v0a\n- v0a --refutes--> v09\n- v09 --supports--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v06\n- v02 --supports--> v04\n- v03 --supports--> v01\n- v0c --supports--> 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":"26a87777940d92a29cbfe8ea6adda2a48e37d9308771dc71aa2c2ff4fe4d83a2","schema":"livegraph.boolean-propagation.v1","spec_hash":"a0ef7b94f37a8eb09d284e630a5bd6f23ed3311e6fadc96dae5e6239c2b2bfb2","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-801d209ff0c6821d1ce5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --supports--> v0b\n- v0b --refutes--> v03\n- v03 --supports--> v07\n- v07 --supports--> v0d\n- v0d --refutes--> v0c\n- v0c --supports--> v02\n- v02 --refutes--> v04\n- v04 --supports--> v0a\n- v0a --refutes--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v06\n- v03 --refutes--> v01\n- v07 --supports--> v09\n- v0b --supports--> v00\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":"1314c56671533538d3821ff5e66026e82c2ab5deb3963c09a6bf6d8719b4ae3c","schema":"livegraph.boolean-propagation.v1","spec_hash":"801d209ff0c6821d1ce544d8024f67dbcb7f8559edcac1d5bebbd4fb96b696d3","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-0251a1e745d74a019342","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --refutes--> v05\n- v05 --supports--> v00\n- v00 --refutes--> v03\n- v03 --supports--> v0d\n- v0d --supports--> v02\n- v02 --supports--> v01\n- v01 --supports--> v07\n- v07 --supports--> v09\n- v09 --supports--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v0a\n- v02 --supports--> v06\n- v07 --supports--> v04\n- v09 --refutes--> v0b\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":"8949558eea1b7e59303e2abceb231bf0535633eb573c21d25fd43cc9f236952d","schema":"livegraph.boolean-propagation.v1","spec_hash":"0251a1e745d74a019342c9c29b691afceb968e40e11c29418ee4359699797a3b","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-2789fc7ffd38248698fa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --refutes--> v0c\n- v0c --refutes--> v07\n- v07 --refutes--> v01\n- v01 --supports--> v0b\n- v0b --supports--> v02\n- v02 --refutes--> v0a\n- v0a --supports--> v03\n- v03 --supports--> v05\n- v05 --supports--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v06\n- v02 --refutes--> v08\n- v07 --refutes--> v04\n- v0c --refutes--> v09\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":"4efff2bb15633fd7f0e1126fa5814287013fd1a1f1c76ea5324601da8b1ddee2","schema":"livegraph.boolean-propagation.v1","spec_hash":"2789fc7ffd38248698fa465c04bf0c46969c93fb4e61718f654d5d4d1ee62f5f","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-c97f2d9ee773d5bbff41","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --supports--> v03\n- v03 --refutes--> v0b\n- v0b --supports--> v00\n- v00 --supports--> v02\n- v02 --supports--> v0a\n- v0a --supports--> v06\n- v06 --refutes--> v0d\n- v0d --supports--> v09\n- v09 --refutes--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v0c\n- v02 --refutes--> v07\n- v09 --refutes--> v05\n- v0b --supports--> 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":"80d3143ac16f34cad595a79f4ed4fec62d9c7ff2487a91efb3824421e1a1f133","schema":"livegraph.boolean-propagation.v1","spec_hash":"c97f2d9ee773d5bbff41e8f5c72ae2d6da6ce111ae2eebc1acbfa28f54dcd208","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-01fd562603f62bfd9864","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --supports--> v05\n- v05 --supports--> v06\n- v06 --supports--> v0c\n- v0c --refutes--> v0a\n- v0a --refutes--> v00\n- v00 --refutes--> v0b\n- v0b --refutes--> v0d\n- v0d --refutes--> v04\n- v04 --refutes--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v03\n- v05 --supports--> v08\n- v0b --refutes--> v07\n- v0c --supports--> v09\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":"43c16f67b0af7e627a58af74e7ee10569759c608e01690dac1d746d18b64fae9","schema":"livegraph.boolean-propagation.v1","spec_hash":"01fd562603f62bfd98641b2bd2d007b414edce7bff1c171a3421d4c639e31a6c","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-f5cf86c0642d1bfbe011","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --refutes--> v0b\n- v0b --refutes--> v0a\n- v0a --supports--> v09\n- v09 --refutes--> v03\n- v03 --refutes--> v01\n- v01 --refutes--> v0c\n- v0c --refutes--> v08\n- v08 --supports--> v07\n- v07 --refutes--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v0d\n- v03 --supports--> v00\n- v05 --refutes--> v06\n- v0b --supports--> 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":"50c5e4cbc16b5b7f28f8a01cda9ef629315ee0f67c23910d973c7347594056da","schema":"livegraph.boolean-propagation.v1","spec_hash":"f5cf86c0642d1bfbe01165a1524bfaceca4be3973ce9e3979fc8071e9f0245ba","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-37de77f80903d36415f0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --supports--> v02\n- v02 --supports--> v07\n- v07 --refutes--> v06\n- v06 --supports--> v09\n- v09 --supports--> v0a\n- v0a --supports--> v0c\n- v0c --supports--> v08\n- v08 --supports--> v03\n- v03 --supports--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v07 --refutes--> v04\n- v09 --refutes--> v01\n- v0a --supports--> v0b\n- v0c --refutes--> v00\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":"d3f9d56433ee0758148292c07b098be87bf3836db3776953d399fe764420a9d6","schema":"livegraph.boolean-propagation.v1","spec_hash":"37de77f80903d36415f0ed737ae9f766ca4ba714c9edb7f0e5a2b31d5bd48667","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":9,"item_id":"lggrpo1-fbcf40542ec790ca2153","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --refutes--> v0c\n- v0c --supports--> v02\n- v02 --supports--> v0a\n- v0a --supports--> v06\n- v06 --refutes--> v0d\n- v0d --refutes--> v01\n- v01 --refutes--> v00\n- v00 --refutes--> v0b\n- v0b --refutes--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v04\n- v01 --refutes--> v08\n- v02 --refutes--> v07\n- v06 --refutes--> v09\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":"8c3b4fadbf61feea5b684333b8849aa0a21ef942cac22976616bc98933a44348","schema":"livegraph.boolean-propagation.v1","spec_hash":"fbcf40542ec790ca2153c1438d688121c166c2794fa6c563e80c3d97fd165975","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":9,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":9,"item_id":"lggrpo1-9e8f701430c1101be7b3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --refutes--> v0a\n- v0a --refutes--> v02\n- v02 --refutes--> v00\n- v00 --supports--> v08\n- v08 --refutes--> v05\n- v05 --supports--> v01\n- v01 --supports--> v07\n- v07 --supports--> v04\n- v04 --refutes--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v03\n- v00 --supports--> v09\n- v07 --refutes--> v0b\n- v0a --supports--> v06\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":"60755f4b204c9eed172291abbebd8a7d5cade46b74263703c19477c34f773195","schema":"livegraph.boolean-propagation.v1","spec_hash":"9e8f701430c1101be7b3af6738f344181ba324f91093a51b7ddc387b994ea6c4","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"}],\"hop\":9,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-85a2d00b8d22d878bbad","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --supports--> v0b\n- v0b --refutes--> v0f\n- v0f --refutes--> v05\n- v05 --refutes--> v0c\n- v0c --refutes--> v0d\n- v0d --refutes--> v00\n- v00 --refutes--> v08\n- v08 --refutes--> v07\n- v07 --refutes--> v03\n- v03 --supports--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v08 --refutes--> v01\n- v08 --supports--> v04\n- v0c --refutes--> v06\n- v0d --refutes--> v0e\n- v0f --refutes--> 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":"a55ca62aac13820d6a37d4c2ff861705615f0068ace7ecde2bef9d973e825a48","schema":"livegraph.boolean-propagation.v1","spec_hash":"85a2d00b8d22d878bbadc9c36ea9130a5a349a925ad87a11b0e7969523dceea3","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-8be4ef026d8c14aa52d0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --refutes--> v08\n- v08 --refutes--> v0c\n- v0c --refutes--> v04\n- v04 --refutes--> v0f\n- v0f --supports--> v0b\n- v0b --refutes--> v0e\n- v0e --supports--> v05\n- v05 --supports--> v03\n- v03 --refutes--> v0a\n- v0a --refutes--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v09\n- v03 --supports--> v07\n- v0c --supports--> v01\n- v0f --supports--> v00\n- v0f --supports--> v06\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":"eadcbe84ff258bc0be39a6c15816fb51a6364208423c00fe177c215d4834b350","schema":"livegraph.boolean-propagation.v1","spec_hash":"8be4ef026d8c14aa52d04e7d4c2ee08725826b75124fb643790479a77ce0616e","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-ebfee817628a4e807006","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v09\n- v09 --refutes--> v07\n- v07 --refutes--> v0c\n- v0c --supports--> v0a\n- v0a --supports--> v0d\n- v0d --refutes--> v05\n- v05 --supports--> v03\n- v03 --refutes--> v0b\n- v0b --refutes--> v0f\n- v0f --supports--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v0e\n- v03 --supports--> v04\n- v07 --refutes--> v01\n- v0a --refutes--> v08\n- v0a --supports--> v02\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":"8337de2a4a12f90eb228cce8c3d2eda536c6282e0059b23546829544cc03c0aa","schema":"livegraph.boolean-propagation.v1","spec_hash":"ebfee817628a4e80700695953879fc3210e09522ee2f250a2e02b0b633e0eb5f","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-b4fdf6cdf1b56146effa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --refutes--> v05\n- v05 --refutes--> v01\n- v01 --refutes--> v06\n- v06 --refutes--> v08\n- v08 --refutes--> v03\n- v03 --supports--> v00\n- v00 --supports--> v04\n- v04 --refutes--> v0b\n- v0b --refutes--> v0f\n- v0f --refutes--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v0e\n- v02 --supports--> v07\n- v06 --supports--> v0a\n- v0b --supports--> v0c\n- v0f --refutes--> v0d\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":"9a65bb10b5626b52d29e2ba53d06a042d7122001545187320e5f80c6400bdbd0","schema":"livegraph.boolean-propagation.v1","spec_hash":"b4fdf6cdf1b56146effa3b025c9bb5ebc14f63873b18993afe2f5b10d398ac87","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-fbb1cc2b1333e596a57b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v0f\n- v0f --refutes--> v0a\n- v0a --supports--> v04\n- v04 --supports--> v0c\n- v0c --refutes--> v02\n- v02 --refutes--> v01\n- v01 --refutes--> v09\n- v09 --refutes--> v03\n- v03 --supports--> v00\n- v00 --supports--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v0d\n- v04 --refutes--> v08\n- v06 --refutes--> v05\n- v0a --supports--> v07\n- v0c --supports--> v0e\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":"da124590ef2c9047f774856e9a0a6032835e80e4e84861a104efc1098f121c35","schema":"livegraph.boolean-propagation.v1","spec_hash":"fbb1cc2b1333e596a57b6e812e45caf69f3d7309b3dff2d546edd620c5c04a56","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-d49a8a4e77af9d21e473","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --refutes--> v06\n- v06 --supports--> v04\n- v04 --supports--> v0b\n- v0b --refutes--> v0c\n- v0c --refutes--> v0e\n- v0e --refutes--> v01\n- v01 --refutes--> v05\n- v05 --supports--> v02\n- v02 --refutes--> v0f\n- v0f --refutes--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v03\n- v05 --supports--> v08\n- v05 --supports--> v0d\n- v0e --refutes--> v00\n- v0f --refutes--> v07\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":"2f564fb5beb1d8bfda921d07191418868624e2d89e7e655a94a3f62cd1c0f54e","schema":"livegraph.boolean-propagation.v1","spec_hash":"d49a8a4e77af9d21e473ad0b2beed40d133a364a14016f4c9a78ad77a15878b5","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-a1930aa9482925710af2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --refutes--> v03\n- v03 --supports--> v0f\n- v0f --supports--> v0b\n- v0b --refutes--> v0a\n- v0a --refutes--> v04\n- v04 --refutes--> v0c\n- v0c --supports--> v02\n- v02 --refutes--> v0e\n- v0e --supports--> v09\n- v09 --refutes--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v06\n- v04 --supports--> v07\n- v08 --supports--> v00\n- v0a --supports--> v0d\n- v0b --refutes--> v01\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":"65c4f6da66d4750b2d908cb8ef2b637357cc1cbf5a0d7e8ecdb7a9a119f9ed99","schema":"livegraph.boolean-propagation.v1","spec_hash":"a1930aa9482925710af2a414e461c3e51a00260f99f7204d59fb61a114badd0d","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-9ae794f4c364eabd4676","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --refutes--> v0e\n- v0e --refutes--> v0f\n- v0f --refutes--> v00\n- v00 --supports--> v06\n- v06 --supports--> v05\n- v05 --refutes--> v0c\n- v0c --refutes--> v0d\n- v0d --supports--> v0b\n- v0b --supports--> v08\n- v08 --supports--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v03\n- v09 --supports--> v07\n- v0b --refutes--> v01\n- v0b --supports--> v0a\n- v0d --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":"1ebea834312e61bbc6b9cd1203e6993b2f4339ef4a97f265675fdc73cd7524dd","schema":"livegraph.boolean-propagation.v1","spec_hash":"9ae794f4c364eabd4676949320859f810090b6f5a6084dca56daff9e09dee0eb","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-fd4460dfe736a1d3a7d9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v00\n- v00 --supports--> v06\n- v06 --supports--> v0f\n- v0f --refutes--> v07\n- v07 --refutes--> v0a\n- v0a --refutes--> v09\n- v09 --supports--> v05\n- v05 --refutes--> v08\n- v08 --refutes--> v0d\n- v0d --supports--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v02\n- v00 --supports--> v0c\n- v04 --supports--> v03\n- v05 --supports--> v0e\n- v0d --refutes--> v01\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":"508aeedc4a5ac869ac9b4209b051dcfe4b113837357bd7fb22a58c435a3b8881","schema":"livegraph.boolean-propagation.v1","spec_hash":"fd4460dfe736a1d3a7d9775879b8b35402a89d6eada0b47a3071ecb5782ce38a","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-0712f49d4eb76166dc71","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --supports--> v07\n- v07 --refutes--> v03\n- v03 --refutes--> v09\n- v09 --refutes--> v02\n- v02 --refutes--> v01\n- v01 --refutes--> v0d\n- v0d --supports--> v0b\n- v0b --supports--> v00\n- v00 --refutes--> v06\n- v06 --supports--> v0c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v08\n- v03 --refutes--> v0e\n- v03 --supports--> v0f\n- v0a --refutes--> v04\n- v0b --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":"1e639f4a9a227ed0222e7ff7d575edebb1ab985f9acc0380da22c3c8b33d1210","schema":"livegraph.boolean-propagation.v1","spec_hash":"0712f49d4eb76166dc71a42fce85339414bb4f54fcd4ec9d2b617b5a6d0d6f03","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":10,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-076a32e7865341113fc5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --supports--> v01\n- v01 --supports--> v08\n- v08 --refutes--> v0f\n- v0f --supports--> v09\n- v09 --supports--> v03\n- v03 --supports--> v05\n- v05 --supports--> v0b\n- v0b --supports--> v0d\n- v0d --supports--> v04\n- v04 --supports--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v06\n- v03 --supports--> v0a\n- v05 --refutes--> v0e\n- v09 --refutes--> v07\n- v0b --refutes--> v02\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":"eabde8242a1435f65f8359970d4afffc070b20b8937e7f2c69ad86fa5e5bb39e","schema":"livegraph.boolean-propagation.v1","spec_hash":"076a32e7865341113fc527e7f596289ce34136e136635e9ff93be00635c1c5da","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-12c668a1ce9c3a7ef529","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --supports--> v0a\n- v0a --refutes--> v0e\n- v0e --refutes--> v0d\n- v0d --refutes--> v0c\n- v0c --refutes--> v01\n- v01 --supports--> v09\n- v09 --supports--> v0b\n- v0b --refutes--> v05\n- v05 --refutes--> v08\n- v08 --supports--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v07\n- v05 --supports--> v00\n- v0d --refutes--> v02\n- v0d --refutes--> v06\n- v0e --supports--> v0f\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":"b2f62658f24e3d53bfedfb3447774acb2178a06c08cefd789e020921f8a7a43c","schema":"livegraph.boolean-propagation.v1","spec_hash":"12c668a1ce9c3a7ef5295359e447a49c367802901981233653c2414ef4bf4649","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-771b3e76e7b8d51b4574","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0f --supports--> v0d\n- v0d --refutes--> v0b\n- v0b --supports--> v0c\n- v0c --refutes--> v08\n- v08 --refutes--> v07\n- v07 --supports--> v06\n- v06 --refutes--> v00\n- v00 --supports--> v04\n- v04 --refutes--> v01\n- v01 --refutes--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v05\n- v06 --refutes--> v09\n- v06 --supports--> v03\n- v07 --supports--> v0e\n- v0b --refutes--> v0a\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":"6aacf98b326db5fb8302b2a812b5e1a48ee49a556a363943a9e9ad393348de5d","schema":"livegraph.boolean-propagation.v1","spec_hash":"771b3e76e7b8d51b4574e33bc3162d10330f7b63181a3175e6037a781e635b5e","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-bcc0f88dde89d1006d12","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0e --refutes--> v0d\n- v0d --refutes--> v0b\n- v0b --supports--> v01\n- v01 --refutes--> v00\n- v00 --supports--> v02\n- v02 --refutes--> v03\n- v03 --supports--> v0f\n- v0f --supports--> v0c\n- v0c --refutes--> v04\n- v04 --refutes--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v05\n- v03 --supports--> v08\n- v04 --supports--> v09\n- v0b --supports--> v07\n- v0f --supports--> 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":"05ac90f8ec90693b4c7e45ea5ef97da086220e8e4c17dfe84cde1cf269968f53","schema":"livegraph.boolean-propagation.v1","spec_hash":"bcc0f88dde89d1006d126eefaea14c442e8dddbd1d95d87e750f0dcc5d7b2dad","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-b94015bf474018512fad","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0f --supports--> v0d\n- v0d --refutes--> v05\n- v05 --refutes--> v07\n- v07 --refutes--> v04\n- v04 --refutes--> v06\n- v06 --supports--> v08\n- v08 --supports--> v0c\n- v0c --refutes--> v00\n- v00 --supports--> v0b\n- v0b --supports--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v01\n- v00 --supports--> v0e\n- v04 --refutes--> v02\n- v0b --refutes--> v0a\n- v0f --supports--> v03\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":"c292cc4595c230ff204d4d7454733e9e1038d8bc3a0bc6f297f3c1a5d5e359f8","schema":"livegraph.boolean-propagation.v1","spec_hash":"b94015bf474018512fad1dc6646cefddbff00e1da379fcff9d9ccd9ea04bb694","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-a940ef31c5f028e001ac","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0f --supports--> v06\n- v06 --refutes--> v00\n- v00 --refutes--> v0c\n- v0c --refutes--> v08\n- v08 --refutes--> v02\n- v02 --refutes--> v03\n- v03 --supports--> v04\n- v04 --refutes--> v0a\n- v0a --supports--> v09\n- v09 --refutes--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v01\n- v04 --supports--> v05\n- v09 --supports--> v0d\n- v09 --supports--> v0e\n- v0a --refutes--> v0b\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":"76ca715b3f2112c8e8e869d453394ce81d24f242055ccfbd71bbfdbbfc7f562e","schema":"livegraph.boolean-propagation.v1","spec_hash":"a940ef31c5f028e001ac5a9ce2c3ef985bb35422e16b0cb1da4d0067abcd1dc3","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-b7986a18d67a65e77c5c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --supports--> v02\n- v02 --supports--> v0c\n- v0c --supports--> v08\n- v08 --supports--> v04\n- v04 --refutes--> v0e\n- v0e --supports--> v0b\n- v0b --supports--> v00\n- v00 --supports--> v0f\n- v0f --refutes--> v09\n- v09 --supports--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v08 --refutes--> v06\n- v09 --refutes--> v0a\n- v09 --supports--> v0d\n- v0e --refutes--> v05\n- v0f --supports--> v07\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":"6587ae0f9f57df83e2a0031c9403c5ed809b714b39bcaec2f0990be54d265265","schema":"livegraph.boolean-propagation.v1","spec_hash":"b7986a18d67a65e77c5c5deb576667393dd37983df2883bbcdd6e985eba1b993","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-21a974d6dedb5fd52900","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0f --supports--> v03\n- v03 --supports--> v01\n- v01 --supports--> v09\n- v09 --supports--> v0a\n- v0a --supports--> v07\n- v07 --refutes--> v00\n- v00 --supports--> v0b\n- v0b --refutes--> v0d\n- v0d --supports--> v08\n- v08 --refutes--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v07 --supports--> v02\n- v08 --refutes--> v04\n- v08 --refutes--> v0e\n- v0a --refutes--> v06\n- v0d --refutes--> v0c\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":"863eaeb12d8c0e7ead77a3fb6cfd7f87c31674b348c3f4613336561f90dfecc9","schema":"livegraph.boolean-propagation.v1","spec_hash":"21a974d6dedb5fd52900c13ba5b56a06e5da7ed03394d61c1f11ed7bda83c1d9","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-770468e2f988b70a4421","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v01\n- v01 --refutes--> v0d\n- v0d --supports--> v03\n- v03 --refutes--> v05\n- v05 --supports--> v0c\n- v0c --supports--> v0a\n- v0a --refutes--> v0b\n- v0b --refutes--> v0e\n- v0e --refutes--> v09\n- v09 --supports--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v0b --supports--> v06\n- v0c --refutes--> v04\n- v0c --supports--> v07\n- v0d --refutes--> v08\n- v0e --refutes--> v0f\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":"fc7e2f615d7169559458d2be8edf064c95d9b575c07f702a14503f4c05163255","schema":"livegraph.boolean-propagation.v1","spec_hash":"770468e2f988b70a4421616072769a4c2189beb3036044df3606718c0978d8a8","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-6a660dab85ea33e7e777","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --supports--> v07\n- v07 --refutes--> v04\n- v04 --refutes--> v06\n- v06 --supports--> v09\n- v09 --refutes--> v02\n- v02 --refutes--> v0e\n- v0e --refutes--> v0b\n- v0b --refutes--> v0d\n- v0d --supports--> v01\n- v01 --supports--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v0c\n- v06 --supports--> v05\n- v07 --refutes--> v0f\n- v0d --refutes--> v00\n- v0e --refutes--> v0a\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":"d68b36bc21c4f166bafdc14b4def93161137dc7f4b1d9ce4b11419eb334d1d72","schema":"livegraph.boolean-propagation.v1","spec_hash":"6a660dab85ea33e7e77763c12a4f09d889ad96c68240ea497305bb762991f700","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-22ead2c7e243046a75d2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --refutes--> v0a\n- v0a --refutes--> v0f\n- v0f --refutes--> v0c\n- v0c --supports--> v0d\n- v0d --supports--> v0b\n- v0b --supports--> v09\n- v09 --refutes--> v04\n- v04 --supports--> v08\n- v08 --refutes--> v06\n- v06 --supports--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v01\n- v06 --refutes--> v0e\n- v09 --refutes--> v02\n- v09 --supports--> v00\n- v0c --supports--> v07\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":"473e2d2e869cd44b353904b22c99bba0bd4cf373ddc9c1a0c48382d5ff9df97c","schema":"livegraph.boolean-propagation.v1","spec_hash":"22ead2c7e243046a75d22b1cd78efb0a557957154685bab35046faa5e5f2ffbb","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-100074ae6b98f73255dc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --supports--> v04\n- v04 --supports--> v0e\n- v0e --supports--> v07\n- v07 --supports--> v0f\n- v0f --refutes--> v0b\n- v0b --refutes--> v03\n- v03 --refutes--> v0d\n- v0d --supports--> v00\n- v00 --supports--> v09\n- v09 --refutes--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v01\n- v07 --supports--> v06\n- v0a --refutes--> v02\n- v0d --refutes--> v0c\n- v0f --refutes--> v08\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":"076c20db80ec693e437eca6b5d0246c7aa6826b748ecfc7cfffe39775a33ac4c","schema":"livegraph.boolean-propagation.v1","spec_hash":"100074ae6b98f73255dca9255053d99851f83f1810d11ef84d4b3562d078a740","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-f840221aa42f9bf9d67a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0f --supports--> v0d\n- v0d --refutes--> v0b\n- v0b --refutes--> v08\n- v08 --refutes--> v06\n- v06 --supports--> v0c\n- v0c --refutes--> v0e\n- v0e --supports--> v0a\n- v0a --supports--> v03\n- v03 --supports--> v07\n- v07 --refutes--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v02\n- v07 --refutes--> v05\n- v07 --supports--> v09\n- v0a --refutes--> v04\n- v0b --supports--> v00\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":"cd59ffdf8ac2c0aef73a5f1bf262a610b04136f7a12b74d1f33e54c0057de1bc","schema":"livegraph.boolean-propagation.v1","spec_hash":"f840221aa42f9bf9d67a688183bb0ed7e5130315aa1307873892db3a8ef3c29e","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-29b67d4262cf25b20018","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0d --supports--> v01\n- v01 --refutes--> v05\n- v05 --refutes--> v04\n- v04 --refutes--> v07\n- v07 --supports--> v0b\n- v0b --refutes--> v06\n- v06 --supports--> v0a\n- v0a --supports--> v08\n- v08 --supports--> v0e\n- v0e --supports--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v02\n- v04 --supports--> v09\n- v07 --refutes--> v0c\n- v08 --refutes--> v00\n- v0d --refutes--> v0f\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":"c30d17b1af79314c648cb3d0ef8e120f5cb2ef3ba8a3e4e22d5038fb966fda7f","schema":"livegraph.boolean-propagation.v1","spec_hash":"29b67d4262cf25b2001815d558b2ac1adf231049867b5387601892cccb4211e4","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-e3f41fd663b1432283de","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --supports--> v0a\n- v0a --supports--> v03\n- v03 --supports--> v05\n- v05 --refutes--> v04\n- v04 --supports--> v0e\n- v0e --refutes--> v06\n- v06 --refutes--> v0d\n- v0d --refutes--> v0b\n- v0b --supports--> v00\n- v00 --supports--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v0f\n- v03 --refutes--> v01\n- v05 --supports--> v0c\n- v06 --refutes--> v07\n- v0a --supports--> v09\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":"ab6ace16a53bf352b9679e7f554eef9d2364da1cbea635ff0135e61409b9b255","schema":"livegraph.boolean-propagation.v1","spec_hash":"e3f41fd663b1432283de89bccd1d349cfa60200015f7783493e845908de529ea","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-f464d7e051585ea6484b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --supports--> v00\n- v00 --refutes--> v09\n- v09 --refutes--> v04\n- v04 --supports--> v0e\n- v0e --refutes--> v07\n- v07 --supports--> v05\n- v05 --supports--> v0f\n- v0f --refutes--> v0d\n- v0d --supports--> v02\n- v02 --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v01\n- v04 --supports--> v0c\n- v06 --refutes--> v0a\n- v07 --supports--> v0b\n- v0f --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":"d5c06556b817ebad028b2bc327d966b8e0700a028776d8e63b430ffaeb601a30","schema":"livegraph.boolean-propagation.v1","spec_hash":"f464d7e051585ea6484b8213938d47cee89948eb7c6bcd28ad50556b3e3a9405","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-030c68ac6afe22bc381e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --supports--> v09\n- v09 --refutes--> v06\n- v06 --refutes--> v03\n- v03 --refutes--> v01\n- v01 --supports--> v05\n- v05 --supports--> v0a\n- v0a --refutes--> v0b\n- v0b --refutes--> v07\n- v07 --supports--> v02\n- v02 --supports--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v00\n- v03 --supports--> v08\n- v06 --supports--> v04\n- v09 --refutes--> v0e\n- v0b --refutes--> v0f\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":"69ddda15a1a4555b49be61c6ecad2056d68ec70efcd62a4da8a352faaad4c855","schema":"livegraph.boolean-propagation.v1","spec_hash":"030c68ac6afe22bc381e1a2982a97b0c2780dceec2e681340da5499867229414","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":10,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-ba026c2145d877af51f6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --refutes--> v0d\n- v0d --supports--> v0b\n- v0b --supports--> v04\n- v04 --refutes--> v05\n- v05 --refutes--> v0e\n- v0e --supports--> v06\n- v06 --supports--> v02\n- v02 --refutes--> v08\n- v08 --supports--> v03\n- v03 --supports--> v0c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v0f\n- v08 --supports--> v09\n- v08 --supports--> v0a\n- v0d --refutes--> v00\n- v0d --supports--> 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":"9a05c0df1a0dbe17be0058e8875086dd20be2de4447437a611910298dbe7ff8a","schema":"livegraph.boolean-propagation.v1","spec_hash":"ba026c2145d877af51f613963630785322d66c17a2087a05aae7254de831b227","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-1d3e44eba6a6c6775bb6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --supports--> v0e\n- v0e --refutes--> v05\n- v05 --refutes--> v02\n- v02 --supports--> v08\n- v08 --refutes--> v0a\n- v0a --refutes--> v01\n- v01 --refutes--> v0d\n- v0d --supports--> v0b\n- v0b --supports--> v0f\n- v0f --supports--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v07 --refutes--> v03\n- v0a --refutes--> v09\n- v0d --refutes--> v0c\n- v0e --refutes--> v04\n- v0f --refutes--> v06\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":"530ae4ac23f5f762e79643033bfc5f45c3a876b578aad17807e8d3c5d8b70b33","schema":"livegraph.boolean-propagation.v1","spec_hash":"1d3e44eba6a6c6775bb645309364a40620914980c432af9a3aeca99b80bc7eb5","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-e36ccfce79d92c8712d2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --supports--> v01\n- v01 --refutes--> v09\n- v09 --refutes--> v00\n- v00 --refutes--> v03\n- v03 --supports--> v02\n- v02 --supports--> v0e\n- v0e --supports--> v0d\n- v0d --supports--> v07\n- v07 --supports--> v05\n- v05 --supports--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v0a\n- v05 --refutes--> v06\n- v05 --refutes--> v0b\n- v07 --refutes--> v0c\n- v0e --refutes--> v04\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":"bc29f91404833b98f0f7c886b9cf7d7ea3c8a1c24d67b1b2217026cef04c8060","schema":"livegraph.boolean-propagation.v1","spec_hash":"e36ccfce79d92c8712d289d8498f6848c16c95d87c36f35a9f9fe90183014208","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-3694d6cff02432784194","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0f --refutes--> v02\n- v02 --refutes--> v06\n- v06 --refutes--> v0b\n- v0b --refutes--> v0a\n- v0a --refutes--> v04\n- v04 --supports--> v07\n- v07 --refutes--> v05\n- v05 --supports--> v0e\n- v0e --refutes--> v03\n- v03 --supports--> v0c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v08\n- v03 --refutes--> v00\n- v03 --supports--> v0d\n- v0f --refutes--> v01\n- v0f --supports--> v09\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":"0a196840e099ccc6fe3f60477e8d1f67e675a7589afcf5e2a63558d571cb762c","schema":"livegraph.boolean-propagation.v1","spec_hash":"3694d6cff02432784194f841799536b7d3bae86810d43d60ac58f8d688faeb1d","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-975ec96b2aea55dadd60","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0b --refutes--> v0d\n- v0d --supports--> v05\n- v05 --refutes--> v07\n- v07 --refutes--> v01\n- v01 --supports--> v03\n- v03 --refutes--> v0e\n- v0e --refutes--> v0a\n- v0a --refutes--> v0c\n- v0c --supports--> v08\n- v08 --supports--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v0f\n- v0a --supports--> v00\n- v0a --supports--> v09\n- v0b --refutes--> v04\n- v0b --refutes--> v06\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":"6d7d1c7f671c1d15530485d313390db510d69b7044264c90eef0fbd03fff5eeb","schema":"livegraph.boolean-propagation.v1","spec_hash":"975ec96b2aea55dadd600bed8632e5c241f26f51508f5a3c27245b5f06b9cb46","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-5614fd8979c55d8b13f4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --supports--> v08\n- v08 --refutes--> v00\n- v00 --supports--> v03\n- v03 --supports--> v0f\n- v0f --supports--> v0c\n- v0c --refutes--> v0d\n- v0d --supports--> v01\n- v01 --refutes--> v06\n- v06 --supports--> v04\n- v04 --supports--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v07\n- v00 --refutes--> v09\n- v03 --supports--> v0e\n- v06 --supports--> v02\n- v08 --refutes--> v0a\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":"d3c8be2fa88df04b6a8b8f15688c5f56a61a0893db7f2ee9832ab46932c7edd1","schema":"livegraph.boolean-propagation.v1","spec_hash":"5614fd8979c55d8b13f490ff4ac7ada798e2e805a50b642ca4d40810cdf7c5ff","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-df3435e9e2eef7963d3a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --supports--> v07\n- v07 --refutes--> v0e\n- v0e --refutes--> v0d\n- v0d --refutes--> v00\n- v00 --refutes--> v0b\n- v0b --refutes--> v0c\n- v0c --supports--> v08\n- v08 --supports--> v0f\n- v0f --supports--> v09\n- v09 --refutes--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v01\n- v00 --supports--> v02\n- v08 --refutes--> v06\n- v0f --refutes--> v04\n- v0f --supports--> v05\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":"50af3f32f793739e67ba6da7a444a71b9da8eff566c51b38127698aefc4211fb","schema":"livegraph.boolean-propagation.v1","spec_hash":"df3435e9e2eef7963d3a8c712eeca200063cd617f57999095ca8d75807dd2e35","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-832c9d1628dabdd3de6c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --refutes--> v0d\n- v0d --supports--> v08\n- v08 --refutes--> v05\n- v05 --supports--> v01\n- v01 --refutes--> v0a\n- v0a --supports--> v0b\n- v0b --supports--> v0f\n- v0f --refutes--> v09\n- v09 --supports--> v07\n- v07 --refutes--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v06\n- v09 --refutes--> v02\n- v09 --supports--> v0e\n- v0b --refutes--> v03\n- v0d --supports--> v00\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":"6abce50dbc4c8ba21019e8c8ba244c543785137ef186daefc35ecaf66abe9cb6","schema":"livegraph.boolean-propagation.v1","spec_hash":"832c9d1628dabdd3de6cff7c87cee5b2d793bff6a20946d33f32ea96eb4b06f4","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-f120273e8709eed3cad9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --supports--> v0f\n- v0f --supports--> v08\n- v08 --supports--> v0d\n- v0d --supports--> v05\n- v05 --refutes--> v06\n- v06 --supports--> v09\n- v09 --refutes--> v07\n- v07 --refutes--> v0b\n- v0b --refutes--> v0a\n- v0a --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v00\n- v05 --refutes--> v0c\n- v07 --refutes--> v01\n- v09 --refutes--> v0e\n- v0f --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":"38d6680ed7c41fa69edca12716ebd5ff9c19bfd5866d6d9347fd81b973e2771f","schema":"livegraph.boolean-propagation.v1","spec_hash":"f120273e8709eed3cad9f0418d4a730ec5ebbf85b4b4aeefc37c2cf6b65325bc","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-cd12a00ba9121a98e19a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --supports--> v0e\n- v0e --supports--> v04\n- v04 --refutes--> v05\n- v05 --refutes--> v08\n- v08 --supports--> v02\n- v02 --supports--> v09\n- v09 --refutes--> v0f\n- v0f --supports--> v0d\n- v0d --refutes--> v07\n- v07 --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v0c\n- v05 --refutes--> v0a\n- v05 --supports--> v00\n- v08 --refutes--> v0b\n- v0f --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":"dbd6e240e8a69d015b3d07c17811517b70b980689cd47c01817f34c940e24102","schema":"livegraph.boolean-propagation.v1","spec_hash":"cd12a00ba9121a98e19a522eafd66e664b11572d520ee367942a425475b9dac3","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-837e1e4ad1a5fcc2f94e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --refutes--> v05\n- v05 --supports--> v0b\n- v0b --refutes--> v0e\n- v0e --refutes--> v0c\n- v0c --refutes--> v06\n- v06 --supports--> v07\n- v07 --supports--> v02\n- v02 --refutes--> v0a\n- v0a --refutes--> v08\n- v08 --supports--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v07 --supports--> v0d\n- v08 --refutes--> v00\n- v0a --supports--> v01\n- v0c --refutes--> v09\n- v0e --supports--> v0f\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":"d243c931a38126a6efcbf171ea3270ae0fd22d3e5572c17dc736f8fe2f838086","schema":"livegraph.boolean-propagation.v1","spec_hash":"837e1e4ad1a5fcc2f94e3940aa2f70a595c972d5057e8329f5ffc6924d4d8018","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-4aa32cbc55c7c611d7c6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0d --supports--> v07\n- v07 --supports--> v0e\n- v0e --supports--> v04\n- v04 --supports--> v08\n- v08 --refutes--> v09\n- v09 --supports--> v01\n- v01 --refutes--> v03\n- v03 --refutes--> v05\n- v05 --refutes--> v0c\n- v0c --refutes--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v0f\n- v04 --refutes--> v06\n- v09 --supports--> v0b\n- v0d --supports--> v00\n- v0d --supports--> v0a\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":"0cacc0a515aac5f53e02b70be4ac4bbad3319a86f5ac6fa2c0a8d343e37bc601","schema":"livegraph.boolean-propagation.v1","spec_hash":"4aa32cbc55c7c611d7c63cc699f0c66976e6e58cb4f688e5a526425d3a0bc0fb","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-e4688bdfddf37bd5813a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --supports--> v08\n- v08 --supports--> v00\n- v00 --supports--> v0f\n- v0f --supports--> v06\n- v06 --supports--> v07\n- v07 --refutes--> v0e\n- v0e --refutes--> v04\n- v04 --refutes--> v01\n- v01 --supports--> v03\n- v03 --refutes--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v02\n- v07 --supports--> v0c\n- v08 --refutes--> v09\n- v0e --refutes--> v0d\n- v0f --supports--> v0a\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":"64fec630b1b7255890bc9fd25e89204db65fe810f6dc7e2d1ca47772776af306","schema":"livegraph.boolean-propagation.v1","spec_hash":"e4688bdfddf37bd5813a3c7c5795336398123bf8b3f44a216dc9c2820b3af418","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-c36fd7c5dbc17f15cfec","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --refutes--> v09\n- v09 --supports--> v0e\n- v0e --supports--> v0f\n- v0f --refutes--> v04\n- v04 --supports--> v03\n- v03 --refutes--> v0c\n- v0c --supports--> v0d\n- v0d --supports--> v01\n- v01 --supports--> v02\n- v02 --supports--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v06\n- v02 --refutes--> v05\n- v02 --refutes--> v0a\n- v09 --refutes--> v08\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":"eb7d064cf03459bb4a4d49869d54a3712aa9db385d002a955f3655aa7d8f13aa","schema":"livegraph.boolean-propagation.v1","spec_hash":"c36fd7c5dbc17f15cfec49ad24c4bcc960214db8b6c797ff7163677a2fef5c54","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-9ed9e9b716013f375c41","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --supports--> v07\n- v07 --supports--> v05\n- v05 --refutes--> v00\n- v00 --refutes--> v03\n- v03 --supports--> v02\n- v02 --refutes--> v01\n- v01 --refutes--> v06\n- v06 --supports--> v0b\n- v0b --refutes--> v0a\n- v0a --refutes--> v0e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v04\n- v08 --refutes--> v09\n- v0a --refutes--> v0c\n- v0a --refutes--> v0f\n- v0b --refutes--> v0d\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":"cf318a32b02baf3025496f3fcabc9d287e04cdfc5139657ef520dcf1ee0fd82a","schema":"livegraph.boolean-propagation.v1","spec_hash":"9ed9e9b716013f375c41ae0e12c6fc24841c8097fbbd190b37bc888769314198","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":10,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-37b749fdeb2665a08f8a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --refutes--> v0b\n- v0b --refutes--> v01\n- v01 --supports--> v09\n- v09 --refutes--> v03\n- v03 --supports--> v06\n- v06 --refutes--> v00\n- v00 --supports--> v05\n- v05 --supports--> v0a\n- v0a --supports--> v04\n- v04 --refutes--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v02\n- v05 --refutes--> v0d\n- v06 --supports--> v0f\n- v08 --refutes--> v0e\n- v0a --refutes--> v0c\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":"7415ca49adc80ea97a3c8002b108e76899d4124024817b83ad55f3ab61a66e41","schema":"livegraph.boolean-propagation.v1","spec_hash":"37b749fdeb2665a08f8a1e3cc4c742f39646f4a49acea25298485be4b7ca5b10","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"}],\"hop\":10,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-b754a80696dc9a0f2574","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --refutes--> v01\n- v01 --supports--> v0e\n- v0e --supports--> v06\n- v06 --refutes--> v05\n- v05 --supports--> v09\n- v09 --refutes--> v07\n- v07 --refutes--> v00\n- v00 --refutes--> v0a\n- v0a --refutes--> v0f\n- v0f --refutes--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v02\n- v00 --refutes--> v0c\n- v00 --supports--> v0d\n- v09 --supports--> v0b\n- v0a --supports--> 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":"35cba687df55b6b919c19258403c1909d8ca143adab8a6503ba0d3e2cd7693b9","schema":"livegraph.boolean-propagation.v1","spec_hash":"b754a80696dc9a0f25749b4651b82667d1b5354bf947757f86b8b4b524a4478c","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-ea70516763be9dd66e93","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0b --supports--> v09\n- v09 --supports--> v01\n- v01 --supports--> v08\n- v08 --supports--> v04\n- v04 --supports--> v05\n- v05 --refutes--> v0e\n- v0e --supports--> v0a\n- v0a --refutes--> v07\n- v07 --refutes--> v0c\n- v0c --refutes--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v06\n- v04 --refutes--> v03\n- v08 --refutes--> v0f\n- v08 --supports--> v00\n- v0e --supports--> v02\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":"465f0834cbf91f50045883979d5b0a2ddd00cddb0231d2756ebe17056e66aa68","schema":"livegraph.boolean-propagation.v1","spec_hash":"ea70516763be9dd66e93702b15413fc15f4f0c45f6c6f8c25f5dd9afa5f7eb56","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-f436249c9d0486658a23","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0e --refutes--> v03\n- v03 --supports--> v09\n- v09 --refutes--> v04\n- v04 --refutes--> v05\n- v05 --refutes--> v07\n- v07 --refutes--> v06\n- v06 --refutes--> v00\n- v00 --refutes--> v0d\n- v0d --refutes--> v0a\n- v0a --refutes--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0f\n- v04 --refutes--> v08\n- v0d --refutes--> v02\n- v0e --refutes--> v01\n- v0e --refutes--> v0c\n\n[VERIFIED ROOT]\nv0e = 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":"89a98055b4301b396c04e8c09cc4909b2c03e0962ef2e6a5af6767a96fd8ba96","schema":"livegraph.boolean-propagation.v1","spec_hash":"f436249c9d0486658a23253e9c8996309600befc0bcaf688cae972b11ba0f07d","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-28871afbc1b952efcee1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --supports--> v0c\n- v0c --refutes--> v00\n- v00 --supports--> v04\n- v04 --refutes--> v06\n- v06 --supports--> v0f\n- v0f --supports--> v0e\n- v0e --supports--> v0a\n- v0a --supports--> v03\n- v03 --supports--> v0b\n- v0b --refutes--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v07\n- v09 --supports--> v02\n- v0c --supports--> v01\n- v0f --supports--> v05\n- v0f --supports--> v08\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":"0c57c2511932049c1f0757d0cec11259ebee399546c272e450ed322d11a02295","schema":"livegraph.boolean-propagation.v1","spec_hash":"28871afbc1b952efcee151bfdd3b9fe748a0a5525eedf60c243dc2faf9af24a3","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-0bc7c6a7362ef8815f63","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --supports--> v00\n- v00 --supports--> v07\n- v07 --refutes--> v03\n- v03 --refutes--> v06\n- v06 --refutes--> v0e\n- v0e --supports--> v0d\n- v0d --refutes--> v04\n- v04 --refutes--> v08\n- v08 --supports--> v0f\n- v0f --refutes--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v0c\n- v03 --supports--> v0b\n- v04 --refutes--> v09\n- v06 --refutes--> v0a\n- v0f --refutes--> v01\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":"9c0f8b06f1f7d4b797cb0380497c8143add90b3778785ae2ccf37efd3a51b84d","schema":"livegraph.boolean-propagation.v1","spec_hash":"0bc7c6a7362ef8815f63de596ef90016d79d0a32357147bd0f2df23960a67540","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-eea0bd4d9071e53a510c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --supports--> v05\n- v05 --supports--> v00\n- v00 --supports--> v0d\n- v0d --refutes--> v0b\n- v0b --refutes--> v03\n- v03 --refutes--> v06\n- v06 --supports--> v01\n- v01 --supports--> v0f\n- v0f --supports--> v02\n- v02 --supports--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v0e\n- v01 --supports--> v04\n- v05 --supports--> v0a\n- v08 --refutes--> v0c\n- v0b --refutes--> v09\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":"734b6efa55d29b881c6181c00954a8e2de48706d290bace79cf69648ec557370","schema":"livegraph.boolean-propagation.v1","spec_hash":"eea0bd4d9071e53a510c8cc7bb9baff375cd714df580d949fa34699b07d0a888","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-01d297e6da4d670f56fc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --supports--> v07\n- v07 --supports--> v00\n- v00 --refutes--> v02\n- v02 --refutes--> v09\n- v09 --refutes--> v0b\n- v0b --supports--> v05\n- v05 --supports--> v0a\n- v0a --supports--> v03\n- v03 --refutes--> v0c\n- v0c --supports--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v01\n- v09 --refutes--> v04\n- v0a --refutes--> v0d\n- v0a --supports--> v0e\n- v0c --refutes--> v08\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":"4616f87377cac98d61491a40c2322e17f0909396088129cf27ea5a95e35c8f12","schema":"livegraph.boolean-propagation.v1","spec_hash":"01d297e6da4d670f56fc59d50b8355237088d30f3e86dfdf6ab0332e2047d53b","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"}],\"hop\":10,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-bbdb36087660e06b21fa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --refutes--> v07\n- v07 --supports--> v0f\n- v0f --refutes--> v02\n- v02 --supports--> v03\n- v03 --supports--> v04\n- v04 --refutes--> v09\n- v09 --refutes--> v06\n- v06 --supports--> v0e\n- v0e --refutes--> v05\n- v05 --refutes--> v0c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v08 --refutes--> v01\n- v08 --supports--> v00\n- v08 --supports--> v0d\n- v0e --refutes--> v0b\n- v0f --supports--> v0a\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":"0e3157ff745a3152b6baf03d876eeb72d58a9db12951a0f18568ac4ea978aa8c","schema":"livegraph.boolean-propagation.v1","spec_hash":"bbdb36087660e06b21fa8c1c50de2a7c42d5e511748ab0bef4d94f6c4ef64ecd","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-c7bc67002a3604f55c64","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --supports--> v07\n- v07 --supports--> v0e\n- v0e --supports--> v06\n- v06 --supports--> v04\n- v04 --refutes--> v08\n- v08 --refutes--> v0a\n- v0a --supports--> v01\n- v01 --refutes--> v0b\n- v0b --refutes--> v0d\n- v0d --refutes--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v0a --supports--> v09\n- v0b --refutes--> v0f\n- v0d --refutes--> v02\n- v0e --refutes--> v00\n- v0e --supports--> v03\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":"790cf05ef92d123f17c487abe38c4c3a1e6d1f9ad1309bcd3915005f2a1342c6","schema":"livegraph.boolean-propagation.v1","spec_hash":"c7bc67002a3604f55c64db0b15320d73cf30cdd48dfa196d42712cbddbbc1bcd","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-b2bb4bc783887f961e84","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --refutes--> v05\n- v05 --supports--> v06\n- v06 --supports--> v0e\n- v0e --refutes--> v07\n- v07 --supports--> v0f\n- v0f --supports--> v0d\n- v0d --supports--> v08\n- v08 --refutes--> v04\n- v04 --supports--> v0a\n- v0a --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v08 --refutes--> v02\n- v0a --refutes--> v09\n- v0d --refutes--> v0c\n- v0e --refutes--> v00\n- v0f --supports--> v0b\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":"6c6739a934cc7b32015f7dcb87c572e1c16c3c244f841acd518281af2ef71d35","schema":"livegraph.boolean-propagation.v1","spec_hash":"b2bb4bc783887f961e845dfe4181b99f01da558a3046bf47a38374b495834737","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-2e7516d12c8f80e568e5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0f --supports--> v04\n- v04 --supports--> v0c\n- v0c --supports--> v06\n- v06 --refutes--> v03\n- v03 --supports--> v05\n- v05 --refutes--> v0b\n- v0b --supports--> v0d\n- v0d --supports--> v02\n- v02 --supports--> v0e\n- v0e --supports--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v00\n- v05 --refutes--> v01\n- v0b --refutes--> v0a\n- v0d --refutes--> v08\n- v0e --refutes--> v09\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":"309fb0d7a246a5e7af5d3b9ed6e2ab7ac2e19bd165365712e72ebbb76d05ff38","schema":"livegraph.boolean-propagation.v1","spec_hash":"2e7516d12c8f80e568e519ea8d81051ebf15fa0f01147d635b38466bd721ade2","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-06e37eb74ecd56e1925b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --supports--> v09\n- v09 --refutes--> v00\n- v00 --refutes--> v03\n- v03 --refutes--> v08\n- v08 --supports--> v0d\n- v0d --refutes--> v0e\n- v0e --refutes--> v05\n- v05 --supports--> v0c\n- v0c --refutes--> v07\n- v07 --refutes--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v06\n- v09 --supports--> v0f\n- v0a --supports--> v02\n- v0d --supports--> v01\n- v0e --supports--> v0b\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":"f9c88839642bc1cb053d0ee0692e8534a7733ccf3c9917ab58df9c3de69cebfe","schema":"livegraph.boolean-propagation.v1","spec_hash":"06e37eb74ecd56e1925bb625f53656f5b251160cca065c52c88a6cde6e544a6b","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":10,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-e43dc24f0cbb04e468bf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --refutes--> v0a\n- v0a --supports--> v0b\n- v0b --supports--> v09\n- v09 --supports--> v01\n- v01 --refutes--> v08\n- v08 --refutes--> v0f\n- v0f --supports--> v05\n- v05 --refutes--> v06\n- v06 --supports--> v0c\n- v0c --refutes--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v0d\n- v07 --supports--> v04\n- v0a --supports--> v00\n- v0b --supports--> v0e\n- v0f --refutes--> v03\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":"d9fc1578a9b328cae28e58b87904a05a10a783f7dd960900f9b27b7c003d3809","schema":"livegraph.boolean-propagation.v1","spec_hash":"e43dc24f0cbb04e468bf27d96007f64001e4a8391def8ca92cad251365fcc08c","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-370656cd8081f19cc3d1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v05\n- v05 --refutes--> v0a\n- v0a --refutes--> v0b\n- v0b --refutes--> v09\n- v09 --refutes--> v02\n- v02 --refutes--> v00\n- v00 --refutes--> v0c\n- v0c --refutes--> v03\n- v03 --supports--> v0d\n- v0d --supports--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v0f\n- v05 --refutes--> v08\n- v09 --refutes--> v04\n- v09 --supports--> v0e\n- v0d --supports--> 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":"3575b57776d45ee1a35ce911e8a3ac2cfe522573a0a9a764c578b667e7a9d8cd","schema":"livegraph.boolean-propagation.v1","spec_hash":"370656cd8081f19cc3d16ccf1131a885efe8eee7cc8bc9323ef263853931dd89","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":10,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-0d5509f42e01ad6c9ab0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --supports--> v0d\n- v0d --refutes--> v0c\n- v0c --supports--> v01\n- v01 --supports--> v00\n- v00 --refutes--> v02\n- v02 --supports--> v03\n- v03 --supports--> v0a\n- v0a --supports--> v04\n- v04 --supports--> v09\n- v09 --supports--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0e\n- v00 --supports--> v06\n- v01 --refutes--> v0b\n- v04 --supports--> v0f\n- v0d --refutes--> v05\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":"351bb24e97e0a106318d313388f90b77057374d53ae4ed0ebcda3dd86a87081f","schema":"livegraph.boolean-propagation.v1","spec_hash":"0d5509f42e01ad6c9ab0f49229f419fc4b168d41144df735fb59f6a858a9db81","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":10,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-b26a3ca97f69318bfbd5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v01\n- v01 --supports--> v0d\n- v0d --refutes--> v0a\n- v0a --supports--> v0b\n- v0b --supports--> v09\n- v09 --supports--> v07\n- v07 --refutes--> v08\n- v08 --refutes--> v0f\n- v0f --supports--> v05\n- v05 --supports--> v0c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v06\n- v05 --refutes--> v0e\n- v05 --supports--> v03\n- v07 --supports--> v00\n- v07 --supports--> 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":"455475bcb73b1304e72a181ec0574ecae0c8958a443894724414759f6ef127e8","schema":"livegraph.boolean-propagation.v1","spec_hash":"b26a3ca97f69318bfbd5263e343f54306b9172d697b4776618ea6846c0c26bf4","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-ff6c7056f7e3c921c0a7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v0a\n- v0a --supports--> v07\n- v07 --supports--> v09\n- v09 --supports--> v03\n- v03 --supports--> v01\n- v01 --refutes--> v02\n- v02 --supports--> v08\n- v08 --refutes--> v00\n- v00 --supports--> v0d\n- v0d --refutes--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v0f\n- v07 --supports--> v0c\n- v08 --supports--> v05\n- v08 --supports--> v0e\n- v09 --refutes--> v06\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":"3ec44dff7e9265cd7174fda165c394098d93d50b370c40cb25e1035426f073bb","schema":"livegraph.boolean-propagation.v1","spec_hash":"ff6c7056f7e3c921c0a71a6bcb58a930aac1ddb142f297b8455c69449d0c052e","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"}],\"hop\":10,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-f88b6f16dd6d70248bcf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --supports--> v0d\n- v0d --refutes--> v07\n- v07 --refutes--> v02\n- v02 --refutes--> v0f\n- v0f --refutes--> v03\n- v03 --refutes--> v08\n- v08 --supports--> v01\n- v01 --refutes--> v0e\n- v0e --refutes--> v0c\n- v0c --supports--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v00\n- v02 --supports--> v0b\n- v0c --refutes--> v06\n- v0c --supports--> v04\n- v0e --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":"f923a2ac886ae1b00b5bd13ebea1bd123c86e40c503a046efbd12987bd176743","schema":"livegraph.boolean-propagation.v1","spec_hash":"f88b6f16dd6d70248bcfab0c9d74b5991d90ec2a7e4b5a0c17c5fc9c7b57bc96","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-3364352832300c426a55","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --supports--> v0e\n- v0e --refutes--> v00\n- v00 --supports--> v0a\n- v0a --refutes--> v0d\n- v0d --supports--> v05\n- v05 --refutes--> v08\n- v08 --supports--> v0f\n- v0f --supports--> v0c\n- v0c --refutes--> v04\n- v04 --supports--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v01\n- v0c --refutes--> v09\n- v0c --supports--> v02\n- v0c --supports--> v03\n- v0f --refutes--> v06\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":"e0dd206c39ac36890f368f5451198e9bb9b38a5ebc2b3f88fbd7b84939ef5fee","schema":"livegraph.boolean-propagation.v1","spec_hash":"3364352832300c426a55de461b0af7267ea45e4b2a09f9f69eab4d62f9216b32","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":10,"item_id":"lggrpo1-e02e949229417b4865f5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --refutes--> v0a\n- v0a --supports--> v09\n- v09 --refutes--> v0d\n- v0d --supports--> v08\n- v08 --supports--> v07\n- v07 --supports--> v0f\n- v0f --refutes--> v0b\n- v0b --refutes--> v03\n- v03 --refutes--> v0c\n- v0c --refutes--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v0c --refutes--> v00\n- v0c --supports--> v02\n- v0c --supports--> v04\n- v0d --supports--> v05\n- v0f --refutes--> v0e\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":"956fe445e9c732796567ad7fb9d8956f9016eae9be48cf9c9ce487df214a1cb0","schema":"livegraph.boolean-propagation.v1","spec_hash":"e02e949229417b4865f5a9a34eda8e7f47bf4f6caa4735e4ed7c18f82260a85e","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":10,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":10,"item_id":"lggrpo1-3af1721c2afdd6a5ca32","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --refutes--> v0d\n- v0d --supports--> v09\n- v09 --refutes--> v04\n- v04 --supports--> v05\n- v05 --refutes--> v0c\n- v0c --supports--> v0a\n- v0a --supports--> v0b\n- v0b --supports--> v01\n- v01 --supports--> v08\n- v08 --refutes--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v03\n- v01 --refutes--> v02\n- v01 --supports--> v07\n- v0c --refutes--> v0e\n- v0d --supports--> v06\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":"6816286656dd2bdb20ff83d1e5864c835905db18cd6a914423244384c5ac2bb6","schema":"livegraph.boolean-propagation.v1","spec_hash":"3af1721c2afdd6a5ca3250d9b840d40cedab4ab1c051027e2d4ac4019d7f0840","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"}],\"hop\":10,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-920a148f043dbbb6994b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --supports--> v0a\n- v0a --supports--> v02\n- v02 --supports--> v0e\n- v0e --refutes--> v03\n- v03 --supports--> v10\n- v10 --supports--> v07\n- v07 --refutes--> v06\n- v06 --refutes--> v01\n- v01 --supports--> v00\n- v00 --supports--> v0c\n- v0c --refutes--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v05\n- v01 --refutes--> v0d\n- v0e --refutes--> v08\n- v0e --supports--> v0b\n- v10 --refutes--> v0f\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":"d74d4e102d643f7fd98dcfa3245da14d58588bf3f155fd594c59f53293545dfa","schema":"livegraph.boolean-propagation.v1","spec_hash":"920a148f043dbbb6994be523a7c9d6f6f86cd4add7f1aba53d450804d69aaf27","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-ca2a94e005543ab9d298","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0d --supports--> v05\n- v05 --supports--> v06\n- v06 --refutes--> v0b\n- v0b --refutes--> v00\n- v00 --refutes--> v0f\n- v0f --refutes--> v07\n- v07 --refutes--> v02\n- v02 --supports--> v0a\n- v0a --supports--> v10\n- v10 --refutes--> v09\n- v09 --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v04\n- v06 --refutes--> v0e\n- v0d --refutes--> v08\n- v0d --refutes--> v0c\n- v0f --refutes--> v01\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":"88fc8fa661d3dac76744358eb202b204374c2e276ab933ff54af2c78a94d6f99","schema":"livegraph.boolean-propagation.v1","spec_hash":"ca2a94e005543ab9d2986ce09292648b881b0044073c6ac6239451bd19f2ae89","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-6fce28d31e73f9fb7f18","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0e --supports--> v02\n- v02 --supports--> v04\n- v04 --refutes--> v0f\n- v0f --refutes--> v06\n- v06 --refutes--> v0b\n- v0b --supports--> v0c\n- v0c --refutes--> v00\n- v00 --supports--> v0a\n- v0a --refutes--> v05\n- v05 --refutes--> v09\n- v09 --refutes--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v03\n- v02 --supports--> v07\n- v06 --refutes--> v0d\n- v09 --refutes--> v01\n- v0e --supports--> v10\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":"bc160782467d3baaed2e45ccf806c5d9df547bfb830519f5b7f97239f81482f1","schema":"livegraph.boolean-propagation.v1","spec_hash":"6fce28d31e73f9fb7f18b00695417c7975d34e24cf4cbd18e72900ee23a8562e","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-4037b657da02b073d9e3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --refutes--> v0f\n- v0f --supports--> v03\n- v03 --supports--> v04\n- v04 --supports--> v02\n- v02 --supports--> v05\n- v05 --supports--> v0c\n- v0c --supports--> v00\n- v00 --refutes--> v06\n- v06 --supports--> v0d\n- v0d --supports--> v07\n- v07 --supports--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v09\n- v03 --supports--> v10\n- v04 --refutes--> v0a\n- v05 --supports--> v0b\n- v06 --refutes--> v0e\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":"986a6b4c0bc1864ac6b41a8380615514c73f79acfe4e4f07030e68366e576a8c","schema":"livegraph.boolean-propagation.v1","spec_hash":"4037b657da02b073d9e3f10506ca26f23c23f127e5e4b1ac48072fe2e562353a","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-d960e9ed804462f4a15a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0b --supports--> v07\n- v07 --refutes--> v04\n- v04 --supports--> v00\n- v00 --refutes--> v08\n- v08 --refutes--> v0a\n- v0a --supports--> v10\n- v10 --refutes--> v0d\n- v0d --supports--> v01\n- v01 --supports--> v03\n- v03 --refutes--> v05\n- v05 --refutes--> v0e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v02\n- v01 --supports--> v0c\n- v03 --refutes--> v09\n- v05 --supports--> v06\n- v0b --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":"d7af4afa531e66b7f75a1719b0c76c567443ccacb680fce8b42f2d47eed876f8","schema":"livegraph.boolean-propagation.v1","spec_hash":"d960e9ed804462f4a15af66a4cbbe85552c603c3f5375fc7737333bd565787c0","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-0dc09bd36da70efd91bb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --supports--> v02\n- v02 --refutes--> v0f\n- v0f --refutes--> v00\n- v00 --supports--> v08\n- v08 --refutes--> v0a\n- v0a --refutes--> v04\n- v04 --supports--> v01\n- v01 --refutes--> v10\n- v10 --refutes--> v0b\n- v0b --refutes--> v03\n- v03 --supports--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v09\n- v0a --refutes--> v0e\n- v0a --supports--> v05\n- v0c --refutes--> v07\n- v0c --supports--> v0d\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":"3454ca37c607218ae5031bb5c374093982e991eba7477a9de80d7a2cbdaa6738","schema":"livegraph.boolean-propagation.v1","spec_hash":"0dc09bd36da70efd91bb79220304099804df822005c9ea2546eec6de7ca0606f","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-db7f763c8e85bfebd7c9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --supports--> v10\n- v10 --supports--> v0e\n- v0e --supports--> v03\n- v03 --refutes--> v0d\n- v0d --refutes--> v04\n- v04 --refutes--> v0c\n- v0c --refutes--> v07\n- v07 --refutes--> v0f\n- v0f --refutes--> v06\n- v06 --refutes--> v05\n- v05 --supports--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v02\n- v0c --refutes--> v0b\n- v0c --supports--> v09\n- v0e --refutes--> v01\n- v10 --refutes--> v08\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":"da42d4edb98183390d783d34b1b54e8ad4016ba10e40e30da9cd425a86f29aa6","schema":"livegraph.boolean-propagation.v1","spec_hash":"db7f763c8e85bfebd7c9f0152419091bdafe4f4cb25a818e56e751530baae26a","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-fc4b89c8346ad9db244d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0b --supports--> v00\n- v00 --supports--> v0d\n- v0d --supports--> v06\n- v06 --refutes--> v0e\n- v0e --supports--> v0a\n- v0a --refutes--> v03\n- v03 --refutes--> v08\n- v08 --supports--> v05\n- v05 --supports--> v0c\n- v0c --refutes--> v0f\n- v0f --refutes--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v02\n- v06 --refutes--> v07\n- v0c --supports--> v09\n- v0e --refutes--> v10\n- v0f --refutes--> 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":"75fb5729173cf8b5c4ac93572304b1e3d628c2c7837358399dc17364362590a1","schema":"livegraph.boolean-propagation.v1","spec_hash":"fc4b89c8346ad9db244dc26a0864f44d70520741701870ef05e71c5164aca437","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-7bcafde91901ff861e6b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0e --supports--> v0c\n- v0c --supports--> v05\n- v05 --refutes--> v10\n- v10 --refutes--> v0f\n- v0f --supports--> v0a\n- v0a --refutes--> v03\n- v03 --supports--> v06\n- v06 --supports--> v01\n- v01 --refutes--> v0b\n- v0b --refutes--> v09\n- v09 --supports--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v00\n- v09 --refutes--> v08\n- v0a --refutes--> v04\n- v0a --refutes--> v0d\n- v0f --supports--> v07\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":"3f24c7839bb00168ecd46a8746b9c5fe78c9701e847be9b0c376ca1ab4a80a5d","schema":"livegraph.boolean-propagation.v1","spec_hash":"7bcafde91901ff861e6befc67dad289cbd952acc84d98c076f8241c42222eddf","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-5abdab010180f2a2c2bf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --refutes--> v04\n- v04 --refutes--> v08\n- v08 --refutes--> v03\n- v03 --refutes--> v10\n- v10 --refutes--> v09\n- v09 --refutes--> v0b\n- v0b --refutes--> v0a\n- v0a --refutes--> v0e\n- v0e --supports--> v0f\n- v0f --refutes--> v0c\n- v0c --supports--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v00\n- v0a --refutes--> v0d\n- v0e --refutes--> v01\n- v0e --refutes--> v05\n- v0f --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":"eef327f6577f07e35a4901d646a23522d7d22e9f8b462528d7728a62ac426b81","schema":"livegraph.boolean-propagation.v1","spec_hash":"5abdab010180f2a2c2bf66b46654fa97a73e0b64d9504476a8b291e2c0e146c3","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-090046ab154870cdc13e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --supports--> v0d\n- v0d --refutes--> v07\n- v07 --refutes--> v00\n- v00 --refutes--> v0e\n- v0e --refutes--> v05\n- v05 --supports--> v0a\n- v0a --supports--> v09\n- v09 --refutes--> v08\n- v08 --refutes--> v01\n- v01 --refutes--> v02\n- v02 --supports--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v04\n- v00 --refutes--> v06\n- v07 --refutes--> v0f\n- v0c --supports--> v10\n- v0d --supports--> v0b\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":"07ed3d3160655e967d75484f2d2f5f5f63d3586fd9e10af23c3067367aaf7c1a","schema":"livegraph.boolean-propagation.v1","spec_hash":"090046ab154870cdc13e507fa53e670740cfc2135e743356b8f575ca0cc82a84","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"}],\"hop\":11,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-86fbf3d8a3a65397890e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0f --supports--> v03\n- v03 --refutes--> v0e\n- v0e --supports--> v05\n- v05 --refutes--> v06\n- v06 --supports--> v0d\n- v0d --supports--> v04\n- v04 --refutes--> v0c\n- v0c --supports--> v10\n- v10 --refutes--> v07\n- v07 --refutes--> v02\n- v02 --supports--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v0a\n- v04 --supports--> v09\n- v05 --refutes--> v08\n- v0e --supports--> v0b\n- v0f --refutes--> v01\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":"9f11268513158de5e27a042ce6ae668a58b3808ecacb7b4765b5057d1a0e8e91","schema":"livegraph.boolean-propagation.v1","spec_hash":"86fbf3d8a3a65397890e8cc98078099184ad1d93fd28b1305bb4cc50022fca6b","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-dec7c9b1a6f6b8a624af","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --refutes--> v0a\n- v0a --supports--> v0c\n- v0c --refutes--> v10\n- v10 --refutes--> v04\n- v04 --supports--> v0f\n- v0f --refutes--> v08\n- v08 --supports--> v09\n- v09 --refutes--> v0b\n- v0b --supports--> v00\n- v00 --refutes--> v0e\n- v0e --supports--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v02\n- v08 --refutes--> v07\n- v0a --refutes--> v05\n- v0a --supports--> v0d\n- v0b --refutes--> v06\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":"949d0f7a7598a6616f81089b4d6d4e454ff19236d22dd54c4717267a24ce868d","schema":"livegraph.boolean-propagation.v1","spec_hash":"dec7c9b1a6f6b8a624af612216156da6998a0912eff6c9d15da1d4b42d4d0ff7","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-f3a3243eecb7c12772f8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --supports--> v00\n- v00 --refutes--> v08\n- v08 --refutes--> v0d\n- v0d --refutes--> v10\n- v10 --refutes--> v04\n- v04 --supports--> v09\n- v09 --supports--> v0e\n- v0e --supports--> v03\n- v03 --refutes--> v07\n- v07 --refutes--> v0b\n- v0b --refutes--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v07 --refutes--> v02\n- v0b --refutes--> v01\n- v0e --refutes--> v0c\n- v0e --supports--> v05\n- v0e --supports--> v06\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":"fbfaf272e0eec9b28ff0dd164481cbc9150aa544d3dd41e2642c3577e94ad376","schema":"livegraph.boolean-propagation.v1","spec_hash":"f3a3243eecb7c12772f83a015beb2700f75b38f96e12c93d4114c50aa199d44e","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-f2f4ffc1f6e33585b1f9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --refutes--> v0e\n- v0e --supports--> v04\n- v04 --supports--> v07\n- v07 --refutes--> v08\n- v08 --supports--> v09\n- v09 --refutes--> v0c\n- v0c --refutes--> v00\n- v00 --supports--> v02\n- v02 --refutes--> v03\n- v03 --refutes--> v01\n- v01 --supports--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v06\n- v02 --supports--> v0a\n- v05 --supports--> v10\n- v0c --refutes--> v0f\n- v0e --refutes--> v0b\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":"372db84498117e799d683e9b3dd31d2c80425b503d2e09527c3a74c09826a737","schema":"livegraph.boolean-propagation.v1","spec_hash":"f2f4ffc1f6e33585b1f94da968232f54ef97de39472b033f3e7b42a9196e140c","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":11,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-843a414df6a9fc47ffb5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --refutes--> v07\n- v07 --supports--> v0d\n- v0d --refutes--> v0e\n- v0e --refutes--> v05\n- v05 --refutes--> v0a\n- v0a --refutes--> v01\n- v01 --refutes--> v00\n- v00 --supports--> v03\n- v03 --supports--> v0b\n- v0b --supports--> v04\n- v04 --supports--> v10\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v08\n- v09 --refutes--> v02\n- v09 --refutes--> v06\n- v0e --supports--> v0c\n- v0e --supports--> v0f\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":"c0d42c7285cd73ead0adf4645c00b90896d629c54cd31b8368d073f4e3108c17","schema":"livegraph.boolean-propagation.v1","spec_hash":"843a414df6a9fc47ffb5d443ce2d19cb7b8f1dbec23523ba40567147bed2bf82","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":11,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-8bf7d2d1312d1fc7af0a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --supports--> v09\n- v09 --refutes--> v03\n- v03 --supports--> v0d\n- v0d --refutes--> v08\n- v08 --refutes--> v01\n- v01 --refutes--> v0c\n- v0c --refutes--> v07\n- v07 --supports--> v10\n- v10 --supports--> v0f\n- v0f --supports--> v04\n- v04 --refutes--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v0e\n- v03 --refutes--> v05\n- v03 --refutes--> v0b\n- v07 --supports--> v00\n- v08 --supports--> 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":"864ac7a16d57d7fabd3198ad8134c32cbc59e3352c45b29edad300c9139a0a27","schema":"livegraph.boolean-propagation.v1","spec_hash":"8bf7d2d1312d1fc7af0ad3148708b27436abeed6a1ed12eba3ab9d468660216a","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-b4394820743e78f302eb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --supports--> v04\n- v04 --supports--> v0c\n- v0c --supports--> v05\n- v05 --supports--> v0d\n- v0d --supports--> v02\n- v02 --refutes--> v09\n- v09 --supports--> v0a\n- v0a --supports--> v0f\n- v0f --refutes--> v06\n- v06 --refutes--> v0e\n- v0e --supports--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v03\n- v04 --refutes--> v00\n- v04 --supports--> v01\n- v06 --supports--> v0b\n- v0c --refutes--> v10\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":"0627c2d7c72d5e92d0a37859f33780a684075c495664a570bf63fe5e5f22ad18","schema":"livegraph.boolean-propagation.v1","spec_hash":"b4394820743e78f302eb1d7a6cc7bb6f55b71872230b18ba77fb273381159710","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-9ea49ef9ef7107ba8dc8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --supports--> v0b\n- v0b --refutes--> v05\n- v05 --refutes--> v07\n- v07 --supports--> v01\n- v01 --refutes--> v10\n- v10 --supports--> v0a\n- v0a --supports--> v08\n- v08 --supports--> v0e\n- v0e --refutes--> v03\n- v03 --refutes--> v00\n- v00 --refutes--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v02\n- v01 --supports--> v04\n- v05 --supports--> v0d\n- v06 --refutes--> v0c\n- v0e --refutes--> v0f\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":"c75191cb7a48f93a3c5618c94b4f5fa7eacc45071bef92e0b35f306796b5b88d","schema":"livegraph.boolean-propagation.v1","spec_hash":"9ea49ef9ef7107ba8dc8b7c667f6202ad34de04d7f1215b5413f24e72bdb0f99","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-da54bfae392e059e9503","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --refutes--> v10\n- v10 --refutes--> v01\n- v01 --refutes--> v0b\n- v0b --refutes--> v02\n- v02 --refutes--> v0f\n- v0f --supports--> v03\n- v03 --refutes--> v0a\n- v0a --supports--> v09\n- v09 --refutes--> v06\n- v06 --supports--> v0e\n- v0e --refutes--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v0d\n- v06 --refutes--> v0c\n- v0b --supports--> v05\n- v0f --refutes--> v07\n- v10 --supports--> v00\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":"a0ee5fc96da59eef6503fbd3b93120c9a386958d35c7350ba68c7771b5bb1e33","schema":"livegraph.boolean-propagation.v1","spec_hash":"da54bfae392e059e9503d4a28b44a2311d540cf7af50fc1a0b0718914d1cc800","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-9a21f127e9d234466537","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0f --supports--> v0c\n- v0c --supports--> v10\n- v10 --supports--> v04\n- v04 --refutes--> v07\n- v07 --supports--> v00\n- v00 --refutes--> v0d\n- v0d --supports--> v05\n- v05 --supports--> v01\n- v01 --refutes--> v02\n- v02 --supports--> v03\n- v03 --refutes--> v0e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v06\n- v01 --refutes--> v09\n- v07 --refutes--> v08\n- v0c --refutes--> v0a\n- v0d --refutes--> v0b\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":"9ece68d7520f9596747d82cdd97e1ac37ff884dd125c1c2bdb3d92b0b60d8782","schema":"livegraph.boolean-propagation.v1","spec_hash":"9a21f127e9d2344665374f436d84358e9a0c7672a81476659993ce4263206018","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-6d4d9f2c8fe836ac20d4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --refutes--> v01\n- v01 --supports--> v07\n- v07 --refutes--> v0c\n- v0c --supports--> v03\n- v03 --refutes--> v09\n- v09 --supports--> v04\n- v04 --supports--> v0e\n- v0e --supports--> v00\n- v00 --supports--> v06\n- v06 --supports--> v08\n- v08 --supports--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v10\n- v06 --refutes--> v0a\n- v07 --refutes--> v0d\n- v09 --refutes--> v0b\n- v0e --refutes--> v05\n\n[VERIFIED ROOT]\nv02 = 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":"44bbd2541acba806b267204154fcfa8d05f43030cb3da53376f12fd0439efeb2","schema":"livegraph.boolean-propagation.v1","spec_hash":"6d4d9f2c8fe836ac20d4d393320ab8aac91b88c85fd9759c7c34aa425c6456ef","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":11,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-c38b03bffd3cbb3ea9db","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --refutes--> v09\n- v09 --refutes--> v10\n- v10 --supports--> v0b\n- v0b --refutes--> v06\n- v06 --refutes--> v02\n- v02 --supports--> v0f\n- v0f --refutes--> v0d\n- v0d --refutes--> v00\n- v00 --refutes--> v08\n- v08 --refutes--> v0c\n- v0c --supports--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v05\n- v02 --supports--> v0e\n- v08 --refutes--> v03\n- v0c --supports--> v01\n- v0c --supports--> v04\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":"3507ea439f1d078893aede6019e01949f6add6a7b5928b83996dc19e6258678a","schema":"livegraph.boolean-propagation.v1","spec_hash":"c38b03bffd3cbb3ea9db2457500ac63c8bbd5915987028c98ba99e6c4e8a7c73","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-dfc99ed29bd3d7701e25","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --supports--> v08\n- v08 --supports--> v0b\n- v0b --refutes--> v0d\n- v0d --supports--> v03\n- v03 --supports--> v09\n- v09 --refutes--> v06\n- v06 --supports--> v02\n- v02 --supports--> v04\n- v04 --refutes--> v0f\n- v0f --refutes--> v0e\n- v0e --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v07\n- v09 --supports--> v0c\n- v0b --refutes--> v0a\n- v0b --supports--> v01\n- v0b --supports--> v10\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":"7c14d8599764a607ddf45d990a932faee38468e2357c277943551c1182786364","schema":"livegraph.boolean-propagation.v1","spec_hash":"dfc99ed29bd3d7701e25623626b7759314f25092b0678105e0c01c82b3b300c7","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-4db060fc90087e4e8123","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v10\n- v10 --supports--> v05\n- v05 --supports--> v0e\n- v0e --supports--> v04\n- v04 --refutes--> v07\n- v07 --supports--> v00\n- v00 --refutes--> v08\n- v08 --refutes--> v09\n- v09 --supports--> v0d\n- v0d --refutes--> v03\n- v03 --supports--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v0b\n- v00 --supports--> v0f\n- v03 --supports--> v01\n- v05 --supports--> v0a\n- v07 --supports--> v0c\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":"e9573674620528ee46a259a1eddade6ec1b8798cb93a7ba519849dfc63a186d2","schema":"livegraph.boolean-propagation.v1","spec_hash":"4db060fc90087e4e8123d29c58d61eabab0a6d207031fb0a0e674da4f57894d1","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-f408603781c9daa34ac5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --refutes--> v0c\n- v0c --supports--> v08\n- v08 --refutes--> v0d\n- v0d --supports--> v07\n- v07 --refutes--> v06\n- v06 --supports--> v0b\n- v0b --refutes--> v09\n- v09 --supports--> v0f\n- v0f --supports--> v05\n- v05 --refutes--> v10\n- v10 --supports--> v0e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v00\n- v07 --refutes--> v04\n- v0c --refutes--> v01\n- v0d --refutes--> v03\n- v0f --supports--> v02\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":"04ee0e4523d84f636333a9f1533ed2e5987da7ed767c6fe3a48d430d2e1a2691","schema":"livegraph.boolean-propagation.v1","spec_hash":"f408603781c9daa34ac54d838394134fd0afaa851b8bd1c8a19bf028470f6a06","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-c32c0e314b6c5328264a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --supports--> v08\n- v08 --supports--> v00\n- v00 --refutes--> v02\n- v02 --supports--> v07\n- v07 --supports--> v05\n- v05 --refutes--> v10\n- v10 --supports--> v0b\n- v0b --refutes--> v06\n- v06 --refutes--> v03\n- v03 --supports--> v04\n- v04 --refutes--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v0d\n- v08 --refutes--> v01\n- v08 --refutes--> v09\n- v0a --refutes--> v0e\n- v10 --supports--> v0c\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":"27b7e43970d9291e6a6017c4b5160ae32f00872d44c28c4a00d77adde36ad9ea","schema":"livegraph.boolean-propagation.v1","spec_hash":"c32c0e314b6c5328264a01b87b10b5b6603efa1671e02b2dbc1b66c112a0f177","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-21e74acc475fbee2cb34","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --supports--> v04\n- v04 --refutes--> v08\n- v08 --refutes--> v10\n- v10 --refutes--> v06\n- v06 --refutes--> v02\n- v02 --supports--> v0c\n- v0c --refutes--> v0b\n- v0b --supports--> v05\n- v05 --refutes--> v09\n- v09 --refutes--> v0e\n- v0e --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v01\n- v04 --refutes--> v0f\n- v04 --supports--> v03\n- v0b --supports--> v07\n- v0b --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":"0380a9489e2e49ee9ddeac96bd7aaa38f6cedcf8b14ac21c1a773241565fd98b","schema":"livegraph.boolean-propagation.v1","spec_hash":"21e74acc475fbee2cb34808b4b507985bac9ca35b5b941e55b341ba9b5d7f726","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-82f1b68a3a043e0a0685","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --refutes--> v0d\n- v0d --refutes--> v08\n- v08 --supports--> v0b\n- v0b --refutes--> v0c\n- v0c --supports--> v00\n- v00 --supports--> v0e\n- v0e --refutes--> v04\n- v04 --supports--> v10\n- v10 --refutes--> v0f\n- v0f --refutes--> v0a\n- v0a --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v01\n- v00 --supports--> v06\n- v0a --refutes--> v07\n- v0b --supports--> v02\n- v10 --refutes--> v09\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":"2eb56ba10e4aee734b9448c7d8c7e8b885dd5bf3c3ac1b004dd47752ce39f848","schema":"livegraph.boolean-propagation.v1","spec_hash":"82f1b68a3a043e0a068522b61d095c5e45c48f58c48b57ce93df21f212cd1f4b","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-051f5ee81a39e53ecef3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --refutes--> v07\n- v07 --refutes--> v0e\n- v0e --refutes--> v0f\n- v0f --supports--> v05\n- v05 --supports--> v04\n- v04 --supports--> v06\n- v06 --supports--> v03\n- v03 --supports--> v0a\n- v0a --refutes--> v10\n- v10 --supports--> v08\n- v08 --supports--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v09\n- v03 --supports--> v01\n- v03 --supports--> v02\n- v06 --supports--> v0c\n- v0e --supports--> v0b\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":"380fadf9eb604358c866470ad1e264e5e62f1bad9fa9f91eace4162196965967","schema":"livegraph.boolean-propagation.v1","spec_hash":"051f5ee81a39e53ecef35b21d958df8008ee6cdc98099f55ef016fd52b347426","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-ddce2b39059b0b3e9f73","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --supports--> v0c\n- v0c --supports--> v07\n- v07 --supports--> v10\n- v10 --refutes--> v0b\n- v0b --supports--> v01\n- v01 --supports--> v09\n- v09 --supports--> v04\n- v04 --supports--> v08\n- v08 --supports--> v0d\n- v0d --supports--> v0f\n- v0f --refutes--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v0e\n- v01 --supports--> v00\n- v04 --supports--> v0a\n- v0d --refutes--> v05\n- v10 --refutes--> 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":"c8f0517a8c9988b965c9711e1c74c065714153a30dbd8b7b77f14c4fa34ce013","schema":"livegraph.boolean-propagation.v1","spec_hash":"ddce2b39059b0b3e9f73624b1dc23aca4254aee185c36cae5b88a97b6d26ccdf","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-51cd3665c558012d8854","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v10 --supports--> v05\n- v05 --refutes--> v06\n- v06 --supports--> v0f\n- v0f --supports--> v03\n- v03 --refutes--> v08\n- v08 --refutes--> v0e\n- v0e --supports--> v07\n- v07 --supports--> v01\n- v01 --refutes--> v04\n- v04 --refutes--> v02\n- v02 --supports--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v09\n- v04 --supports--> v0c\n- v04 --supports--> v0d\n- v08 --supports--> v00\n- v10 --supports--> v0b\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":"d7cf92e20e19ec88b8106a364e0cf0d1f61174b593ede297aaf2a59d845980ef","schema":"livegraph.boolean-propagation.v1","spec_hash":"51cd3665c558012d8854202f0829b107a6f88090ff5612a5807a8f9e3482bad1","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-03cc6b4e0d6684545c26","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0d --refutes--> v04\n- v04 --refutes--> v00\n- v00 --refutes--> v09\n- v09 --refutes--> v06\n- v06 --refutes--> v0c\n- v0c --refutes--> v0a\n- v0a --supports--> v08\n- v08 --refutes--> v03\n- v03 --supports--> v0b\n- v0b --supports--> v10\n- v10 --supports--> v0e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v02\n- v08 --supports--> v01\n- v09 --refutes--> v05\n- v0c --refutes--> v0f\n- v0d --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":"99ce3d58438abba8124b09c0b1b46806072d1fc2e8bc0ca9a7086d6864e51d78","schema":"livegraph.boolean-propagation.v1","spec_hash":"03cc6b4e0d6684545c265a57a8a6acdbc9e58cba2b67df0d9c6d08a8a73eeaa5","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-bedcd64d3da28c41dae3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v0d\n- v0d --refutes--> v06\n- v06 --refutes--> v0e\n- v0e --supports--> v10\n- v10 --supports--> v0a\n- v0a --refutes--> v0f\n- v0f --refutes--> v0c\n- v0c --refutes--> v01\n- v01 --supports--> v09\n- v09 --supports--> v07\n- v07 --supports--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v05\n- v07 --refutes--> v03\n- v0e --refutes--> v08\n- v10 --refutes--> v02\n- v10 --supports--> v0b\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":"f0d48656f912a10e096f6f9e354165c58b01057cc2f5a81a2dbefcedaee3c25d","schema":"livegraph.boolean-propagation.v1","spec_hash":"bedcd64d3da28c41dae3ddc87d624d95391ec771958f90149c64063af9f7b858","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-c3fb43e02ac63bb6e849","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v10 --refutes--> v0c\n- v0c --refutes--> v00\n- v00 --refutes--> v07\n- v07 --refutes--> v05\n- v05 --supports--> v0e\n- v0e --supports--> v0b\n- v0b --supports--> v0f\n- v0f --refutes--> v01\n- v01 --supports--> v06\n- v06 --supports--> v0d\n- v0d --refutes--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v04\n- v06 --refutes--> v03\n- v06 --supports--> v0a\n- v07 --supports--> v08\n- v0f --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":"226b52e62ad83865ac84b8f492c10f5d4e2efc2cf41687522de9edbe87e14c15","schema":"livegraph.boolean-propagation.v1","spec_hash":"c3fb43e02ac63bb6e8491f6967d1fb8bb18b5735efd24d11ef009144747b2f7f","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-603368578578e168136a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v02\n- v02 --supports--> v05\n- v05 --supports--> v0c\n- v0c --supports--> v0e\n- v0e --supports--> v04\n- v04 --refutes--> v0d\n- v0d --refutes--> v08\n- v08 --refutes--> v0f\n- v0f --supports--> v09\n- v09 --supports--> v10\n- v10 --supports--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v03\n- v06 --refutes--> v01\n- v08 --refutes--> v00\n- v0c --refutes--> v0a\n- v0c --refutes--> v0b\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":"4a7c720474c775c4560f89d7dc3054ccec31b6085d44d78422e232ab3ca41d8b","schema":"livegraph.boolean-propagation.v1","spec_hash":"603368578578e168136abb2032872236d0b4f7c237a50114d55e4b57396f09cd","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-23045147b17035185f61","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --supports--> v04\n- v04 --refutes--> v08\n- v08 --supports--> v09\n- v09 --supports--> v0e\n- v0e --supports--> v01\n- v01 --refutes--> v05\n- v05 --supports--> v0f\n- v0f --refutes--> v06\n- v06 --refutes--> v0d\n- v0d --supports--> v02\n- v02 --refutes--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v0b\n- v08 --refutes--> v00\n- v08 --refutes--> v10\n- v08 --supports--> v03\n- v0d --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":"114bcbef347a82096ea87168d6d0dce5775661b6e15070c5317a0b744ed09095","schema":"livegraph.boolean-propagation.v1","spec_hash":"23045147b17035185f6128bd2375b98aaf10e3cf84de22560b0ac89176848ad6","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-70641b2521b86275cc18","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0e --refutes--> v0c\n- v0c --supports--> v01\n- v01 --refutes--> v06\n- v06 --refutes--> v0b\n- v0b --refutes--> v03\n- v03 --refutes--> v04\n- v04 --refutes--> v08\n- v08 --supports--> v00\n- v00 --supports--> v09\n- v09 --refutes--> v0f\n- v0f --supports--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v10\n- v04 --supports--> v0d\n- v0f --refutes--> v05\n- v0f --refutes--> v07\n- v0f --refutes--> v0a\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":"bf211e66a35a121f0d942cbfde5bb650c0ec7787d3025b4dd3fea1a3d8fbbc36","schema":"livegraph.boolean-propagation.v1","spec_hash":"70641b2521b86275cc18b30a52b3654c6b085268926aba8e0db4db1d6b2d9ed8","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-ec64066de9121c3af474","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --supports--> v01\n- v01 --supports--> v07\n- v07 --supports--> v08\n- v08 --refutes--> v04\n- v04 --refutes--> v06\n- v06 --refutes--> v0a\n- v0a --supports--> v09\n- v09 --refutes--> v02\n- v02 --supports--> v00\n- v00 --supports--> v0b\n- v0b --refutes--> v0c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v0e\n- v07 --refutes--> v10\n- v08 --supports--> v03\n- v09 --refutes--> v0f\n- v0b --supports--> v0d\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":"831a38d58478f45641453dd969402bb9721aed78348dfc8b9cabfe9cb4031dc9","schema":"livegraph.boolean-propagation.v1","spec_hash":"ec64066de9121c3af474b09f2268ca41b395e27b2fbd64b85c1e570da6b888a7","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"}],\"hop\":11,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-c43743266183a0ae545d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --supports--> v0c\n- v0c --refutes--> v00\n- v00 --refutes--> v09\n- v09 --refutes--> v0f\n- v0f --refutes--> v0b\n- v0b --supports--> v0a\n- v0a --refutes--> v08\n- v08 --refutes--> v07\n- v07 --supports--> v10\n- v10 --supports--> v02\n- v02 --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v06\n- v01 --supports--> v0d\n- v09 --refutes--> v04\n- v09 --refutes--> v0e\n- v10 --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":"46c9635e81116428d4f0e6e8dd843115b53f4d2015183bd41847cbdd05d3a1e1","schema":"livegraph.boolean-propagation.v1","spec_hash":"c43743266183a0ae545ddc59e374236ce96f456c779f7ce9921413d78a0a7d5d","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-8f1b7ba2e085cb3dc0b3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --refutes--> v0f\n- v0f --supports--> v0b\n- v0b --refutes--> v06\n- v06 --supports--> v0a\n- v0a --refutes--> v09\n- v09 --supports--> v03\n- v03 --refutes--> v01\n- v01 --supports--> v08\n- v08 --refutes--> v0d\n- v0d --refutes--> v02\n- v02 --supports--> v10\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v05\n- v0b --supports--> v04\n- v0d --refutes--> v0e\n- v0d --supports--> v07\n- v0d --supports--> v0c\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":"55a06730c21c94192643edbf733497de401d8f85e7c16c643f2377715a8437c8","schema":"livegraph.boolean-propagation.v1","spec_hash":"8f1b7ba2e085cb3dc0b341e3fcf997f8c00600b6ab57dabb7c1156e46efdabf2","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-a5e38255181d97587e6d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v04\n- v04 --supports--> v0a\n- v0a --refutes--> v09\n- v09 --refutes--> v07\n- v07 --refutes--> v01\n- v01 --supports--> v03\n- v03 --refutes--> v08\n- v08 --refutes--> v06\n- v06 --supports--> v05\n- v05 --supports--> v0e\n- v0e --supports--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v0c\n- v03 --supports--> v0f\n- v08 --refutes--> v10\n- v09 --refutes--> v0b\n- v09 --refutes--> 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":"54d543645ac2abd0dfc5ae7b3d0cf55a2df6af0437a5d744b6a4cfa71d5ff2a8","schema":"livegraph.boolean-propagation.v1","spec_hash":"a5e38255181d97587e6d021c24b97e63196b882f2473d59d6597c702d8f9ce9d","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"}],\"hop\":11,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-fac157b25f64f79c4853","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --refutes--> v02\n- v02 --refutes--> v08\n- v08 --supports--> v0b\n- v0b --refutes--> v03\n- v03 --supports--> v0f\n- v0f --supports--> v05\n- v05 --refutes--> v0e\n- v0e --refutes--> v06\n- v06 --refutes--> v04\n- v04 --supports--> v10\n- v10 --supports--> v0c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v0a\n- v06 --supports--> v00\n- v0b --supports--> v07\n- v0b --supports--> v0d\n- v10 --supports--> v01\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":"650af4048e1b6b2b77dd30e4f4775e8b2da4e269ef8bb0871c18ad5bbad1a6c5","schema":"livegraph.boolean-propagation.v1","spec_hash":"fac157b25f64f79c485323fb599af337d508f6c677082f9ef662a81a2a1fa10e","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-a9015d6e98998b45da39","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --refutes--> v05\n- v05 --refutes--> v02\n- v02 --supports--> v04\n- v04 --refutes--> v0f\n- v0f --supports--> v0b\n- v0b --supports--> v08\n- v08 --supports--> v07\n- v07 --refutes--> v0e\n- v0e --supports--> v0a\n- v0a --refutes--> v09\n- v09 --refutes--> v10\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0c\n- v05 --refutes--> v0d\n- v07 --supports--> v03\n- v09 --supports--> v06\n- v0a --supports--> v01\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":"a8284ee15a0386db4dc00c121f043e5967f56255221c32fabbdf7f8963e2174e","schema":"livegraph.boolean-propagation.v1","spec_hash":"a9015d6e98998b45da397b8aca7a548132beed9d2ca0a8b9245cefca27bbe58d","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-032b55d9b3d2fa12a610","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0d --refutes--> v0c\n- v0c --refutes--> v05\n- v05 --supports--> v0b\n- v0b --supports--> v00\n- v00 --supports--> v0f\n- v0f --supports--> v03\n- v03 --refutes--> v10\n- v10 --supports--> v04\n- v04 --refutes--> v08\n- v08 --supports--> v0a\n- v0a --refutes--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v06\n- v0d --refutes--> v02\n- v0f --refutes--> v09\n- v0f --supports--> v0e\n- v10 --refutes--> v07\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":"c0ea364edaaed4150b1ff0f32c8dd1cd35e580549a67bf4443e0b9af446c6034","schema":"livegraph.boolean-propagation.v1","spec_hash":"032b55d9b3d2fa12a610277e12a16afc8453b73530a9104fd0c3ff562c9f5c83","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-16d2fba1e79e027c540d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --supports--> v09\n- v09 --supports--> v05\n- v05 --refutes--> v0c\n- v0c --refutes--> v0e\n- v0e --supports--> v02\n- v02 --refutes--> v07\n- v07 --refutes--> v01\n- v01 --refutes--> v0d\n- v0d --supports--> v0f\n- v0f --refutes--> v00\n- v00 --refutes--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v10\n- v07 --supports--> v0a\n- v08 --supports--> v03\n- v08 --supports--> v0b\n- v0c --refutes--> v04\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":"ca9ef74a25356e58c8e78928bfbd086a8ba082f425e8734939f31e5167f263d2","schema":"livegraph.boolean-propagation.v1","spec_hash":"16d2fba1e79e027c540d238759b4e46edf526a0ebc4d2a589ef6b7bc4e53411e","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-5ee644d2137841dccaa5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --supports--> v09\n- v09 --supports--> v00\n- v00 --refutes--> v07\n- v07 --refutes--> v0a\n- v0a --refutes--> v04\n- v04 --refutes--> v10\n- v10 --refutes--> v08\n- v08 --supports--> v01\n- v01 --refutes--> v0f\n- v0f --supports--> v0e\n- v0e --supports--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v02\n- v07 --supports--> v0d\n- v0a --refutes--> v0b\n- v0f --supports--> v0c\n- v10 --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":"66d92bb0ec6c133c6537e7c5f8c774da20b0fb287ff71177821a457ad55778dd","schema":"livegraph.boolean-propagation.v1","spec_hash":"5ee644d2137841dccaa564e512e8aff1e1c8f5a4e411dfb8545cd8603e17de8d","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-d9354dae659c1d5f8cd9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --refutes--> v09\n- v09 --supports--> v0b\n- v0b --refutes--> v00\n- v00 --refutes--> v10\n- v10 --supports--> v0a\n- v0a --supports--> v07\n- v07 --refutes--> v0e\n- v0e --supports--> v05\n- v05 --refutes--> v02\n- v02 --refutes--> v03\n- v03 --supports--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v08\n- v05 --supports--> v04\n- v09 --refutes--> v0d\n- v0a --supports--> v06\n- v0e --refutes--> v0c\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":"5b8d95a434cf77bc1030c8e1d60591ad8b3a749bc2a23ce1eaa7880ddeb671a8","schema":"livegraph.boolean-propagation.v1","spec_hash":"d9354dae659c1d5f8cd944b20335c216f7113ae8b72d74313ab39f8bf6e34e43","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-6896216e08401183e4e8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --refutes--> v03\n- v03 --supports--> v0d\n- v0d --supports--> v0f\n- v0f --refutes--> v09\n- v09 --supports--> v0b\n- v0b --supports--> v00\n- v00 --supports--> v05\n- v05 --refutes--> v07\n- v07 --refutes--> v04\n- v04 --refutes--> v0e\n- v0e --supports--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v06\n- v08 --refutes--> v02\n- v09 --supports--> v0a\n- v0d --supports--> v10\n- v0f --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":"bf4ea9e0d2d31fb7755d2649f2af524ea60813a05bc026f3a75ef26565b7532f","schema":"livegraph.boolean-propagation.v1","spec_hash":"6896216e08401183e4e8ce1595e77af27fdf073e63fa42d583fefb932508e2d7","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-f229c925fb627a81304e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --refutes--> v08\n- v08 --refutes--> v02\n- v02 --refutes--> v0d\n- v0d --supports--> v09\n- v09 --supports--> v0f\n- v0f --supports--> v0e\n- v0e --supports--> v0a\n- v0a --supports--> v01\n- v01 --supports--> v00\n- v00 --supports--> v0b\n- v0b --refutes--> v0c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v07\n- v0a --supports--> v10\n- v0b --supports--> v04\n- v0d --refutes--> v05\n- v0f --refutes--> v06\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":"4d2c3095a39c6db538f6f39006bd507ff5b77f80cfdf8a4b88e722b6bb2b69c4","schema":"livegraph.boolean-propagation.v1","spec_hash":"f229c925fb627a81304e9712c56fcc1df1789edc24ccff99bf1d254786b7d497","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-232ca37ef82d408c58a0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --supports--> v0c\n- v0c --supports--> v0e\n- v0e --refutes--> v04\n- v04 --refutes--> v01\n- v01 --supports--> v10\n- v10 --refutes--> v08\n- v08 --refutes--> v0b\n- v0b --refutes--> v0d\n- v0d --refutes--> v00\n- v00 --supports--> v06\n- v06 --refutes--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v0a\n- v02 --supports--> v0f\n- v06 --supports--> v03\n- v08 --refutes--> v07\n- v10 --refutes--> v09\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":"787da2020a78720789b8c6955c09beb34c19c1c300c8d758b88c9a6e99827d44","schema":"livegraph.boolean-propagation.v1","spec_hash":"232ca37ef82d408c58a05520ef65e508cb2347f49ca05eaf9bb18fe01de97f21","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-9f60bd83c299c177ea91","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v10 --refutes--> v09\n- v09 --supports--> v04\n- v04 --refutes--> v0d\n- v0d --supports--> v05\n- v05 --refutes--> v07\n- v07 --supports--> v0c\n- v0c --refutes--> v0b\n- v0b --supports--> v0a\n- v0a --refutes--> v03\n- v03 --supports--> v06\n- v06 --supports--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v07 --supports--> v0f\n- v09 --supports--> v01\n- v0a --supports--> v0e\n- v0c --refutes--> v02\n- v0d --supports--> v00\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":"a3c01207cfa268027441928ae0c84e53824266ec016c71eaffbdcd4209d4f689","schema":"livegraph.boolean-propagation.v1","spec_hash":"9f60bd83c299c177ea91f6c7e28bcd34920b5edfbee719bd2d0216749a1faa14","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-952680fce7eff991f73a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --refutes--> v0b\n- v0b --refutes--> v04\n- v04 --supports--> v05\n- v05 --supports--> v0a\n- v0a --supports--> v10\n- v10 --refutes--> v0e\n- v0e --supports--> v0d\n- v0d --refutes--> v0c\n- v0c --supports--> v06\n- v06 --supports--> v07\n- v07 --refutes--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v08\n- v0a --supports--> v02\n- v0a --supports--> v03\n- v0e --supports--> v00\n- v0e --supports--> v0f\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":"c25e546bfd85c2fcd6033519869b7ff0a16093975ebb90d355204c553689ef37","schema":"livegraph.boolean-propagation.v1","spec_hash":"952680fce7eff991f73a5cac24e64f9a25ac96404a893d55cbc0038676e70096","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-2b297d8c6f52820de9f1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0d --supports--> v01\n- v01 --refutes--> v0a\n- v0a --refutes--> v0f\n- v0f --supports--> v05\n- v05 --supports--> v04\n- v04 --supports--> v08\n- v08 --supports--> v02\n- v02 --supports--> v0c\n- v0c --refutes--> v0e\n- v0e --supports--> v0b\n- v0b --refutes--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v09\n- v0d --supports--> v07\n- v0e --refutes--> v00\n- v0e --supports--> v10\n- v0f --supports--> v03\n\n[VERIFIED ROOT]\nv0d = 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":"882d5df346682f6ea1e6eba68615fed1685286f307c987246404e194a523787a","schema":"livegraph.boolean-propagation.v1","spec_hash":"2b297d8c6f52820de9f10c060cf70ffcbfd36fd35757ba7048f40f2412ccfc43","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-29a1a70dfd5e3a4d4acf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --refutes--> v09\n- v09 --supports--> v05\n- v05 --supports--> v0e\n- v0e --refutes--> v06\n- v06 --supports--> v00\n- v00 --refutes--> v01\n- v01 --refutes--> v0f\n- v0f --refutes--> v03\n- v03 --refutes--> v0d\n- v0d --supports--> v04\n- v04 --supports--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v0c\n- v05 --supports--> v02\n- v0a --supports--> v0b\n- v0d --supports--> v10\n- v0f --refutes--> v08\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":"fcd6e3aa3c6555688334c2c880c84125c0a7cea5614c5cc4cbd904d67d7b5632","schema":"livegraph.boolean-propagation.v1","spec_hash":"29a1a70dfd5e3a4d4acfb7509fa591359729fd99d11346b0dadb5cf9f7e6bbb4","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-84394e612c14010e5c93","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --refutes--> v00\n- v00 --supports--> v05\n- v05 --supports--> v0f\n- v0f --refutes--> v07\n- v07 --refutes--> v01\n- v01 --refutes--> v06\n- v06 --supports--> v10\n- v10 --refutes--> v0d\n- v0d --supports--> v04\n- v04 --refutes--> v0c\n- v0c --supports--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v02\n- v00 --supports--> v0a\n- v01 --supports--> v08\n- v03 --supports--> v0e\n- v0c --supports--> v0b\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":"3e949836dbdc6663f1f1092787e7caf861dc1785bc4884c907d94944d733eb0d","schema":"livegraph.boolean-propagation.v1","spec_hash":"84394e612c14010e5c9397dc5a6779e0a1e7271e971ec698ab5016dc412e3347","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-a3b96caaafb9e6ccc30a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --refutes--> v06\n- v06 --refutes--> v0e\n- v0e --supports--> v08\n- v08 --refutes--> v0b\n- v0b --supports--> v07\n- v07 --refutes--> v0d\n- v0d --refutes--> v10\n- v10 --refutes--> v01\n- v01 --refutes--> v00\n- v00 --refutes--> v05\n- v05 --refutes--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v0a\n- v01 --refutes--> v0c\n- v05 --supports--> v09\n- v07 --refutes--> v02\n- v08 --refutes--> v0f\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":"273507850035b125058b4ea2b497074fa72a1edcf47e70d3fb4fec071be24fdc","schema":"livegraph.boolean-propagation.v1","spec_hash":"a3b96caaafb9e6ccc30adf8aa47e64216f24416bd1ca893a5b37434ce88b4be5","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-a6d3908eae7c256c19df","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0d --supports--> v06\n- v06 --supports--> v0a\n- v0a --refutes--> v08\n- v08 --supports--> v10\n- v10 --supports--> v01\n- v01 --supports--> v09\n- v09 --refutes--> v07\n- v07 --supports--> v02\n- v02 --supports--> v04\n- v04 --supports--> v0e\n- v0e --supports--> v0c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v0f\n- v04 --refutes--> v03\n- v04 --refutes--> v05\n- v06 --refutes--> v0b\n- v08 --supports--> v00\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":"f2dbbccefdb66092627e836e0b320eb457f5eb540735e8bf289a3fb2f2db9cb7","schema":"livegraph.boolean-propagation.v1","spec_hash":"a6d3908eae7c256c19df78daef3a5287cda33a74d2cd3ff927c9d8e644684af9","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-1b9d94046eefb4b432ba","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --supports--> v0b\n- v0b --supports--> v09\n- v09 --refutes--> v06\n- v06 --supports--> v0d\n- v0d --supports--> v10\n- v10 --refutes--> v0e\n- v0e --supports--> v0c\n- v0c --refutes--> v02\n- v02 --supports--> v0f\n- v0f --refutes--> v03\n- v03 --supports--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v05\n- v06 --refutes--> v04\n- v0b --supports--> v07\n- v0f --refutes--> v01\n- v10 --refutes--> 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":"7f2b074e13960d4a7109214610271f330fda5c955824774536566a5a1ddda645","schema":"livegraph.boolean-propagation.v1","spec_hash":"1b9d94046eefb4b432ba34770f135b45f13486e7ece3c541cf00d62339f84acc","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-b099fd141cbaf67570a1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --refutes--> v0f\n- v0f --supports--> v04\n- v04 --refutes--> v0e\n- v0e --supports--> v0c\n- v0c --refutes--> v0d\n- v0d --refutes--> v06\n- v06 --supports--> v08\n- v08 --refutes--> v07\n- v07 --supports--> v00\n- v00 --refutes--> v0a\n- v0a --supports--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v0b\n- v08 --refutes--> v10\n- v08 --supports--> v02\n- v09 --refutes--> v05\n- v0a --refutes--> v01\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":"10e47d90f50fb97c3b0ffa7ca3d3336e0f2e698edf3f250368d5f9a7a16ccb6a","schema":"livegraph.boolean-propagation.v1","spec_hash":"b099fd141cbaf67570a1f2903a822e267a9dec5c72f680d2f72d45083c5961e0","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-dbffcb68a0b0602a5629","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0e --supports--> v0f\n- v0f --refutes--> v02\n- v02 --supports--> v05\n- v05 --supports--> v08\n- v08 --refutes--> v10\n- v10 --refutes--> v0d\n- v0d --supports--> v0a\n- v0a --refutes--> v07\n- v07 --refutes--> v06\n- v06 --supports--> v04\n- v04 --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v01\n- v04 --supports--> v0c\n- v08 --supports--> v0b\n- v0d --supports--> v00\n- v0f --refutes--> v09\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":"8c5f08bf1106d16556756f96e07975c98bcc2028b3354da6b6dce751a4165451","schema":"livegraph.boolean-propagation.v1","spec_hash":"dbffcb68a0b0602a56290bc16b5fd0e5a5df3e097b2c889ad4e5b3deb4bfd154","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-5472ee9d40f35c6dacf7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0b --supports--> v0f\n- v0f --refutes--> v06\n- v06 --refutes--> v00\n- v00 --refutes--> v07\n- v07 --refutes--> v04\n- v04 --supports--> v0d\n- v0d --refutes--> v10\n- v10 --supports--> v0c\n- v0c --refutes--> v01\n- v01 --refutes--> v05\n- v05 --supports--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v03\n- v06 --supports--> v08\n- v07 --refutes--> v02\n- v0b --refutes--> v09\n- v0b --supports--> v0e\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":"ab262418551d57f2b82c498fd83c431882c8da9bf0ad2fc9a5fdc05db2cba349","schema":"livegraph.boolean-propagation.v1","spec_hash":"5472ee9d40f35c6dacf7d8b1f3a343ce52deb9c758578e3a95f697385684652a","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":11,"item_id":"lggrpo1-19cab6753f7d74c8005b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --supports--> v06\n- v06 --supports--> v0c\n- v0c --supports--> v0b\n- v0b --supports--> v0d\n- v0d --supports--> v00\n- v00 --supports--> v03\n- v03 --refutes--> v02\n- v02 --supports--> v08\n- v08 --refutes--> v0e\n- v0e --supports--> v0f\n- v0f --supports--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v10\n- v06 --refutes--> v09\n- v08 --refutes--> v07\n- v0b --refutes--> v01\n- v0c --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":"f81afe021253935c7fa840cdb57f49cb2c5c707fb49b7be58ebf0032f0bfa47b","schema":"livegraph.boolean-propagation.v1","spec_hash":"19cab6753f7d74c8005b7572d77212c2a38cfaa8041c1ec719f09ae00bd1cb5b","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":11,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":11,"item_id":"lggrpo1-4089ce0b12ccd0bb4ff4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --refutes--> v00\n- v00 --supports--> v10\n- v10 --refutes--> v02\n- v02 --supports--> v0f\n- v0f --supports--> v07\n- v07 --refutes--> v04\n- v04 --refutes--> v05\n- v05 --supports--> v0d\n- v0d --supports--> v06\n- v06 --refutes--> v0a\n- v0a --refutes--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v08\n- v02 --refutes--> v0e\n- v05 --refutes--> v03\n- v0d --refutes--> v0c\n- v0f --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":"b54f7c89429f8a0cf3928c265178bbc5117470e23daab1ee9857a21e2011a427","schema":"livegraph.boolean-propagation.v1","spec_hash":"4089ce0b12ccd0bb4ff4123e0058bfb556125440698d5a68837f4cc5202ef8f5","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":11,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-334c4321884c57d5705f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v10\n- v10 --supports--> v0c\n- v0c --refutes--> v03\n- v03 --refutes--> v05\n- v05 --refutes--> v0e\n- v0e --supports--> v07\n- v07 --supports--> v04\n- v04 --refutes--> v01\n- v01 --supports--> v0f\n- v0f --refutes--> v0b\n- v0b --supports--> v02\n- v02 --supports--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v06\n- v03 --supports--> v0a\n- v04 --supports--> v12\n- v0b --refutes--> v08\n- v0c --refutes--> v11\n- v10 --refutes--> v0d\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":"fa4170dd553194914ecaa15c04923c6d9394db0f12c5611726be1a5a52ef40e8","schema":"livegraph.boolean-propagation.v1","spec_hash":"334c4321884c57d5705f76e7bc8e562852b7a3499fd130cd242543a5f8e0f34d","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-3951f99db88f79814f66","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0d --refutes--> v0e\n- v0e --refutes--> v0b\n- v0b --refutes--> v03\n- v03 --supports--> v07\n- v07 --supports--> v09\n- v09 --refutes--> v0f\n- v0f --supports--> v0c\n- v0c --refutes--> v00\n- v00 --refutes--> v05\n- v05 --supports--> v10\n- v10 --supports--> v12\n- v12 --supports--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v0b --supports--> v0a\n- v0c --refutes--> v01\n- v0e --supports--> v11\n- v0f --refutes--> v04\n- v10 --supports--> v02\n- v10 --supports--> v06\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":"cb0c7746ef244f09eaba9cc85d91015a8f892d82b3520d67090be338baed6040","schema":"livegraph.boolean-propagation.v1","spec_hash":"3951f99db88f79814f66d0b1a2b05a6861e712c652a566df475dffc2e4752a50","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-d4564e1a4fdd589c0e99","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0f --refutes--> v0d\n- v0d --supports--> v08\n- v08 --supports--> v11\n- v11 --refutes--> v0a\n- v0a --refutes--> v01\n- v01 --refutes--> v02\n- v02 --supports--> v12\n- v12 --refutes--> v0c\n- v0c --refutes--> v06\n- v06 --refutes--> v09\n- v09 --refutes--> v05\n- v05 --refutes--> v10\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v07\n- v09 --refutes--> v03\n- v09 --refutes--> v0e\n- v0a --refutes--> v00\n- v0f --supports--> v04\n- v11 --refutes--> 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":"d3eb79c7ac762923a5018b420ec25e2a0c3eb1114d52d1fde395d7469cfba7c4","schema":"livegraph.boolean-propagation.v1","spec_hash":"d4564e1a4fdd589c0e997df8cf974b7a70d2020647aa3067ab950884f441100d","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-4a28d7d733b49db70ba1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --refutes--> v06\n- v06 --refutes--> v05\n- v05 --refutes--> v0c\n- v0c --supports--> v00\n- v00 --supports--> v0f\n- v0f --refutes--> v12\n- v12 --supports--> v0a\n- v0a --supports--> v04\n- v04 --refutes--> v11\n- v11 --refutes--> v0d\n- v0d --supports--> v09\n- v09 --supports--> v10\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v01\n- v09 --supports--> v0e\n- v0c --supports--> v03\n- v0d --supports--> v07\n- v11 --refutes--> v02\n- v12 --refutes--> v0b\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":"2e38d957e7035a881cc159abc4a6da5f3b320e004f6a1cc4f201db62602510db","schema":"livegraph.boolean-propagation.v1","spec_hash":"4a28d7d733b49db70ba191bdbb5daba416195080bdae8bcbb9568811a9586e0b","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-7faffb018c79f1f0d8cd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v10 --refutes--> v06\n- v06 --supports--> v03\n- v03 --refutes--> v0b\n- v0b --supports--> v12\n- v12 --refutes--> v01\n- v01 --supports--> v09\n- v09 --supports--> v08\n- v08 --refutes--> v11\n- v11 --supports--> v0f\n- v0f --supports--> v0d\n- v0d --refutes--> v00\n- v00 --supports--> v0e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v04\n- v01 --supports--> v05\n- v03 --refutes--> v07\n- v09 --supports--> v0c\n- v0b --refutes--> v02\n- v11 --refutes--> v0a\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":"bac318c73fc44af554490218ad9fd9787179206ba487c8f9879c3c587a2667f8","schema":"livegraph.boolean-propagation.v1","spec_hash":"7faffb018c79f1f0d8cdc74cafcfa3be8e9480d3de424af56bdfd5def14904e9","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-e0bd7559f98886cabba7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v11 --supports--> v03\n- v03 --supports--> v12\n- v12 --supports--> v07\n- v07 --supports--> v10\n- v10 --supports--> v02\n- v02 --refutes--> v0d\n- v0d --supports--> v00\n- v00 --supports--> v0c\n- v0c --supports--> v0e\n- v0e --refutes--> v0a\n- v0a --refutes--> v0f\n- v0f --refutes--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v04\n- v02 --supports--> v01\n- v03 --refutes--> v06\n- v03 --refutes--> v0b\n- v10 --supports--> v05\n- v11 --supports--> v09\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":"2cd5e14bca9ddf60db17f75eee7f7c7e11d9fc10179e53d88f8b6faaad6108e2","schema":"livegraph.boolean-propagation.v1","spec_hash":"e0bd7559f98886cabba7a1aa623ecb7d881bb851071085fd776c545857dc5835","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-53b4be2f74cbd1e9d70f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --supports--> v12\n- v12 --supports--> v0a\n- v0a --supports--> v04\n- v04 --supports--> v0e\n- v0e --refutes--> v10\n- v10 --refutes--> v00\n- v00 --refutes--> v09\n- v09 --refutes--> v02\n- v02 --supports--> v0f\n- v0f --supports--> v03\n- v03 --refutes--> v06\n- v06 --refutes--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v08\n- v03 --refutes--> v0b\n- v07 --refutes--> v05\n- v07 --refutes--> v0c\n- v09 --supports--> v11\n- v12 --supports--> v0d\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":"cf7ee50305b759e3370a574608e67ab477dbc9d602aa544eb4d82cee3d3265ef","schema":"livegraph.boolean-propagation.v1","spec_hash":"53b4be2f74cbd1e9d70fc7ff0088df629b01c153078c2c4f4af93f98b026e36e","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-a18b7e79249998a607a0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0d --supports--> v0b\n- v0b --supports--> v0c\n- v0c --refutes--> v05\n- v05 --refutes--> v00\n- v00 --supports--> v09\n- v09 --supports--> v08\n- v08 --supports--> v01\n- v01 --supports--> v03\n- v03 --supports--> v0f\n- v0f --supports--> v06\n- v06 --refutes--> v11\n- v11 --refutes--> v12\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v07\n- v03 --refutes--> v0a\n- v03 --supports--> v10\n- v0c --refutes--> v0e\n- v0c --supports--> v02\n- v0d --supports--> v04\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":"b02b32f8f9c4c467a966274fea15b89c73356fdd32e3d1593dfbe5f933a7d140","schema":"livegraph.boolean-propagation.v1","spec_hash":"a18b7e79249998a607a0c631b19e47a3d490f88583c9a849bed2fccba58a76a1","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-5671fe4837609ab30685","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --refutes--> v03\n- v03 --refutes--> v07\n- v07 --supports--> v08\n- v08 --supports--> v12\n- v12 --refutes--> v01\n- v01 --supports--> v10\n- v10 --refutes--> v0b\n- v0b --supports--> v02\n- v02 --supports--> v00\n- v00 --supports--> v0e\n- v0e --supports--> v06\n- v06 --supports--> v11\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v05\n- v03 --refutes--> v04\n- v03 --supports--> v0d\n- v07 --refutes--> v09\n- v0b --refutes--> v0c\n- v10 --refutes--> v0f\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":"246680705b44ae8c46972a99bab9af7fbb6a3ae0e83c7231157828859bf6ec30","schema":"livegraph.boolean-propagation.v1","spec_hash":"5671fe4837609ab306858a014d29b38e7dd1416c0056bc2ab1c6d1bb0a2cf7ba","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-6b963b5bc6982941624e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --refutes--> v01\n- v01 --refutes--> v0f\n- v0f --supports--> v04\n- v04 --supports--> v06\n- v06 --supports--> v11\n- v11 --supports--> v0e\n- v0e --refutes--> v09\n- v09 --refutes--> v07\n- v07 --supports--> v0b\n- v0b --supports--> v08\n- v08 --supports--> v0d\n- v0d --refutes--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v12\n- v05 --refutes--> v03\n- v06 --supports--> v10\n- v0b --refutes--> v00\n- v0b --refutes--> v0c\n- v0d --supports--> 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":"cabc1ca030d9db2bda99d62b8f58c162453c4866db8d25dfed7b066f2610d880","schema":"livegraph.boolean-propagation.v1","spec_hash":"6b963b5bc6982941624e1c20c03a11d602f9f050e4e956ef3e2fd669eec26444","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-90dc855d29f772c59b9d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0b --refutes--> v01\n- v01 --supports--> v09\n- v09 --supports--> v07\n- v07 --supports--> v04\n- v04 --refutes--> v03\n- v03 --supports--> v0a\n- v0a --supports--> v02\n- v02 --supports--> v0c\n- v0c --supports--> v08\n- v08 --supports--> v05\n- v05 --refutes--> v0f\n- v0f --supports--> v11\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v00\n- v04 --refutes--> v12\n- v05 --refutes--> v06\n- v07 --refutes--> v0d\n- v08 --refutes--> v0e\n- v0b --refutes--> v10\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":"9c0818f176ba4f92657f9db3a60fdefbd831a7696dfec5a03ed92de202d9f79a","schema":"livegraph.boolean-propagation.v1","spec_hash":"90dc855d29f772c59b9d6b74bf228fcf0f090063cd6df63d67f497c48ff7f1cb","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":12,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-3bfb125c73511973fbf3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --supports--> v01\n- v01 --refutes--> v0e\n- v0e --supports--> v02\n- v02 --refutes--> v0b\n- v0b --refutes--> v0a\n- v0a --refutes--> v06\n- v06 --refutes--> v11\n- v11 --supports--> v03\n- v03 --refutes--> v10\n- v10 --supports--> v00\n- v00 --refutes--> v05\n- v05 --refutes--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0f\n- v03 --supports--> v04\n- v03 --supports--> v08\n- v0c --supports--> v0d\n- v0e --refutes--> v12\n- v11 --supports--> v09\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":"69d1f9c2cf78afea92b95546d7dc2d398111660b97b3b752da59dd8b85c0c4d0","schema":"livegraph.boolean-propagation.v1","spec_hash":"3bfb125c73511973fbf38c9ecc841892cfcb2724d71e4b05fd65fdead239bc9d","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-09d3c4809fea732260d3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --refutes--> v05\n- v05 --refutes--> v12\n- v12 --refutes--> v0e\n- v0e --supports--> v0b\n- v0b --refutes--> v0a\n- v0a --refutes--> v11\n- v11 --refutes--> v04\n- v04 --refutes--> v01\n- v01 --refutes--> v08\n- v08 --refutes--> v0d\n- v0d --refutes--> v09\n- v09 --refutes--> v0c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v06\n- v05 --refutes--> v07\n- v08 --refutes--> v0f\n- v08 --supports--> v03\n- v0e --refutes--> v10\n- v12 --supports--> v02\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":"8da8202f82c103265ac0011c1c742c06c33092c07fd6cbea396bdcc623529c22","schema":"livegraph.boolean-propagation.v1","spec_hash":"09d3c4809fea732260d3e688aed920d7dc0227e942580724d4ba8e03d084d375","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-672a928e2f664c7e10d6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0f --supports--> v07\n- v07 --refutes--> v12\n- v12 --supports--> v08\n- v08 --supports--> v05\n- v05 --refutes--> v03\n- v03 --refutes--> v06\n- v06 --refutes--> v01\n- v01 --supports--> v10\n- v10 --refutes--> v04\n- v04 --supports--> v00\n- v00 --supports--> v0d\n- v0d --refutes--> v0c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v11\n- v05 --supports--> v02\n- v06 --refutes--> v0a\n- v08 --supports--> v0e\n- v0f --supports--> v0b\n- v12 --supports--> v09\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":"b43afe4f41a5e64b7160ab44776ff9c6368b2cb2b1cc2f7b79416871303b3d84","schema":"livegraph.boolean-propagation.v1","spec_hash":"672a928e2f664c7e10d6d654a1a10c25ed7ca9e5fab0034b6db6602c5429aa23","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-9989016664fde20c771f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --refutes--> v01\n- v01 --refutes--> v0a\n- v0a --refutes--> v12\n- v12 --refutes--> v11\n- v11 --refutes--> v0f\n- v0f --refutes--> v10\n- v10 --supports--> v0c\n- v0c --supports--> v02\n- v02 --refutes--> v06\n- v06 --supports--> v07\n- v07 --refutes--> v05\n- v05 --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v04\n- v01 --supports--> v0b\n- v06 --refutes--> v0e\n- v06 --supports--> v0d\n- v0c --supports--> v09\n- v12 --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":"4cb585e343f5ff6d5f8bc12c2824fb3b08b0e8c3b6bb5c550d4c1586b54a8f2e","schema":"livegraph.boolean-propagation.v1","spec_hash":"9989016664fde20c771f254d162ce77b3c0d93519ea07398c61208f56dd315d3","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-4f53a55527a7094ee780","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v12\n- v12 --supports--> v03\n- v03 --refutes--> v08\n- v08 --supports--> v0d\n- v0d --refutes--> v10\n- v10 --refutes--> v07\n- v07 --supports--> v06\n- v06 --refutes--> v11\n- v11 --supports--> v09\n- v09 --refutes--> v0f\n- v0f --supports--> v02\n- v02 --supports--> v0c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v08 --refutes--> v01\n- v08 --refutes--> v0e\n- v0d --refutes--> v0a\n- v0d --refutes--> v0b\n- v11 --supports--> v05\n- v12 --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":"051f029b6ee84138a2b674f8aec5252cfc43dc7008b90d1364f79baca4f93164","schema":"livegraph.boolean-propagation.v1","spec_hash":"4f53a55527a7094ee78023a1aad33a6ff7c45ab7fd98bb30eae7c2cc3271ec03","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-f31bf45a0b7dd305a96b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v10 --refutes--> v04\n- v04 --refutes--> v09\n- v09 --refutes--> v06\n- v06 --supports--> v0d\n- v0d --supports--> v01\n- v01 --refutes--> v0e\n- v0e --refutes--> v08\n- v08 --refutes--> v00\n- v00 --supports--> v05\n- v05 --supports--> v07\n- v07 --supports--> v0f\n- v0f --supports--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v0b\n- v07 --supports--> v03\n- v08 --supports--> v11\n- v09 --refutes--> v12\n- v09 --supports--> v02\n- v10 --refutes--> v0c\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":"d0a7dc36a571e313d19c0ff301cd424e93458500753718f86f5993084f3693cc","schema":"livegraph.boolean-propagation.v1","spec_hash":"f31bf45a0b7dd305a96b82b6291bff396b843b95b74459f874297bcbbe27790f","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-a35fb3e817fd8140a507","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0e --refutes--> v07\n- v07 --supports--> v03\n- v03 --refutes--> v08\n- v08 --refutes--> v0f\n- v0f --refutes--> v09\n- v09 --supports--> v00\n- v00 --refutes--> v0d\n- v0d --supports--> v01\n- v01 --refutes--> v05\n- v05 --supports--> v02\n- v02 --supports--> v10\n- v10 --refutes--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v04\n- v00 --refutes--> v0a\n- v00 --supports--> v12\n- v03 --refutes--> v11\n- v05 --supports--> v06\n- v0e --supports--> v0c\n\n[VERIFIED ROOT]\nv0e = 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":"7c6f9ac84388613d732875bdb88324936b464c53caacd6c4a17ef117eff99f92","schema":"livegraph.boolean-propagation.v1","spec_hash":"a35fb3e817fd8140a50723f09ab8709771b35b8ff2c604b1ffd38b06e64114fc","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-26d8e9213c42e75b00b9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0b --supports--> v04\n- v04 --supports--> v09\n- v09 --refutes--> v03\n- v03 --supports--> v00\n- v00 --supports--> v0e\n- v0e --supports--> v0c\n- v0c --refutes--> v07\n- v07 --refutes--> v0a\n- v0a --supports--> v08\n- v08 --refutes--> v10\n- v10 --supports--> v11\n- v11 --refutes--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v02\n- v04 --refutes--> v0f\n- v07 --supports--> v0d\n- v09 --supports--> v05\n- v0a --refutes--> v12\n- v0c --supports--> v01\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":"ce1c52c485acc9e38d15eee965b3a96f72107a300231a410f80c7777ba1065cd","schema":"livegraph.boolean-propagation.v1","spec_hash":"26d8e9213c42e75b00b969a3f3639e9534fcac6ba81e080826fd06f65e32d018","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-125853f8521e1f7f8326","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --refutes--> v08\n- v08 --supports--> v06\n- v06 --supports--> v04\n- v04 --supports--> v07\n- v07 --supports--> v10\n- v10 --supports--> v0b\n- v0b --refutes--> v0a\n- v0a --refutes--> v01\n- v01 --refutes--> v05\n- v05 --supports--> v03\n- v03 --supports--> v0e\n- v0e --refutes--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v0f\n- v04 --refutes--> v0d\n- v0a --supports--> v00\n- v0c --refutes--> v11\n- v0e --supports--> v09\n- v10 --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":"3abca9b59508db290070b45b25e6a715b1d24ba9c37d91f851a4c6d016b1e04c","schema":"livegraph.boolean-propagation.v1","spec_hash":"125853f8521e1f7f83262de02c2066be1d71bf63cd918a29a111fe17a9ad61fa","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-9e265c3e41a539a78e79","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0b --supports--> v05\n- v05 --refutes--> v09\n- v09 --supports--> v07\n- v07 --refutes--> v0e\n- v0e --refutes--> v08\n- v08 --refutes--> v06\n- v06 --refutes--> v00\n- v00 --refutes--> v03\n- v03 --supports--> v02\n- v02 --supports--> v04\n- v04 --supports--> v11\n- v11 --refutes--> v12\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v0a\n- v02 --refutes--> v10\n- v02 --supports--> v0c\n- v07 --refutes--> v01\n- v09 --supports--> v0f\n- v0e --supports--> v0d\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":"40fd7379b77e10cc3d5ee46902c7c41225d4caaeea3493189ba5d45b2422b4c0","schema":"livegraph.boolean-propagation.v1","spec_hash":"9e265c3e41a539a78e790c5dd698fae4111c2bd1756a79c72841e154000bd815","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-ff8bdc58fe21d0c1ab95","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --refutes--> v07\n- v07 --supports--> v03\n- v03 --refutes--> v02\n- v02 --supports--> v10\n- v10 --supports--> v00\n- v00 --refutes--> v08\n- v08 --supports--> v0a\n- v0a --supports--> v04\n- v04 --refutes--> v12\n- v12 --supports--> v0f\n- v0f --refutes--> v01\n- v01 --supports--> v0c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v06\n- v08 --refutes--> v0e\n- v08 --refutes--> v11\n- v0a --supports--> v05\n- v10 --supports--> v0d\n- v12 --refutes--> v0b\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":"4c21b281e632bd8a1ddc8d8cb84398e2441835313d4a99a596535ca7bdd4d316","schema":"livegraph.boolean-propagation.v1","spec_hash":"ff8bdc58fe21d0c1ab95b6e13704b6950b999c632aa0d8f4ac545567251afd43","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-b89c202c5f2af0f911f3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --refutes--> v12\n- v12 --supports--> v11\n- v11 --supports--> v10\n- v10 --refutes--> v0d\n- v0d --supports--> v00\n- v00 --supports--> v09\n- v09 --refutes--> v02\n- v02 --refutes--> v03\n- v03 --refutes--> v05\n- v05 --refutes--> v0c\n- v0c --supports--> v0f\n- v0f --supports--> v0e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v0a --refutes--> v04\n- v0d --refutes--> v0b\n- v0f --supports--> v01\n- v11 --supports--> v06\n- v11 --supports--> v08\n- v12 --supports--> v07\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":"90702b7c0173782e49150b415cc1188c7c0b7b8fe1d30f0f6b005f26d4eb5919","schema":"livegraph.boolean-propagation.v1","spec_hash":"b89c202c5f2af0f911f31bc17e44453088c2788923d8e6a631954d0184c33063","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-80dba3f882d7e51510e7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --supports--> v0c\n- v0c --supports--> v01\n- v01 --refutes--> v0a\n- v0a --supports--> v04\n- v04 --refutes--> v09\n- v09 --supports--> v08\n- v08 --refutes--> v05\n- v05 --supports--> v0f\n- v0f --supports--> v02\n- v02 --refutes--> v03\n- v03 --refutes--> v0e\n- v0e --supports--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v08 --refutes--> v06\n- v09 --refutes--> v0d\n- v0a --refutes--> v00\n- v0a --refutes--> v10\n- v0c --supports--> v11\n- v0f --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":"1f3e5f99016b5eac42d9ab70d25f8e0c7b1754317a22f556e794db62bfefbaa2","schema":"livegraph.boolean-propagation.v1","spec_hash":"80dba3f882d7e51510e74d802ca53170d4f0e857cb75f1d2185858aa2af53bad","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0f\"}],\"hop\":12,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-80abbc94eda11b2e81d8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v11 --refutes--> v07\n- v07 --refutes--> v12\n- v12 --supports--> v09\n- v09 --refutes--> v0b\n- v0b --supports--> v0e\n- v0e --refutes--> v0f\n- v0f --supports--> v02\n- v02 --refutes--> v04\n- v04 --refutes--> v0d\n- v0d --refutes--> v0a\n- v0a --supports--> v0c\n- v0c --refutes--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v00\n- v0a --refutes--> v08\n- v0a --supports--> v05\n- v0f --refutes--> v06\n- v0f --refutes--> v10\n- v0f --supports--> v03\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":"ceec040ec62365891e8bf8af2b768994b2eb2d11971f3e2f99a3a649675b3242","schema":"livegraph.boolean-propagation.v1","spec_hash":"80abbc94eda11b2e81d8eb5ffb4f336783148dbf602b9f9e70421694614da34f","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-6a237be3fb8440b54c27","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0b --supports--> v0a\n- v0a --supports--> v0f\n- v0f --supports--> v0e\n- v0e --refutes--> v11\n- v11 --refutes--> v08\n- v08 --refutes--> v0c\n- v0c --refutes--> v00\n- v00 --refutes--> v02\n- v02 --refutes--> v10\n- v10 --refutes--> v12\n- v12 --supports--> v03\n- v03 --supports--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v0d\n- v03 --supports--> v09\n- v0a --supports--> v04\n- v0a --supports--> v07\n- v0f --refutes--> v06\n- v11 --supports--> v05\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":"ccae4aef905abd42fa10ce3cb7da07fd5c5558c5be08a414b5c28b1c8e0d2598","schema":"livegraph.boolean-propagation.v1","spec_hash":"6a237be3fb8440b54c27b9aa3020d9a1f54825b4d845f9e581e363c6f992d3c7","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-d17e4699406e4b52a8b4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --supports--> v0e\n- v0e --refutes--> v02\n- v02 --refutes--> v05\n- v05 --supports--> v03\n- v03 --supports--> v0d\n- v0d --supports--> v0f\n- v0f --refutes--> v09\n- v09 --refutes--> v06\n- v06 --supports--> v0a\n- v0a --supports--> v00\n- v00 --supports--> v0c\n- v0c --supports--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v12\n- v08 --refutes--> v0b\n- v08 --refutes--> v11\n- v0a --refutes--> v04\n- v0a --refutes--> v10\n- v0f --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":"672ceec5c79ff1cb1e1b8fcc6bc7d799c85a539b3d0c0f94fd8db565bd5f79be","schema":"livegraph.boolean-propagation.v1","spec_hash":"d17e4699406e4b52a8b4518a54f09ebbb9afcfc7bfaa896bd5b5353f7b4cbde7","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":12,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-4b19d57088402ca8fc6c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v10 --supports--> v01\n- v01 --refutes--> v0b\n- v0b --refutes--> v04\n- v04 --refutes--> v05\n- v05 --refutes--> v03\n- v03 --supports--> v02\n- v02 --refutes--> v08\n- v08 --refutes--> v11\n- v11 --supports--> v0d\n- v0d --refutes--> v09\n- v09 --refutes--> v06\n- v06 --supports--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v00\n- v04 --supports--> v0e\n- v06 --supports--> v07\n- v0b --refutes--> v12\n- v10 --supports--> v0c\n- v11 --refutes--> v0a\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":"b217ad69f30e0260b477b63d31cba91d152bc632de1b3ffa6822cc19ad49b3aa","schema":"livegraph.boolean-propagation.v1","spec_hash":"4b19d57088402ca8fc6c2a1bc55f714712105fbc543fe4089a40b6873d0295b7","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-193d02dd8dc6357af9a8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0e --supports--> v02\n- v02 --supports--> v03\n- v03 --supports--> v00\n- v00 --refutes--> v08\n- v08 --supports--> v0c\n- v0c --supports--> v10\n- v10 --refutes--> v04\n- v04 --refutes--> v0a\n- v0a --supports--> v05\n- v05 --supports--> v0f\n- v0f --refutes--> v0d\n- v0d --supports--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v07\n- v02 --supports--> v11\n- v04 --supports--> v09\n- v0d --supports--> v0b\n- v0f --refutes--> v06\n- v10 --refutes--> v12\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":"a4a04e17d2461f45fc9d2ad92ee899ef271edc13b25cc45c42ec7f1d52280c7a","schema":"livegraph.boolean-propagation.v1","spec_hash":"193d02dd8dc6357af9a8859a93dce5c447bc6a4b2164b1e3c199dbb48dc95c35","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-e98444fc5589728b94d9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --refutes--> v0d\n- v0d --refutes--> v11\n- v11 --refutes--> v07\n- v07 --supports--> v01\n- v01 --supports--> v10\n- v10 --refutes--> v05\n- v05 --supports--> v0e\n- v0e --refutes--> v02\n- v02 --supports--> v0a\n- v0a --refutes--> v0f\n- v0f --refutes--> v08\n- v08 --supports--> v12\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v0c\n- v02 --refutes--> v06\n- v02 --refutes--> v0b\n- v07 --supports--> v00\n- v0e --refutes--> v09\n- v0e --supports--> v04\n\n[VERIFIED ROOT]\nv03 = 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":"ff64c5705639ca240d51ad9a29cf447affa86a9fc5d04d6f243b41f0a43d9f87","schema":"livegraph.boolean-propagation.v1","spec_hash":"e98444fc5589728b94d97c277168be36108066fd4465c8dbc0f36cbb3f49d6d5","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-7337497f9ba2da06e1f5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --refutes--> v03\n- v03 --supports--> v0f\n- v0f --refutes--> v08\n- v08 --refutes--> v06\n- v06 --refutes--> v02\n- v02 --supports--> v04\n- v04 --supports--> v12\n- v12 --refutes--> v07\n- v07 --supports--> v00\n- v00 --refutes--> v0c\n- v0c --refutes--> v10\n- v10 --refutes--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v05\n- v07 --refutes--> v0a\n- v08 --supports--> v0b\n- v09 --supports--> v0e\n- v12 --refutes--> v0d\n- v12 --refutes--> v11\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":"8c877a231eb3104a004b5dec18a946b2d9357be2973443b143447f490b9f90e6","schema":"livegraph.boolean-propagation.v1","spec_hash":"7337497f9ba2da06e1f5af583497d11dff91cc0e2727ea532428b41f80d26a34","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-f4da69764ba29d6ae048","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --supports--> v01\n- v01 --refutes--> v0c\n- v0c --refutes--> v03\n- v03 --supports--> v12\n- v12 --refutes--> v09\n- v09 --refutes--> v07\n- v07 --refutes--> v02\n- v02 --supports--> v0e\n- v0e --supports--> v10\n- v10 --refutes--> v0d\n- v0d --refutes--> v0f\n- v0f --refutes--> v11\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v0b\n- v05 --supports--> v0a\n- v09 --supports--> v00\n- v0c --supports--> v06\n- v0d --refutes--> v08\n- v0e --supports--> 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":"9724f9dcc4e8d1ad9b69513989aac7c73c9e638b54e819c4dd9e6d4f25a940c1","schema":"livegraph.boolean-propagation.v1","spec_hash":"f4da69764ba29d6ae04831fbd0659401de911a43f4414c993d9db1ff29641656","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-7d73e80d3a879021e2a8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --supports--> v0e\n- v0e --refutes--> v05\n- v05 --refutes--> v0d\n- v0d --refutes--> v08\n- v08 --supports--> v0a\n- v0a --supports--> v07\n- v07 --supports--> v12\n- v12 --refutes--> v03\n- v03 --supports--> v00\n- v00 --refutes--> v02\n- v02 --refutes--> v0f\n- v0f --refutes--> v10\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v01\n- v05 --refutes--> v0b\n- v06 --refutes--> v04\n- v06 --refutes--> v0c\n- v08 --refutes--> v09\n- v0a --supports--> v11\n\n[VERIFIED ROOT]\nv06 = 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":"cabd4d7871bdbb49cf57275ef14296a25000dd4a9318050a3b9e95a29ab9659a","schema":"livegraph.boolean-propagation.v1","spec_hash":"7d73e80d3a879021e2a87f12ce00165e9de32f9257d5c35bab64fa5441f8418a","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-0a4d236b4b87a718c100","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v10 --refutes--> v0c\n- v0c --refutes--> v0a\n- v0a --supports--> v07\n- v07 --supports--> v11\n- v11 --supports--> v09\n- v09 --supports--> v0b\n- v0b --supports--> v02\n- v02 --refutes--> v0d\n- v0d --supports--> v00\n- v00 --refutes--> v04\n- v04 --refutes--> v12\n- v12 --refutes--> v0e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0f\n- v07 --supports--> v05\n- v09 --refutes--> v08\n- v0a --refutes--> v06\n- v0b --refutes--> v03\n- v11 --refutes--> v01\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":"a544b9f949e3d77100040ea38b4d37aedb07fb80a8ceabdf86b03b8aff95d243","schema":"livegraph.boolean-propagation.v1","spec_hash":"0a4d236b4b87a718c1005593b7cc9d8d7b0769c7572eb6ecfa023d3be39afadf","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-12f72a8c495d146dbabb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --refutes--> v00\n- v00 --refutes--> v02\n- v02 --refutes--> v0b\n- v0b --refutes--> v07\n- v07 --supports--> v03\n- v03 --refutes--> v0a\n- v0a --refutes--> v0c\n- v0c --refutes--> v11\n- v11 --refutes--> v04\n- v04 --refutes--> v0e\n- v0e --supports--> v09\n- v09 --refutes--> v12\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v10\n- v02 --refutes--> v0d\n- v07 --supports--> v06\n- v09 --supports--> v05\n- v0a --supports--> v0f\n- v0c --refutes--> 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":"f3d2e75ab95dbcd0c8f1d1359065b175c930eb898621fea04e46ac2cb3855428","schema":"livegraph.boolean-propagation.v1","spec_hash":"12f72a8c495d146dbabb2882935df74e20f174cf445d8704598f88f1f99703c7","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-a64922774ffff4063c14","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --supports--> v05\n- v05 --supports--> v04\n- v04 --supports--> v0e\n- v0e --refutes--> v07\n- v07 --supports--> v0b\n- v0b --supports--> v03\n- v03 --refutes--> v10\n- v10 --supports--> v02\n- v02 --refutes--> v01\n- v01 --supports--> v0c\n- v0c --supports--> v0f\n- v0f --refutes--> v11\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v00\n- v02 --supports--> v0d\n- v05 --refutes--> v08\n- v07 --refutes--> v0a\n- v09 --refutes--> v12\n- v0e --refutes--> v06\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":"b642ac4c26e84bff3e12d908fa7800c01ebd92450b2d6c119b3d2122e7d5b000","schema":"livegraph.boolean-propagation.v1","spec_hash":"a64922774ffff4063c140a66442ecee89920843f358e159b767506bd5b5a21e6","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-2873a64b4094797bf791","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --refutes--> v0d\n- v0d --supports--> v06\n- v06 --supports--> v00\n- v00 --supports--> v03\n- v03 --refutes--> v08\n- v08 --refutes--> v07\n- v07 --refutes--> v05\n- v05 --refutes--> v02\n- v02 --refutes--> v12\n- v12 --refutes--> v10\n- v10 --refutes--> v04\n- v04 --refutes--> v0e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v0c\n- v02 --refutes--> v01\n- v06 --refutes--> v0f\n- v0a --supports--> v0b\n- v10 --refutes--> v11\n- v12 --refutes--> v09\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":"146c2fe9c1f2977f36572d023b386742f8268afb5c9d6c89a7e5f84639a4a1c0","schema":"livegraph.boolean-propagation.v1","spec_hash":"2873a64b4094797bf79173ddd215d147517ed8ec49e8d747dba55f5f64b980a3","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-b9aeeb41160611b8bc45","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --supports--> v01\n- v01 --refutes--> v09\n- v09 --refutes--> v00\n- v00 --supports--> v04\n- v04 --supports--> v05\n- v05 --refutes--> v02\n- v02 --refutes--> v03\n- v03 --refutes--> v10\n- v10 --refutes--> v11\n- v11 --supports--> v0e\n- v0e --supports--> v08\n- v08 --supports--> v0c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v0a\n- v01 --supports--> v0b\n- v03 --refutes--> v07\n- v03 --supports--> v0f\n- v04 --supports--> v12\n- v05 --supports--> v0d\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":"5f4aaac19373dfbc5e20902f952e4b4c4d13e6c341bb38d15579176f866a1849","schema":"livegraph.boolean-propagation.v1","spec_hash":"b9aeeb41160611b8bc45e38b1846ccb624d3722f09490c17440a96e0405dd41e","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-5fb971c339ee67109c0b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0d --refutes--> v07\n- v07 --supports--> v01\n- v01 --refutes--> v09\n- v09 --supports--> v0b\n- v0b --refutes--> v02\n- v02 --refutes--> v0c\n- v0c --refutes--> v05\n- v05 --supports--> v00\n- v00 --refutes--> v08\n- v08 --refutes--> v11\n- v11 --refutes--> v10\n- v10 --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0a\n- v00 --supports--> v04\n- v07 --refutes--> v0f\n- v0c --refutes--> v0e\n- v10 --refutes--> v06\n- v10 --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":"e966e66ae00569dd532d95ebdf6aeedb1aff2ce7176eb9a99fef8c9d0580637f","schema":"livegraph.boolean-propagation.v1","spec_hash":"5fb971c339ee67109c0b0c3851d52d9bd4f5e68fc4b892de029ca558e9b7011e","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-12be124eaf0371ba41ee","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v0c\n- v0c --refutes--> v02\n- v02 --refutes--> v09\n- v09 --supports--> v0f\n- v0f --supports--> v01\n- v01 --supports--> v0b\n- v0b --refutes--> v05\n- v05 --refutes--> v06\n- v06 --supports--> v03\n- v03 --refutes--> v0e\n- v0e --supports--> v12\n- v12 --refutes--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v10\n- v03 --supports--> v0a\n- v0b --supports--> v08\n- v0f --supports--> v00\n- v12 --refutes--> v07\n- v12 --refutes--> v11\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":"be27879ad6efa2fe41c7559ce0dde4297e367b22f61424f135e5cc112708ffe5","schema":"livegraph.boolean-propagation.v1","spec_hash":"12be124eaf0371ba41eeffc87830296dd1908fd4ae1882d6cc03f862b1db5989","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-9a537380efc247af5f27","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --refutes--> v0e\n- v0e --refutes--> v10\n- v10 --supports--> v0f\n- v0f --supports--> v09\n- v09 --refutes--> v0a\n- v0a --supports--> v02\n- v02 --supports--> v00\n- v00 --supports--> v07\n- v07 --refutes--> v0d\n- v0d --supports--> v04\n- v04 --refutes--> v0c\n- v0c --supports--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v01\n- v03 --refutes--> v12\n- v04 --refutes--> v05\n- v04 --refutes--> v08\n- v0c --supports--> v0b\n- v0f --supports--> v11\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":"ee940451a90f41e451fd7f2a94ee0cfd2528cf286c4b99a967ab444dde557b02","schema":"livegraph.boolean-propagation.v1","spec_hash":"9a537380efc247af5f27401cc947cc5aaf51ce6b2c6fe78ebaabcfaf21572846","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-2a77bdefa644b16ad78a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --supports--> v0f\n- v0f --supports--> v11\n- v11 --supports--> v05\n- v05 --refutes--> v03\n- v03 --refutes--> v0a\n- v0a --refutes--> v00\n- v00 --refutes--> v12\n- v12 --supports--> v04\n- v04 --supports--> v0b\n- v0b --refutes--> v02\n- v02 --supports--> v0d\n- v0d --supports--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v0c\n- v0a --refutes--> v0e\n- v0b --refutes--> v10\n- v11 --supports--> v07\n- v12 --refutes--> v06\n- v12 --supports--> 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":"93363640fe48d452450355cc63a20ecb585743d93da14569a01bf5653acb1fcc","schema":"livegraph.boolean-propagation.v1","spec_hash":"2a77bdefa644b16ad78a70cb92d1e29b7fffe628505eca2c0428897b01dd5733","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-02f348d760c76ba46d23","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --refutes--> v05\n- v05 --refutes--> v0f\n- v0f --refutes--> v08\n- v08 --refutes--> v10\n- v10 --refutes--> v0a\n- v0a --supports--> v0c\n- v0c --supports--> v01\n- v01 --refutes--> v0e\n- v0e --refutes--> v0b\n- v0b --supports--> v02\n- v02 --refutes--> v03\n- v03 --refutes--> v12\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v0d\n- v02 --refutes--> v06\n- v08 --refutes--> v04\n- v0a --refutes--> v11\n- v0a --supports--> v09\n- v0f --refutes--> v07\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":"3a58c0fd36fa7b5ec3be607268f727129b0e7db85e34410a5d935710298cdf70","schema":"livegraph.boolean-propagation.v1","spec_hash":"02f348d760c76ba46d238d32ea0c2477cf5cefb468ca6c180866b234f564c5f2","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-ac3f8e19fa83b6697176","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v11 --supports--> v06\n- v06 --supports--> v0e\n- v0e --refutes--> v00\n- v00 --supports--> v04\n- v04 --refutes--> v0d\n- v0d --refutes--> v10\n- v10 --supports--> v09\n- v09 --refutes--> v0a\n- v0a --refutes--> v05\n- v05 --supports--> v12\n- v12 --supports--> v08\n- v08 --refutes--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v01\n- v05 --supports--> v07\n- v0e --refutes--> v03\n- v0e --supports--> v0c\n- v11 --refutes--> v02\n- v11 --supports--> v0b\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":"cb1136b640af3357819792aec30b645d76400dda06c1e81b6724d65d35a761e4","schema":"livegraph.boolean-propagation.v1","spec_hash":"ac3f8e19fa83b6697176ae61ec2293c1a021f6e168a2fb6805bfe9f0985a516c","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-e7b0bc1e29376c4e4e66","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v0d\n- v0d --supports--> v0f\n- v0f --refutes--> v08\n- v08 --refutes--> v00\n- v00 --refutes--> v0b\n- v0b --supports--> v03\n- v03 --refutes--> v10\n- v10 --supports--> v04\n- v04 --supports--> v0c\n- v0c --refutes--> v01\n- v01 --refutes--> v05\n- v05 --supports--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v0b --refutes--> v12\n- v0c --refutes--> v0e\n- v0c --supports--> v02\n- v0d --supports--> v07\n- v10 --refutes--> v0a\n- v10 --refutes--> v11\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":"cd8ddea785dfae67e4212b1c1be668dd45a3a7c38341ddbbb31112ec9809a6e8","schema":"livegraph.boolean-propagation.v1","spec_hash":"e7b0bc1e29376c4e4e6646882c66ff4f4656e1957cfcb40cdbbf73d88b477c1a","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-03a376a034ad3f48f623","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --refutes--> v09\n- v09 --refutes--> v10\n- v10 --refutes--> v05\n- v05 --refutes--> v0e\n- v0e --supports--> v0f\n- v0f --supports--> v00\n- v00 --refutes--> v07\n- v07 --supports--> v06\n- v06 --supports--> v0c\n- v0c --refutes--> v02\n- v02 --supports--> v0d\n- v0d --refutes--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0b\n- v05 --refutes--> v11\n- v05 --refutes--> v12\n- v06 --supports--> v01\n- v0c --refutes--> v03\n- v0c --refutes--> v0a\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":"12ff4324704945234a2db8a9800685e57d1913f6e1870ce0c5b53240ddb3fe35","schema":"livegraph.boolean-propagation.v1","spec_hash":"03a376a034ad3f48f62395b076e36fc73a93ef758b09411e25f3070001ca6226","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-345de218b889475d7894","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --refutes--> v10\n- v10 --supports--> v01\n- v01 --supports--> v02\n- v02 --supports--> v0c\n- v0c --refutes--> v0a\n- v0a --refutes--> v00\n- v00 --supports--> v04\n- v04 --refutes--> v05\n- v05 --supports--> v0e\n- v0e --refutes--> v0f\n- v0f --supports--> v0d\n- v0d --supports--> v11\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v03\n- v05 --supports--> v0b\n- v05 --supports--> v12\n- v0c --supports--> v09\n- v0d --refutes--> v07\n- v0e --supports--> v06\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":"dbab7f02c4e4bd659947001e95505768056ff571948489e7357157c8199b8903","schema":"livegraph.boolean-propagation.v1","spec_hash":"345de218b889475d7894bd6f134e06abc0a1a8c054500b32365b7af653d7111f","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"}],\"hop\":12,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-14bb35fa1ee33577c2e8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0f --refutes--> v0b\n- v0b --refutes--> v04\n- v04 --supports--> v07\n- v07 --supports--> v12\n- v12 --refutes--> v0d\n- v0d --supports--> v00\n- v00 --refutes--> v05\n- v05 --supports--> v10\n- v10 --refutes--> v11\n- v11 --supports--> v09\n- v09 --refutes--> v06\n- v06 --refutes--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v03\n- v0d --supports--> v0a\n- v10 --refutes--> v0e\n- v10 --supports--> v0c\n- v12 --refutes--> v01\n- v12 --refutes--> v08\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":"1f2cd472dc238fcbeb7c6d310fa64dcb4533772e46a0acdd167a1880645d5533","schema":"livegraph.boolean-propagation.v1","spec_hash":"14bb35fa1ee33577c2e8bb56aea59f6e07a52a6882cdc2c2a399fb065d7bba93","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-3d7c2d5096d562a4cab8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v11 --supports--> v04\n- v04 --refutes--> v0e\n- v0e --refutes--> v07\n- v07 --supports--> v0c\n- v0c --supports--> v12\n- v12 --refutes--> v05\n- v05 --supports--> v10\n- v10 --refutes--> v01\n- v01 --supports--> v0b\n- v0b --refutes--> v0a\n- v0a --refutes--> v09\n- v09 --supports--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v06\n- v01 --supports--> v0f\n- v04 --refutes--> v08\n- v0c --refutes--> v0d\n- v10 --refutes--> v00\n- v11 --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":"dbcfebb13d239b0a67371102e136933b98b838173ee386fdb9621f445416f2be","schema":"livegraph.boolean-propagation.v1","spec_hash":"3d7c2d5096d562a4cab8ef30114430494ac964f28ca50bb1a133b1113a057f0d","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-b2cbddfa345b97043574","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v11 --supports--> v06\n- v06 --supports--> v0f\n- v0f --refutes--> v03\n- v03 --supports--> v04\n- v04 --supports--> v0b\n- v0b --refutes--> v0a\n- v0a --supports--> v05\n- v05 --refutes--> v07\n- v07 --refutes--> v00\n- v00 --supports--> v09\n- v09 --supports--> v02\n- v02 --refutes--> v0e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v10\n- v04 --refutes--> v01\n- v05 --supports--> v12\n- v07 --refutes--> v0d\n- v0a --supports--> v0c\n- v0f --refutes--> v08\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":"c465be57e44ba9fb8f09ae9908bc3b48730bdfc0063847f181e4f966071fa170","schema":"livegraph.boolean-propagation.v1","spec_hash":"b2cbddfa345b97043574b513353c7d9967ac80158934002f709c550b0716c728","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-06be9cae56dc92b4ef0b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --supports--> v0d\n- v0d --refutes--> v00\n- v00 --refutes--> v12\n- v12 --supports--> v11\n- v11 --refutes--> v0e\n- v0e --supports--> v02\n- v02 --supports--> v05\n- v05 --refutes--> v08\n- v08 --supports--> v0a\n- v0a --refutes--> v0f\n- v0f --supports--> v06\n- v06 --supports--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0c\n- v02 --refutes--> v09\n- v05 --refutes--> v03\n- v05 --refutes--> v07\n- v0a --supports--> v04\n- v0d --supports--> v10\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":"d015a29e422c217cb1b2d381b6a37ad3feccea9d3a80d75a9bddac7ae25011a2","schema":"livegraph.boolean-propagation.v1","spec_hash":"06be9cae56dc92b4ef0b40188ee825e0d9b42186910f0a15633b2f507d003ba6","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-5f598e30b762a7d4a7f2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --supports--> v08\n- v08 --supports--> v05\n- v05 --supports--> v04\n- v04 --refutes--> v0d\n- v0d --supports--> v0c\n- v0c --supports--> v09\n- v09 --refutes--> v0b\n- v0b --supports--> v00\n- v00 --supports--> v06\n- v06 --supports--> v0f\n- v0f --refutes--> v0e\n- v0e --supports--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v02\n- v03 --refutes--> v07\n- v05 --refutes--> v10\n- v0b --supports--> v12\n- v0e --refutes--> v01\n- v0f --refutes--> v11\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":"d0d8cc9cdacdefc561ac70b6f61e3d0e99b49310787d15bb6ad6f92c6535bbc1","schema":"livegraph.boolean-propagation.v1","spec_hash":"5f598e30b762a7d4a7f2f4b9c5b3536e48b37fe3bcf7230575bf112f9b75d942","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"}],\"hop\":12,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-d516e11b03ce5d1b634d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0e --supports--> v02\n- v02 --supports--> v04\n- v04 --supports--> v11\n- v11 --refutes--> v0d\n- v0d --refutes--> v01\n- v01 --refutes--> v06\n- v06 --supports--> v05\n- v05 --refutes--> v0c\n- v0c --supports--> v09\n- v09 --supports--> v0a\n- v0a --refutes--> v07\n- v07 --refutes--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v0b\n- v05 --supports--> v10\n- v06 --refutes--> v00\n- v07 --refutes--> v12\n- v09 --supports--> v0f\n- v0c --refutes--> v03\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":"36105ca7b54b0bb0b23af6c82680aadc993bf589d65079a8c8c538c429085a97","schema":"livegraph.boolean-propagation.v1","spec_hash":"d516e11b03ce5d1b634df4fa3c0517d9258a68a8b830066f8af63d70f00f752a","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-df7b2b618a7d084bcd33","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v10 --supports--> v0d\n- v0d --refutes--> v03\n- v03 --supports--> v0c\n- v0c --refutes--> v00\n- v00 --refutes--> v04\n- v04 --supports--> v0b\n- v0b --supports--> v11\n- v11 --refutes--> v0a\n- v0a --refutes--> v01\n- v01 --refutes--> v08\n- v08 --supports--> v06\n- v06 --refutes--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v12\n- v03 --supports--> v05\n- v0a --supports--> v09\n- v0b --supports--> v07\n- v0c --supports--> v0f\n- v0d --refutes--> v0e\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":"c6e65d6a3386898f549b818f185a15bad36c9f963ad1671289e65cb2b1374139","schema":"livegraph.boolean-propagation.v1","spec_hash":"df7b2b618a7d084bcd33c3e6f8cd2606859400fbfe5b01197dc28e065b4f0276","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-7c61119991f1a53ab036","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0f --refutes--> v09\n- v09 --supports--> v00\n- v00 --supports--> v10\n- v10 --supports--> v0a\n- v0a --supports--> v11\n- v11 --refutes--> v12\n- v12 --refutes--> v0c\n- v0c --refutes--> v0b\n- v0b --refutes--> v07\n- v07 --supports--> v05\n- v05 --supports--> v0d\n- v0d --refutes--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v07 --refutes--> v04\n- v0a --refutes--> v08\n- v0a --refutes--> v0e\n- v0d --refutes--> v01\n- v0f --refutes--> v02\n- v11 --refutes--> v03\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":"5b8c9582a83bebe3755407631cdf2d66e2574bf5d3206b311250efa34d9bdc63","schema":"livegraph.boolean-propagation.v1","spec_hash":"7c61119991f1a53ab036bfec93da1e6683821f1ea29347507482e5e3602d12a2","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-44ad02c5b590b50f7f48","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --refutes--> v10\n- v10 --supports--> v0f\n- v0f --supports--> v08\n- v08 --refutes--> v11\n- v11 --refutes--> v06\n- v06 --supports--> v00\n- v00 --supports--> v01\n- v01 --supports--> v07\n- v07 --supports--> v0b\n- v0b --supports--> v09\n- v09 --refutes--> v12\n- v12 --refutes--> v0c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0d\n- v01 --supports--> v0a\n- v08 --refutes--> v03\n- v09 --refutes--> v04\n- v11 --refutes--> v02\n- v12 --refutes--> v0e\n\n[VERIFIED ROOT]\nv05 = 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":"15e3f40d50beef55b89c3c43756f4b57fda6d5f3294f0342ab50ad3f0b862115","schema":"livegraph.boolean-propagation.v1","spec_hash":"44ad02c5b590b50f7f48483d514c9039477fef57dc619ae3769710d13cc76ebf","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-d6a585c9109bb9269758","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --supports--> v0f\n- v0f --supports--> v08\n- v08 --refutes--> v0a\n- v0a --refutes--> v0e\n- v0e --refutes--> v03\n- v03 --refutes--> v07\n- v07 --supports--> v0c\n- v0c --supports--> v02\n- v02 --refutes--> v00\n- v00 --refutes--> v12\n- v12 --refutes--> v09\n- v09 --refutes--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v10\n- v02 --refutes--> v0b\n- v07 --supports--> v0d\n- v08 --refutes--> v11\n- v0a --supports--> v06\n- v0e --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":"dd2ac3d10761525e5d8922e0e9aedbb36b60deab3c500829813236b8b007ba4c","schema":"livegraph.boolean-propagation.v1","spec_hash":"d6a585c9109bb9269758a9dc9b40e594c70dd9797185bd0b1a34c547bde0edb3","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-8c09484357113d4114c7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0d --refutes--> v0c\n- v0c --refutes--> v01\n- v01 --supports--> v04\n- v04 --refutes--> v08\n- v08 --refutes--> v09\n- v09 --refutes--> v0e\n- v0e --supports--> v12\n- v12 --supports--> v03\n- v03 --supports--> v02\n- v02 --refutes--> v06\n- v06 --refutes--> v10\n- v10 --refutes--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v0a\n- v0c --refutes--> v0f\n- v0c --supports--> v00\n- v0d --supports--> v05\n- v0d --supports--> v11\n- v12 --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":"573ddbb76e5fa2302b7d457c8274b661614dd8638a3013e12004795f4a588137","schema":"livegraph.boolean-propagation.v1","spec_hash":"8c09484357113d4114c78ac9de9183abeb011e6a4762a65bc7e9f1bfc4f4464d","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-efe89addf046f872737e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v10 --refutes--> v0e\n- v0e --supports--> v0a\n- v0a --supports--> v03\n- v03 --supports--> v09\n- v09 --refutes--> v02\n- v02 --refutes--> v0c\n- v0c --refutes--> v00\n- v00 --supports--> v07\n- v07 --supports--> v0f\n- v0f --supports--> v11\n- v11 --refutes--> v06\n- v06 --supports--> v12\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v04\n- v03 --supports--> v08\n- v0e --refutes--> v01\n- v0e --supports--> v05\n- v0f --supports--> v0b\n- v11 --refutes--> v0d\n\n[VERIFIED ROOT]\nv10 = 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":"7b3eca65d70567073f5f652e421b1773d0c563e9ddadb136bd812f33b526bf48","schema":"livegraph.boolean-propagation.v1","spec_hash":"efe89addf046f872737ea1e6a63b699767e1a6dbcf01f7bbee05dd0402663abe","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-29308c000f1d05138b92","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --supports--> v09\n- v09 --supports--> v0b\n- v0b --supports--> v07\n- v07 --supports--> v12\n- v12 --refutes--> v0c\n- v0c --supports--> v0f\n- v0f --refutes--> v00\n- v00 --refutes--> v03\n- v03 --supports--> v08\n- v08 --refutes--> v05\n- v05 --supports--> v04\n- v04 --supports--> v11\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0a\n- v06 --supports--> v01\n- v08 --refutes--> v02\n- v09 --refutes--> v0d\n- v0b --refutes--> v0e\n- v0c --supports--> v10\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":"00a8d5fb0f02be974beb187b792ddaa61dd24d19c216775b72c342f9fddf3127","schema":"livegraph.boolean-propagation.v1","spec_hash":"29308c000f1d05138b92c6e72beca2d89abdaac1d8f9da12ed6b361e74e90bca","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-28fc825f8d4b0d7b9cd9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --supports--> v0f\n- v0f --refutes--> v02\n- v02 --supports--> v12\n- v12 --supports--> v08\n- v08 --supports--> v0e\n- v0e --refutes--> v0b\n- v0b --supports--> v11\n- v11 --refutes--> v01\n- v01 --supports--> v0a\n- v0a --supports--> v09\n- v09 --supports--> v03\n- v03 --refutes--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v07\n- v03 --supports--> v06\n- v08 --supports--> v0d\n- v0a --refutes--> v0c\n- v0f --supports--> v00\n- v12 --refutes--> 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":"6c7b185a91a3affa8a0370555dad3594ef9c4d46a09afada947250e95f2358d5","schema":"livegraph.boolean-propagation.v1","spec_hash":"28fc825f8d4b0d7b9cd926dac5e80df0c5fccf03d78774078b09accd8105b7b1","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-a2027d98b64f9582706d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --refutes--> v0e\n- v0e --supports--> v10\n- v10 --refutes--> v0d\n- v0d --supports--> v09\n- v09 --refutes--> v01\n- v01 --refutes--> v0b\n- v0b --refutes--> v02\n- v02 --refutes--> v05\n- v05 --supports--> v0c\n- v0c --refutes--> v0f\n- v0f --supports--> v11\n- v11 --supports--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v07\n- v02 --refutes--> v0a\n- v09 --refutes--> v00\n- v0d --supports--> v06\n- v0d --supports--> v12\n- v10 --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":"f8c5c4b256b3fcaa0ea8b39840ff7cc14eac1c3815d06d5390b6c311f325fe23","schema":"livegraph.boolean-propagation.v1","spec_hash":"a2027d98b64f9582706d7858452f818081d831d467d2e71177f9b5fe5aad84ad","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":12,"item_id":"lggrpo1-f2cb557c476d9b120287","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v12 --refutes--> v0f\n- v0f --refutes--> v04\n- v04 --refutes--> v0b\n- v0b --refutes--> v01\n- v01 --supports--> v05\n- v05 --refutes--> v06\n- v06 --supports--> v03\n- v03 --supports--> v07\n- v07 --refutes--> v0a\n- v0a --refutes--> v02\n- v02 --refutes--> v00\n- v00 --refutes--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v08\n- v0b --refutes--> v0c\n- v0b --supports--> v0e\n- v0f --refutes--> v09\n- v0f --refutes--> v11\n- v12 --refutes--> v10\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":"8d8e1309c08c6334bf81e8a95f0d0bb4a9c56d1434732961e10985922355f948","schema":"livegraph.boolean-propagation.v1","spec_hash":"f2cb557c476d9b120287ce8ff5846e4f33aac98e8708739f21fd8d444d8a6017","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":12,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":12,"item_id":"lggrpo1-6b999cbab4970ecaf0a8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0e --refutes--> v02\n- v02 --supports--> v10\n- v10 --supports--> v0f\n- v0f --supports--> v00\n- v00 --refutes--> v04\n- v04 --refutes--> v09\n- v09 --supports--> v06\n- v06 --refutes--> v0c\n- v0c --supports--> v0b\n- v0b --supports--> v11\n- v11 --refutes--> v08\n- v08 --supports--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v05\n- v08 --supports--> v03\n- v0c --refutes--> v01\n- v0e --refutes--> v07\n- v0e --supports--> v0d\n- v10 --refutes--> v12\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":"d3532f442f51e74f2042cb05d10fae7df1841f575dbb8af96765799c65ba6a1b","schema":"livegraph.boolean-propagation.v1","spec_hash":"6b999cbab4970ecaf0a81c4a649b244d3edff891e2591ae15f719df30d52dd4f","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":12,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-321d4cccffd7983014f1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v11 --refutes--> v00\n- v00 --refutes--> v08\n- v08 --refutes--> v02\n- v02 --supports--> v04\n- v04 --refutes--> v09\n- v09 --refutes--> v12\n- v12 --supports--> v13\n- v13 --refutes--> v0b\n- v0b --supports--> v10\n- v10 --refutes--> v0c\n- v0c --supports--> v03\n- v03 --refutes--> v07\n- v07 --supports--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v06\n- v03 --refutes--> v0f\n- v04 --refutes--> v0e\n- v07 --supports--> v0a\n- v11 --supports--> v05\n- v12 --refutes--> v01\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":"55a3935054e2bfbeaee7eee70ad671c055aa84858bca17ee1766b86a76e79b40","schema":"livegraph.boolean-propagation.v1","spec_hash":"321d4cccffd7983014f18c6cc413b20a8e755faba5929c6500b25578f8ac00d1","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-e6aad6540e4e35a3af2b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --refutes--> v13\n- v13 --refutes--> v10\n- v10 --refutes--> v04\n- v04 --supports--> v07\n- v07 --refutes--> v06\n- v06 --refutes--> v0c\n- v0c --supports--> v09\n- v09 --refutes--> v0b\n- v0b --refutes--> v11\n- v11 --supports--> v03\n- v03 --supports--> v0f\n- v0f --supports--> v05\n- v05 --supports--> v12\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v0e\n- v08 --supports--> v00\n- v09 --refutes--> v0a\n- v09 --supports--> v01\n- v0b --supports--> v0d\n- v0f --supports--> v02\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":"bd6fff84b4b0add7d8901b0dc27b85a28f6094f8fe6720cdbf105e9ca4b971e4","schema":"livegraph.boolean-propagation.v1","spec_hash":"e6aad6540e4e35a3af2bc1eca50f67569f34e449b2bb07757c7941f6f9ef4d71","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-e44e1a20cb7174c8d7b0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --refutes--> v0e\n- v0e --refutes--> v0f\n- v0f --refutes--> v01\n- v01 --supports--> v03\n- v03 --supports--> v06\n- v06 --refutes--> v00\n- v00 --supports--> v11\n- v11 --supports--> v0b\n- v0b --refutes--> v12\n- v12 --supports--> v05\n- v05 --supports--> v02\n- v02 --refutes--> v13\n- v13 --supports--> v10\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v09\n- v03 --supports--> v07\n- v06 --refutes--> v0a\n- v0e --supports--> v0c\n- v0e --supports--> v0d\n- v11 --supports--> v04\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":"a9d977663588a05c4e1b3e54877b6920fc2f27441129d176a14d526f2f219943","schema":"livegraph.boolean-propagation.v1","spec_hash":"e44e1a20cb7174c8d7b08f6f0ad993f5c004baaa1f55afdb8bb3da0a92635c92","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-808627a72be72143766e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0f --supports--> v05\n- v05 --supports--> v00\n- v00 --supports--> v0a\n- v0a --supports--> v13\n- v13 --supports--> v03\n- v03 --refutes--> v04\n- v04 --refutes--> v09\n- v09 --refutes--> v10\n- v10 --refutes--> v0c\n- v0c --refutes--> v0e\n- v0e --supports--> v08\n- v08 --supports--> v02\n- v02 --refutes--> v11\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v07\n- v04 --refutes--> v01\n- v04 --refutes--> v0b\n- v04 --supports--> v0d\n- v0e --refutes--> v12\n- v13 --refutes--> 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":"971e4cbf303502b9c4b211cfa2e5b7feddf4381c89d660e2f2a6bf99d07396a7","schema":"livegraph.boolean-propagation.v1","spec_hash":"808627a72be72143766e53890e66c3934fae033ab075674007925020e4f97626","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-b56b897afbb617b86855","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v10 --supports--> v0b\n- v0b --refutes--> v02\n- v02 --supports--> v12\n- v12 --refutes--> v11\n- v11 --refutes--> v06\n- v06 --supports--> v04\n- v04 --supports--> v00\n- v00 --supports--> v0f\n- v0f --supports--> v07\n- v07 --supports--> v0d\n- v0d --supports--> v0e\n- v0e --refutes--> v0a\n- v0a --supports--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v08\n- v04 --supports--> v01\n- v06 --refutes--> v13\n- v0d --supports--> v05\n- v0e --refutes--> v09\n- v0e --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":"c48d685cc8d5cc2f4eddb0a56fe19c36a5cc0cbfbf24b83346168ffa5e10054e","schema":"livegraph.boolean-propagation.v1","spec_hash":"b56b897afbb617b8685512c9b598726c8d9fac29cf062666ac66cc8a32b724a8","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-bf0e4f00023a8573a0f3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0b --supports--> v00\n- v00 --refutes--> v04\n- v04 --supports--> v05\n- v05 --refutes--> v13\n- v13 --supports--> v10\n- v10 --supports--> v0f\n- v0f --supports--> v06\n- v06 --refutes--> v07\n- v07 --refutes--> v02\n- v02 --supports--> v09\n- v09 --refutes--> v0a\n- v0a --supports--> v11\n- v11 --refutes--> v12\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0e\n- v05 --supports--> v0d\n- v06 --refutes--> v01\n- v0a --refutes--> v08\n- v0f --supports--> v03\n- v11 --supports--> v0c\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":"5f26ee80eae65967c39b55a01e6f2f7674b8cbb1872b00b935adc9f3272dec7d","schema":"livegraph.boolean-propagation.v1","spec_hash":"bf0e4f00023a8573a0f3446859e77958e3460da1ae846857d3cf3c20bd68315c","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-e1139d5838468d6fa101","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0f --refutes--> v0b\n- v0b --refutes--> v0e\n- v0e --supports--> v00\n- v00 --supports--> v0c\n- v0c --refutes--> v02\n- v02 --refutes--> v01\n- v01 --supports--> v07\n- v07 --supports--> v09\n- v09 --refutes--> v06\n- v06 --refutes--> v10\n- v10 --refutes--> v11\n- v11 --supports--> v0d\n- v0d --supports--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v05\n- v06 --refutes--> v08\n- v07 --supports--> v0a\n- v0b --refutes--> v12\n- v0e --refutes--> v04\n- v0e --supports--> v13\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":"da30ed55ec9a584e96de91a99683e9f7b9d6869edd47e09ff6874bbf8c6155ac","schema":"livegraph.boolean-propagation.v1","spec_hash":"e1139d5838468d6fa1012b0381b653b47da6acb007d5d4268b5fcbbc2a6cb3dd","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":13,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-d05519a0d6be565cb44f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v03\n- v03 --supports--> v0a\n- v0a --supports--> v0b\n- v0b --refutes--> v11\n- v11 --refutes--> v10\n- v10 --supports--> v12\n- v12 --supports--> v07\n- v07 --supports--> v0f\n- v0f --supports--> v0d\n- v0d --supports--> v13\n- v13 --refutes--> v09\n- v09 --refutes--> v00\n- v00 --refutes--> v0e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v08\n- v04 --refutes--> v01\n- v04 --supports--> v02\n- v04 --supports--> v06\n- v11 --supports--> v0c\n- v13 --supports--> v05\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":"fe36ed549ffa7206e7f3c78d024a09eb57ee392f34de8e6ad7c3df5728f8ee5c","schema":"livegraph.boolean-propagation.v1","spec_hash":"d05519a0d6be565cb44fb0db929a5256c0aa9ce65856d00490d1424ef7840c72","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-b940b49aa82352c62f28","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --refutes--> v07\n- v07 --refutes--> v00\n- v00 --supports--> v0f\n- v0f --refutes--> v09\n- v09 --refutes--> v0c\n- v0c --refutes--> v06\n- v06 --refutes--> v04\n- v04 --refutes--> v08\n- v08 --refutes--> v02\n- v02 --supports--> v0b\n- v0b --supports--> v0d\n- v0d --supports--> v11\n- v11 --supports--> v0e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v13\n- v04 --refutes--> v05\n- v08 --supports--> v03\n- v09 --refutes--> v12\n- v0f --refutes--> v01\n- v0f --supports--> v10\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":"611f4f584f820f826c88ba5ce3b4571e5939cc275b1e1fdbe652382f4648c4fb","schema":"livegraph.boolean-propagation.v1","spec_hash":"b940b49aa82352c62f2800b26f3a505c65656e0f01dd4088e262604d5f68c222","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":13,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-05bc8dd5c4a37890ad20","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v12 --refutes--> v0a\n- v0a --supports--> v03\n- v03 --supports--> v11\n- v11 --supports--> v13\n- v13 --supports--> v08\n- v08 --supports--> v09\n- v09 --supports--> v02\n- v02 --refutes--> v10\n- v10 --refutes--> v0d\n- v0d --refutes--> v01\n- v01 --supports--> v05\n- v05 --refutes--> v06\n- v06 --refutes--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v04\n- v08 --refutes--> v0b\n- v08 --supports--> v0c\n- v09 --supports--> v0e\n- v0d --supports--> v00\n- v12 --supports--> v0f\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":"740076edb414f0116d029c43e7b0578f8f87b563659eeafe36ccbee70c7e0aae","schema":"livegraph.boolean-propagation.v1","spec_hash":"05bc8dd5c4a37890ad208703857fd984da8cda27ecea47d7e894e3cc1cd5c44e","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-88ff6e08c344dda8a57c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --refutes--> v0c\n- v0c --refutes--> v13\n- v13 --refutes--> v10\n- v10 --supports--> v05\n- v05 --supports--> v08\n- v08 --supports--> v04\n- v04 --refutes--> v12\n- v12 --refutes--> v0d\n- v0d --supports--> v00\n- v00 --refutes--> v0f\n- v0f --supports--> v09\n- v09 --supports--> v06\n- v06 --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v02\n- v01 --supports--> v11\n- v04 --refutes--> v0a\n- v05 --refutes--> v07\n- v10 --refutes--> v0b\n- v10 --supports--> v0e\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":"34ebba271a618ec04987c7261ac786d079144d9186a24fa4d1dd19953ce15329","schema":"livegraph.boolean-propagation.v1","spec_hash":"88ff6e08c344dda8a57c35efee3f5f0a548053653d89c3c205d7e19b3286ba86","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-428a692d722974081202","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --refutes--> v13\n- v13 --supports--> v07\n- v07 --refutes--> v0e\n- v0e --refutes--> v0f\n- v0f --supports--> v0c\n- v0c --supports--> v10\n- v10 --supports--> v06\n- v06 --refutes--> v08\n- v08 --refutes--> v0d\n- v0d --refutes--> v09\n- v09 --refutes--> v04\n- v04 --supports--> v01\n- v01 --supports--> v11\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v05\n- v04 --supports--> v03\n- v06 --supports--> v02\n- v08 --supports--> v12\n- v0a --refutes--> v00\n- v0f --supports--> v0b\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":"0648b71f09673042693c7639f3f8dfe718f4c6cb2fb439be853050476f75b6e1","schema":"livegraph.boolean-propagation.v1","spec_hash":"428a692d722974081202c4693025b8130f542f14531db73d988a4a3121090237","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-17628ef0f912aa48e0a3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --supports--> v09\n- v09 --refutes--> v00\n- v00 --supports--> v11\n- v11 --refutes--> v0e\n- v0e --refutes--> v02\n- v02 --refutes--> v0c\n- v0c --refutes--> v06\n- v06 --refutes--> v07\n- v07 --refutes--> v10\n- v10 --supports--> v0d\n- v0d --refutes--> v01\n- v01 --supports--> v04\n- v04 --refutes--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v0f\n- v04 --refutes--> v08\n- v06 --refutes--> v12\n- v06 --supports--> v03\n- v07 --supports--> v13\n- v0a --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":"97a110355871b609ecb59a7574e7c7c51f1cfe513506569ea71ac8a1fbb8792b","schema":"livegraph.boolean-propagation.v1","spec_hash":"17628ef0f912aa48e0a34bf9a659ee705212d9d91830f28569ec174f8e974252","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"}],\"hop\":13,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-118983b5cee717b734b6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0b --refutes--> v06\n- v06 --supports--> v11\n- v11 --supports--> v08\n- v08 --refutes--> v0a\n- v0a --refutes--> v12\n- v12 --refutes--> v04\n- v04 --supports--> v0e\n- v0e --refutes--> v0c\n- v0c --refutes--> v10\n- v10 --supports--> v05\n- v05 --supports--> v07\n- v07 --refutes--> v00\n- v00 --supports--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --supports--> v02\n- v07 --refutes--> v13\n- v08 --supports--> v03\n- v0b --supports--> v0f\n- v0c --refutes--> v01\n- v0e --supports--> v09\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":"78072e4156f641a3a1347b545ea7a6b88a928993b4a8ae37be531813991d0e4b","schema":"livegraph.boolean-propagation.v1","spec_hash":"118983b5cee717b734b6fbf28a6836bf4848cd4c52778f4e94cfe71a141889b2","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-2838df6f9348c1bca605","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v0f\n- v0f --refutes--> v11\n- v11 --refutes--> v01\n- v01 --refutes--> v09\n- v09 --supports--> v05\n- v05 --refutes--> v10\n- v10 --supports--> v0c\n- v0c --refutes--> v0d\n- v0d --refutes--> v12\n- v12 --refutes--> v08\n- v08 --supports--> v07\n- v07 --refutes--> v00\n- v00 --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v04\n- v08 --refutes--> v13\n- v09 --supports--> v0e\n- v0f --refutes--> v0b\n- v10 --refutes--> v0a\n- v12 --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":"56610ecac57533267ba9d335908f3a41967ca0fa912b9bc4c68a8e9288f4e7a6","schema":"livegraph.boolean-propagation.v1","spec_hash":"2838df6f9348c1bca6057037a6699fac6aa23e96ed4e75561ff5a49426738a1c","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-c01ac45427a941ba150d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v13 --supports--> v09\n- v09 --supports--> v00\n- v00 --refutes--> v01\n- v01 --refutes--> v06\n- v06 --refutes--> v04\n- v04 --refutes--> v02\n- v02 --supports--> v0d\n- v0d --supports--> v0f\n- v0f --refutes--> v08\n- v08 --supports--> v07\n- v07 --supports--> v0e\n- v0e --refutes--> v12\n- v12 --refutes--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v0b\n- v00 --supports--> v0c\n- v07 --refutes--> v05\n- v08 --refutes--> v11\n- v09 --refutes--> v10\n- v0e --supports--> v03\n\n[VERIFIED ROOT]\nv13 = 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":"f5d7ea0b03c54bd378081f529178bbbd0bd46248eb29d2b26530c7003132635d","schema":"livegraph.boolean-propagation.v1","spec_hash":"c01ac45427a941ba150d27d21eaa5f00ef6ba260c5200e0e11c2f1d8aec750ca","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v13\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-921009d38724bdafe6ae","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --refutes--> v13\n- v13 --supports--> v02\n- v02 --supports--> v03\n- v03 --refutes--> v06\n- v06 --supports--> v11\n- v11 --refutes--> v07\n- v07 --supports--> v0f\n- v0f --refutes--> v0b\n- v0b --refutes--> v04\n- v04 --refutes--> v09\n- v09 --supports--> v10\n- v10 --supports--> v0a\n- v0a --supports--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v12\n- v03 --refutes--> v0e\n- v0b --refutes--> v05\n- v0f --refutes--> v00\n- v0f --supports--> v0c\n- v11 --supports--> v08\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":"bd47c79d2294631d0be804a4761c308b4e9f8a85763f446143938a48fd2a8f6c","schema":"livegraph.boolean-propagation.v1","spec_hash":"921009d38724bdafe6ae991991ac3293e41efd1577476e9aeeff606fd378f36c","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-6c7f789be519eae7b2ee","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v13 --supports--> v0f\n- v0f --supports--> v01\n- v01 --refutes--> v08\n- v08 --refutes--> v12\n- v12 --refutes--> v10\n- v10 --supports--> v0b\n- v0b --supports--> v0d\n- v0d --refutes--> v02\n- v02 --supports--> v06\n- v06 --supports--> v09\n- v09 --supports--> v0c\n- v0c --supports--> v04\n- v04 --supports--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v00\n- v09 --refutes--> v11\n- v10 --refutes--> v0e\n- v10 --supports--> v03\n- v10 --supports--> v0a\n- v13 --refutes--> v05\n\n[VERIFIED ROOT]\nv13 = 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":"83b733a9d1685788d94a78361f972fdbe65f17ca58d93467f765ed3b316cad44","schema":"livegraph.boolean-propagation.v1","spec_hash":"6c7f789be519eae7b2eed873989f2d681205c6719d419899ca4f238bfca95d47","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-e7a503c6060b4c524ac0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --refutes--> v01\n- v01 --refutes--> v11\n- v11 --supports--> v02\n- v02 --supports--> v13\n- v13 --refutes--> v10\n- v10 --supports--> v0d\n- v0d --supports--> v0c\n- v0c --supports--> v05\n- v05 --refutes--> v08\n- v08 --supports--> v06\n- v06 --refutes--> v0f\n- v0f --refutes--> v04\n- v04 --refutes--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v09\n- v03 --supports--> v0b\n- v05 --refutes--> v00\n- v08 --supports--> v07\n- v08 --supports--> v0e\n- v0f --refutes--> v12\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":"c2318929f41b5adc5c04f9a44f8bab12a6869315f1741514346f579ff160f0fd","schema":"livegraph.boolean-propagation.v1","spec_hash":"e7a503c6060b4c524ac0671cf3d35771ba89617f3a85c136cd0e6eabef2d1884","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-385bf240730c21f50224","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0d --refutes--> v04\n- v04 --refutes--> v09\n- v09 --refutes--> v02\n- v02 --refutes--> v01\n- v01 --refutes--> v0c\n- v0c --refutes--> v05\n- v05 --supports--> v0f\n- v0f --supports--> v0a\n- v0a --supports--> v12\n- v12 --refutes--> v11\n- v11 --refutes--> v13\n- v13 --supports--> v00\n- v00 --refutes--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v03\n- v0a --refutes--> v10\n- v0c --refutes--> v0e\n- v0c --supports--> v0b\n- v11 --refutes--> v08\n- v13 --supports--> v07\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":"123890c28069abe812ae783a68a97e357e5095afb6dbff0e6b48a51792b32de3","schema":"livegraph.boolean-propagation.v1","spec_hash":"385bf240730c21f5022497b65ff158a09b922ed19366844b70a9135579c3587d","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-9f239d046f567abd85f9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --supports--> v0c\n- v0c --supports--> v08\n- v08 --refutes--> v04\n- v04 --refutes--> v0e\n- v0e --supports--> v09\n- v09 --supports--> v12\n- v12 --refutes--> v0a\n- v0a --refutes--> v06\n- v06 --refutes--> v00\n- v00 --supports--> v10\n- v10 --supports--> v07\n- v07 --refutes--> v02\n- v02 --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v0d\n- v10 --refutes--> v05\n- v10 --refutes--> v0b\n- v10 --supports--> v13\n- v12 --refutes--> v0f\n- v12 --supports--> v11\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":"d7ad9309fbcbf2ddb3fa5dbddef5484ba4d03ed36d9db13fc04854ba2c715c43","schema":"livegraph.boolean-propagation.v1","spec_hash":"9f239d046f567abd85f98705cc3ad683fc4de7f7fb40e1f54539df35e475ba60","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-67685ab78d54504828c6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0e --refutes--> v06\n- v06 --supports--> v0c\n- v0c --supports--> v09\n- v09 --refutes--> v0a\n- v0a --refutes--> v03\n- v03 --refutes--> v0d\n- v0d --refutes--> v07\n- v07 --refutes--> v05\n- v05 --refutes--> v0b\n- v0b --refutes--> v04\n- v04 --supports--> v0f\n- v0f --refutes--> v12\n- v12 --supports--> v11\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v00\n- v07 --supports--> v10\n- v0c --supports--> v02\n- v0d --refutes--> v08\n- v12 --refutes--> v13\n- v12 --supports--> v01\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":"66d978ab6369454b24d8c3c2c6a56fb71647e5c1d776ef10f90a3aee60f819c3","schema":"livegraph.boolean-propagation.v1","spec_hash":"67685ab78d54504828c6657c24bf374a0c0e7093ba71d8f2f493c75f472f664d","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-7630c7fbe863aea9745b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v0b\n- v0b --refutes--> v0a\n- v0a --supports--> v09\n- v09 --refutes--> v03\n- v03 --supports--> v05\n- v05 --refutes--> v12\n- v12 --supports--> v11\n- v11 --supports--> v13\n- v13 --supports--> v02\n- v02 --refutes--> v0d\n- v0d --supports--> v0e\n- v0e --refutes--> v01\n- v01 --refutes--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0f\n- v03 --refutes--> v06\n- v05 --supports--> v08\n- v0b --supports--> v0c\n- v11 --refutes--> v10\n- v12 --refutes--> v07\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":"4c71e40dc9db87ab75430a17ffc456c8f7aa13c715e300f4e8b49c60c3db68f3","schema":"livegraph.boolean-propagation.v1","spec_hash":"7630c7fbe863aea9745bcc861f2efca115834df88be05cc8d0193489469253c2","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-d5a8477f170a15c3eafe","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --supports--> v0b\n- v0b --refutes--> v03\n- v03 --refutes--> v0e\n- v0e --supports--> v08\n- v08 --refutes--> v05\n- v05 --supports--> v09\n- v09 --supports--> v13\n- v13 --supports--> v11\n- v11 --refutes--> v06\n- v06 --refutes--> v04\n- v04 --refutes--> v0f\n- v0f --supports--> v07\n- v07 --refutes--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v10\n- v05 --supports--> v02\n- v07 --refutes--> v0d\n- v0b --refutes--> v12\n- v13 --refutes--> v00\n- v13 --refutes--> v0c\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":"61921868233bdf929c6d6a48bfe0d41b68da3442a84f0f9c2e6511a6d056c714","schema":"livegraph.boolean-propagation.v1","spec_hash":"d5a8477f170a15c3eafed2931570e76a4521a4b2078e8d41372e8cb5feb9b040","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-79bfccffe1b9f60755ae","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v11 --supports--> v07\n- v07 --supports--> v10\n- v10 --supports--> v09\n- v09 --refutes--> v02\n- v02 --refutes--> v0d\n- v0d --supports--> v0e\n- v0e --supports--> v04\n- v04 --supports--> v13\n- v13 --supports--> v06\n- v06 --supports--> v0a\n- v0a --refutes--> v12\n- v12 --supports--> v0f\n- v0f --supports--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v0b\n- v04 --refutes--> v0c\n- v07 --refutes--> v03\n- v0a --refutes--> v08\n- v0f --supports--> v05\n- v11 --refutes--> v00\n\n[VERIFIED ROOT]\nv11 = 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":"8b2115e5523e9d713433fcd45a2bf096d86b0499fd115f23af2415f079356ccf","schema":"livegraph.boolean-propagation.v1","spec_hash":"79bfccffe1b9f60755aebf79dc2fa279ca6d4dbd0bb4c2b9f043b4c80b1328ba","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-f6c269a62d754427f4c9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --supports--> v00\n- v00 --supports--> v10\n- v10 --refutes--> v0e\n- v0e --supports--> v06\n- v06 --refutes--> v09\n- v09 --refutes--> v02\n- v02 --supports--> v04\n- v04 --supports--> v0a\n- v0a --refutes--> v08\n- v08 --supports--> v05\n- v05 --refutes--> v0d\n- v0d --supports--> v07\n- v07 --supports--> v11\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v13\n- v05 --supports--> v0f\n- v06 --supports--> v0b\n- v07 --supports--> v0c\n- v09 --refutes--> v12\n- v0a --supports--> v01\n\n[VERIFIED ROOT]\nv03 = 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":"2622c22c872b530c047f9332e6e1be2e55a4e4e796e8f1bcba210968dd3fbd0a","schema":"livegraph.boolean-propagation.v1","spec_hash":"f6c269a62d754427f4c98903ef32f2a2b2106bdb182958ee33d17d3a207b50a0","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":13,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-99e23f9f258d4657dcd2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --refutes--> v0d\n- v0d --refutes--> v0f\n- v0f --refutes--> v07\n- v07 --supports--> v0b\n- v0b --refutes--> v05\n- v05 --supports--> v12\n- v12 --refutes--> v0c\n- v0c --refutes--> v00\n- v00 --supports--> v10\n- v10 --supports--> v11\n- v11 --supports--> v02\n- v02 --supports--> v0a\n- v0a --refutes--> v13\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v08\n- v01 --refutes--> v04\n- v01 --supports--> v0e\n- v05 --refutes--> v09\n- v07 --supports--> v03\n- v0d --refutes--> v06\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":"607c7687909083b4b5c782ac7aaba8009a6f71daf14a2c0a903f4d59e8e4aab6","schema":"livegraph.boolean-propagation.v1","spec_hash":"99e23f9f258d4657dcd24d39a5b0dba33ab5bc19142ca2cf94fcc528dc7aeb71","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-a4b9ac0dc3dd4fcdbe4a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v0d\n- v0d --refutes--> v01\n- v01 --supports--> v02\n- v02 --supports--> v0a\n- v0a --refutes--> v06\n- v06 --refutes--> v0f\n- v0f --supports--> v12\n- v12 --refutes--> v10\n- v10 --supports--> v0c\n- v0c --refutes--> v09\n- v09 --refutes--> v11\n- v11 --refutes--> v0b\n- v0b --refutes--> v0e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v07\n- v02 --refutes--> v04\n- v02 --supports--> v08\n- v09 --supports--> v05\n- v09 --supports--> v13\n- v0a --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":"5d449820a145044951a9ac821dd5b242a1c4d46b58f2ee8344a5495c219e740c","schema":"livegraph.boolean-propagation.v1","spec_hash":"a4b9ac0dc3dd4fcdbe4a9da753b9ad2d561382f0fa86372b6972f94a8e1d49b7","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-fd84e665d497feca582b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0b --supports--> v07\n- v07 --refutes--> v11\n- v11 --supports--> v00\n- v00 --refutes--> v02\n- v02 --supports--> v10\n- v10 --refutes--> v03\n- v03 --supports--> v12\n- v12 --refutes--> v0d\n- v0d --refutes--> v04\n- v04 --refutes--> v09\n- v09 --supports--> v0a\n- v0a --refutes--> v06\n- v06 --refutes--> v13\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v01\n- v03 --supports--> v0f\n- v04 --supports--> v0e\n- v09 --refutes--> v05\n- v0b --refutes--> v08\n- v11 --supports--> v0c\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":"0e55d08d4bad628755b08dbca5ffdc2da0adfff813f9fe85562e3257310000a7","schema":"livegraph.boolean-propagation.v1","spec_hash":"fd84e665d497feca582b79a515ce105de6f1e01c68045cff035f7944b6335a35","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-eb192b934ebe1a928eb3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --refutes--> v04\n- v04 --refutes--> v06\n- v06 --refutes--> v0e\n- v0e --supports--> v08\n- v08 --refutes--> v0b\n- v0b --supports--> v03\n- v03 --supports--> v10\n- v10 --refutes--> v0d\n- v0d --supports--> v0c\n- v0c --refutes--> v0a\n- v0a --supports--> v07\n- v07 --refutes--> v12\n- v12 --supports--> v11\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --supports--> v05\n- v06 --supports--> v0f\n- v0c --refutes--> v13\n- v0d --refutes--> v02\n- v0e --refutes--> v00\n- v0e --refutes--> v01\n\n[VERIFIED ROOT]\nv09 = 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":"3dc77851aed5450165bb464f68fa825c3782ed6adb87a507ed8507a78ab3a7db","schema":"livegraph.boolean-propagation.v1","spec_hash":"eb192b934ebe1a928eb3582f4cce5d31f54cfa41e11e6fabf5384527114ff428","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-4188a94a0ee4f4c80bb7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --refutes--> v11\n- v11 --refutes--> v10\n- v10 --supports--> v0f\n- v0f --supports--> v0b\n- v0b --refutes--> v0e\n- v0e --supports--> v13\n- v13 --supports--> v09\n- v09 --supports--> v01\n- v01 --refutes--> v07\n- v07 --supports--> v05\n- v05 --supports--> v02\n- v02 --refutes--> v08\n- v08 --supports--> v12\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0a\n- v02 --supports--> v0c\n- v05 --refutes--> v06\n- v0b --refutes--> v0d\n- v10 --refutes--> v04\n- v11 --supports--> v03\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":"302032dac0c3cc836c8a3d031c9f33014c16d8f170afd3a4bc904c972782dee0","schema":"livegraph.boolean-propagation.v1","spec_hash":"4188a94a0ee4f4c80bb7aadbb27775a0972139626b46da0dbd2a18f796aaa48e","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-cee1e209c6f17b57e2e2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0f --supports--> v0c\n- v0c --refutes--> v10\n- v10 --refutes--> v0b\n- v0b --refutes--> v05\n- v05 --supports--> v04\n- v04 --refutes--> v12\n- v12 --supports--> v0a\n- v0a --supports--> v0d\n- v0d --refutes--> v11\n- v11 --supports--> v02\n- v02 --supports--> v08\n- v08 --refutes--> v03\n- v03 --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v09\n- v03 --supports--> v06\n- v05 --refutes--> v07\n- v0d --supports--> v0e\n- v0d --supports--> v13\n- v0f --refutes--> v01\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":"8765c7e71c488e4c24c93f3923e644ae4bf0f72923475127a97a509206f4dcb0","schema":"livegraph.boolean-propagation.v1","spec_hash":"cee1e209c6f17b57e2e2f39e608242f1d624acd22a348c9410f2e2b92b80712b","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-daedd5250e1314b47ddb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v11 --supports--> v0f\n- v0f --supports--> v03\n- v03 --supports--> v02\n- v02 --refutes--> v0e\n- v0e --refutes--> v0d\n- v0d --refutes--> v06\n- v06 --supports--> v07\n- v07 --supports--> v12\n- v12 --refutes--> v0a\n- v0a --refutes--> v09\n- v09 --supports--> v05\n- v05 --refutes--> v0c\n- v0c --supports--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v00\n- v06 --refutes--> v08\n- v06 --supports--> v01\n- v0e --refutes--> v13\n- v11 --refutes--> v0b\n- v12 --supports--> v10\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":"5a8be5881111c39d41802697aac71b90970706ecea7003f2d00d0ade4d5ade94","schema":"livegraph.boolean-propagation.v1","spec_hash":"daedd5250e1314b47ddb822308c882c5dfed7f7ddf84b90d1d7cc188f6d57b9c","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-7d2b8a001861097fc6b0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --refutes--> v01\n- v01 --refutes--> v09\n- v09 --refutes--> v10\n- v10 --refutes--> v11\n- v11 --refutes--> v0a\n- v0a --supports--> v02\n- v02 --supports--> v0f\n- v0f --refutes--> v12\n- v12 --supports--> v06\n- v06 --supports--> v13\n- v13 --refutes--> v0d\n- v0d --supports--> v0e\n- v0e --refutes--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v00\n- v06 --refutes--> v0b\n- v0a --supports--> v03\n- v0f --refutes--> v04\n- v10 --supports--> v0c\n- v13 --supports--> v08\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":"403a2a0a18fafb8280651e306b54cd90816408da2d1e3462376b75eebb35eb34","schema":"livegraph.boolean-propagation.v1","spec_hash":"7d2b8a001861097fc6b09d0af4ce9d70d5aeaa2f1e33721981b6511b3e89f06c","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-c482c0856cfa4e89641a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --supports--> v13\n- v13 --refutes--> v0f\n- v0f --supports--> v0a\n- v0a --refutes--> v0d\n- v0d --refutes--> v05\n- v05 --supports--> v09\n- v09 --refutes--> v0e\n- v0e --supports--> v01\n- v01 --supports--> v03\n- v03 --supports--> v06\n- v06 --refutes--> v00\n- v00 --refutes--> v12\n- v12 --refutes--> v11\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v0c\n- v05 --supports--> v04\n- v0e --refutes--> v08\n- v0e --supports--> v10\n- v13 --refutes--> v02\n- v13 --refutes--> v0b\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":"8c03f837c84f89acbbe09ec73b711cab12a19b592785656f6f8a2599846b7a2a","schema":"livegraph.boolean-propagation.v1","spec_hash":"c482c0856cfa4e89641a028311b6d957382a00bcea964bc232ee50e08534354b","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-a50e8c0afe43fd0031cc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v10 --supports--> v07\n- v07 --supports--> v0d\n- v0d --supports--> v00\n- v00 --supports--> v02\n- v02 --refutes--> v11\n- v11 --refutes--> v12\n- v12 --refutes--> v0e\n- v0e --refutes--> v09\n- v09 --supports--> v05\n- v05 --refutes--> v0c\n- v0c --refutes--> v0f\n- v0f --refutes--> v13\n- v13 --supports--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v06\n- v07 --refutes--> v03\n- v09 --refutes--> v04\n- v0f --refutes--> v08\n- v10 --refutes--> v01\n- v10 --refutes--> v0a\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":"80f733a06d9e8c3cedceb2cd5bfb26b6c5c0da15336b0e361c9c70d3b96f2bd3","schema":"livegraph.boolean-propagation.v1","spec_hash":"a50e8c0afe43fd0031cce6042816d09e7eb30aa3e6dd80684f7059dd64739aa3","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-9b0b1f3235f75267d4f8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --supports--> v0a\n- v0a --refutes--> v01\n- v01 --supports--> v0d\n- v0d --refutes--> v04\n- v04 --refutes--> v08\n- v08 --refutes--> v12\n- v12 --supports--> v0c\n- v0c --refutes--> v0b\n- v0b --refutes--> v11\n- v11 --refutes--> v09\n- v09 --supports--> v13\n- v13 --supports--> v10\n- v10 --supports--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v0f\n- v08 --supports--> v03\n- v09 --refutes--> v02\n- v09 --refutes--> v0e\n- v0b --refutes--> v00\n- v0b --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":"361841276f6d4e4e009b1334e47345aaebae737b3924c6409480af6576451ae4","schema":"livegraph.boolean-propagation.v1","spec_hash":"9b0b1f3235f75267d4f82ad96b69ecd71ad2cf9cd19f6e99cec96b5951a4856d","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-9ba2066148640e5d3f9a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --refutes--> v11\n- v11 --refutes--> v04\n- v04 --supports--> v0a\n- v0a --refutes--> v05\n- v05 --refutes--> v09\n- v09 --refutes--> v06\n- v06 --refutes--> v00\n- v00 --supports--> v0d\n- v0d --supports--> v0e\n- v0e --supports--> v01\n- v01 --supports--> v12\n- v12 --supports--> v0c\n- v0c --refutes--> v10\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v07\n- v05 --refutes--> v0f\n- v05 --supports--> v0b\n- v09 --supports--> v08\n- v0a --supports--> v02\n- v11 --refutes--> v13\n\n[VERIFIED ROOT]\nv03 = 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":"338d641252a569d3601bee628b8f37c41cc0c40543135c7713e4613fd104be91","schema":"livegraph.boolean-propagation.v1","spec_hash":"9ba2066148640e5d3f9a563dd96691544aa264d34cdb050727d114fd0c3f8ec6","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-bcfa64ebe648b5052a09","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --supports--> v02\n- v02 --refutes--> v0f\n- v0f --refutes--> v11\n- v11 --supports--> v04\n- v04 --supports--> v09\n- v09 --supports--> v05\n- v05 --refutes--> v0d\n- v0d --supports--> v0a\n- v0a --supports--> v01\n- v01 --supports--> v13\n- v13 --supports--> v0c\n- v0c --refutes--> v08\n- v08 --refutes--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v0b\n- v04 --supports--> v00\n- v05 --supports--> v0e\n- v08 --supports--> v12\n- v0f --supports--> v10\n- v13 --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":"423a111eb4445fd1bba4f1f8db80ebcd78b16cf4d98902654307c0849655487a","schema":"livegraph.boolean-propagation.v1","spec_hash":"bcfa64ebe648b5052a09c9df82e43f9342f28196d0c9c7ffaf008a0a68e6d50e","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-67f33a534dc6efb81eb7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --refutes--> v13\n- v13 --supports--> v12\n- v12 --refutes--> v0a\n- v0a --refutes--> v0d\n- v0d --refutes--> v00\n- v00 --refutes--> v0f\n- v0f --supports--> v02\n- v02 --supports--> v10\n- v10 --refutes--> v08\n- v08 --supports--> v01\n- v01 --refutes--> v05\n- v05 --supports--> v0e\n- v0e --supports--> v11\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v0b\n- v08 --supports--> v07\n- v0a --supports--> v03\n- v0a --supports--> v04\n- v0c --supports--> v09\n- v0d --refutes--> v06\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":"7ea791d2a5b6ef867c5aa085642f9f874eae037e8ad206c65da49016a9a0a636","schema":"livegraph.boolean-propagation.v1","spec_hash":"67f33a534dc6efb81eb7b3d5d2933e5bc6a99679bd24396ba5d7039a8ea49202","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-1913c2bb7fac9bd330aa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --supports--> v0d\n- v0d --supports--> v07\n- v07 --supports--> v05\n- v05 --supports--> v0f\n- v0f --supports--> v09\n- v09 --supports--> v0b\n- v0b --supports--> v08\n- v08 --refutes--> v06\n- v06 --refutes--> v12\n- v12 --supports--> v0e\n- v0e --supports--> v04\n- v04 --supports--> v01\n- v01 --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v00\n- v07 --refutes--> v02\n- v07 --refutes--> v10\n- v09 --supports--> v13\n- v0f --refutes--> v0c\n- v12 --refutes--> v11\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":"a2ad89693a08239c44243be228522209f0fa24b59f8c0ec8fb023660877e170b","schema":"livegraph.boolean-propagation.v1","spec_hash":"1913c2bb7fac9bd330aadfae305089f4d1e8a20edffedc9061ff0c8db0bad544","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-8ccad1d5ca722a48c114","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v13 --supports--> v02\n- v02 --supports--> v00\n- v00 --supports--> v03\n- v03 --supports--> v06\n- v06 --supports--> v0a\n- v0a --refutes--> v0b\n- v0b --refutes--> v0d\n- v0d --supports--> v07\n- v07 --supports--> v08\n- v08 --supports--> v0f\n- v0f --refutes--> v05\n- v05 --refutes--> v11\n- v11 --refutes--> v0e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v04\n- v06 --refutes--> v09\n- v06 --refutes--> v10\n- v08 --refutes--> v0c\n- v08 --supports--> v12\n- v13 --refutes--> v01\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":"101b56e46d08c46df5dd71b359f7421c2b57656f9a3865cce00e9a5e8e0392fb","schema":"livegraph.boolean-propagation.v1","spec_hash":"8ccad1d5ca722a48c1147265dda010a50efaccacf935fc720b88a47e95c1244f","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v13\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-e094aca1e9e8cc3c1c66","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --refutes--> v07\n- v07 --refutes--> v08\n- v08 --refutes--> v09\n- v09 --refutes--> v0c\n- v0c --refutes--> v01\n- v01 --supports--> v0f\n- v0f --supports--> v02\n- v02 --refutes--> v11\n- v11 --supports--> v0d\n- v0d --supports--> v0e\n- v0e --refutes--> v03\n- v03 --refutes--> v10\n- v10 --refutes--> v12\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v00\n- v02 --supports--> v06\n- v03 --refutes--> v04\n- v08 --refutes--> v0a\n- v08 --supports--> v0b\n- v0f --refutes--> v13\n\n[VERIFIED ROOT]\nv05 = 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":"5ebc5d9429c1053cfc38498d7cbc02d402f90d9a99248ed32d3c530bbc31a761","schema":"livegraph.boolean-propagation.v1","spec_hash":"e094aca1e9e8cc3c1c6653022c914a3dc0d4a6085997e18a650e4dd508d6dbd7","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":13,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-a9af1a61d4f99cc7ff0c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --supports--> v0f\n- v0f --supports--> v0e\n- v0e --refutes--> v06\n- v06 --refutes--> v07\n- v07 --supports--> v01\n- v01 --refutes--> v10\n- v10 --refutes--> v02\n- v02 --refutes--> v11\n- v11 --refutes--> v04\n- v04 --refutes--> v03\n- v03 --refutes--> v00\n- v00 --refutes--> v13\n- v13 --refutes--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v12\n- v01 --refutes--> v08\n- v02 --supports--> v05\n- v0f --supports--> v0b\n- v10 --refutes--> v0a\n- v13 --supports--> 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":"d05b8721510650a4425e46b3148b223eee8c5895675e1561f53ef7eefdfbe31f","schema":"livegraph.boolean-propagation.v1","spec_hash":"a9af1a61d4f99cc7ff0cb48c35ed6e7e73ddf5704a1eee36e69c60db2cf4fea1","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-b9ec339af0721bdf6139","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0f --refutes--> v06\n- v06 --supports--> v00\n- v00 --refutes--> v0c\n- v0c --refutes--> v13\n- v13 --refutes--> v0a\n- v0a --refutes--> v0e\n- v0e --refutes--> v11\n- v11 --refutes--> v0b\n- v0b --supports--> v08\n- v08 --refutes--> v02\n- v02 --refutes--> v05\n- v05 --refutes--> v07\n- v07 --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v09\n- v07 --refutes--> v12\n- v0b --supports--> v10\n- v0e --refutes--> v01\n- v11 --refutes--> v04\n- v13 --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":"019efcf3902e117a1a95013ec77f4cd03319c72f3bfac0b41965076c5c389ba4","schema":"livegraph.boolean-propagation.v1","spec_hash":"b9ec339af0721bdf6139f200eb36dbdca1d8badbde2ef2372e74bfce734918f2","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-63b922c42627416140d0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v0a\n- v0a --supports--> v01\n- v01 --supports--> v0b\n- v0b --supports--> v12\n- v12 --refutes--> v0e\n- v0e --supports--> v07\n- v07 --supports--> v0f\n- v0f --supports--> v13\n- v13 --refutes--> v00\n- v00 --supports--> v02\n- v02 --refutes--> v0c\n- v0c --refutes--> v08\n- v08 --supports--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v11\n- v04 --refutes--> v0d\n- v08 --supports--> v05\n- v0c --supports--> v06\n- v12 --supports--> v10\n- v13 --supports--> v03\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":"26370b9b4a29d87279d63ae2a3a3f5fd799aabfbe7f3406c26562a58df89ea9b","schema":"livegraph.boolean-propagation.v1","spec_hash":"63b922c42627416140d0d3459625fc50b53a5f5503a0538986d947afe420ce6c","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-7426a9b94f8247f12164","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --refutes--> v11\n- v11 --refutes--> v01\n- v01 --refutes--> v05\n- v05 --supports--> v0b\n- v0b --supports--> v10\n- v10 --refutes--> v08\n- v08 --refutes--> v02\n- v02 --refutes--> v04\n- v04 --supports--> v03\n- v03 --refutes--> v09\n- v09 --refutes--> v12\n- v12 --supports--> v0c\n- v0c --refutes--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v00\n- v03 --supports--> v13\n- v05 --refutes--> v06\n- v09 --supports--> v0f\n- v10 --supports--> v0a\n- v10 --supports--> v0e\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":"ff80808d1a422fa9625f61bc1a339ef322819814aabb3063a5deb51d9ac4b2b6","schema":"livegraph.boolean-propagation.v1","spec_hash":"7426a9b94f8247f121645f68967048c135fd624774e6482aaae80f43b91fe97e","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-9b8d714a33d158b9d600","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --refutes--> v04\n- v04 --refutes--> v10\n- v10 --supports--> v08\n- v08 --refutes--> v0f\n- v0f --refutes--> v12\n- v12 --refutes--> v07\n- v07 --refutes--> v06\n- v06 --supports--> v01\n- v01 --refutes--> v0a\n- v0a --supports--> v0b\n- v0b --refutes--> v0c\n- v0c --supports--> v0d\n- v0d --supports--> v0e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v13\n- v04 --refutes--> v11\n- v07 --supports--> v00\n- v07 --supports--> v03\n- v08 --refutes--> v09\n- v0b --refutes--> v02\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":"36b574e079cbaed4b17bd6c87edc7ebd43f5939b7c378c60693e6ecf03fcdd7b","schema":"livegraph.boolean-propagation.v1","spec_hash":"9b8d714a33d158b9d600140c414d4d584c53f01326e7d2fc5d75f3f376b9286d","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-bfea8a4abd715e318f14","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0d --supports--> v0e\n- v0e --supports--> v10\n- v10 --refutes--> v0c\n- v0c --supports--> v00\n- v00 --refutes--> v01\n- v01 --refutes--> v09\n- v09 --refutes--> v06\n- v06 --supports--> v02\n- v02 --supports--> v13\n- v13 --refutes--> v0f\n- v0f --supports--> v04\n- v04 --refutes--> v0b\n- v0b --supports--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v03\n- v01 --refutes--> v12\n- v0d --supports--> v11\n- v0f --supports--> v07\n- v0f --supports--> v08\n- v10 --supports--> v0a\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":"da3fbe3a7b34ca3fe909ce57e2d8c1ef7d88add94dedd6511e2b5463975ca839","schema":"livegraph.boolean-propagation.v1","spec_hash":"bfea8a4abd715e318f14477de0330f83e906ef0d0cb0a0bbccbb052a72b96762","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-3f5e733e1efb3b83202e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --supports--> v12\n- v12 --supports--> v06\n- v06 --supports--> v02\n- v02 --refutes--> v0b\n- v0b --refutes--> v13\n- v13 --refutes--> v10\n- v10 --refutes--> v04\n- v04 --refutes--> v0e\n- v0e --refutes--> v11\n- v11 --supports--> v0d\n- v0d --refutes--> v0c\n- v0c --refutes--> v03\n- v03 --refutes--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v05\n- v04 --supports--> v01\n- v06 --supports--> v07\n- v0e --refutes--> v09\n- v11 --refutes--> v0f\n- v12 --supports--> v00\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":"00c3892794aaed942d42f784ae8b8393b06a3581a0854297e1454bac49c97c82","schema":"livegraph.boolean-propagation.v1","spec_hash":"3f5e733e1efb3b83202e0ed06beacf07652f2d31d500f50e3b05613593e9de8c","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-d318e6101df11e3f90df","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v12 --refutes--> v10\n- v10 --supports--> v03\n- v03 --refutes--> v0f\n- v0f --supports--> v0d\n- v0d --supports--> v00\n- v00 --supports--> v0b\n- v0b --refutes--> v05\n- v05 --refutes--> v07\n- v07 --supports--> v06\n- v06 --refutes--> v13\n- v13 --refutes--> v11\n- v11 --supports--> v02\n- v02 --supports--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v0a\n- v06 --refutes--> v0e\n- v07 --supports--> v08\n- v0d --refutes--> v09\n- v10 --refutes--> v0c\n- v13 --refutes--> v04\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":"27c687e8329cb54da9cb9b695da40d8e2597f1f3a71fc0cdf25fcc4fd42daf2d","schema":"livegraph.boolean-propagation.v1","spec_hash":"d318e6101df11e3f90dfbbb411baccc0c4c11c37dda37af96f23ac5212c18626","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v12\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-3f0670daccd68a287990","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v12 --supports--> v0c\n- v0c --supports--> v04\n- v04 --supports--> v06\n- v06 --supports--> v09\n- v09 --supports--> v01\n- v01 --supports--> v0a\n- v0a --supports--> v00\n- v00 --refutes--> v10\n- v10 --supports--> v02\n- v02 --refutes--> v07\n- v07 --refutes--> v05\n- v05 --supports--> v0d\n- v0d --supports--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v0b\n- v02 --supports--> v11\n- v04 --supports--> v13\n- v07 --refutes--> v0e\n- v09 --supports--> v03\n- v0a --refutes--> v08\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":"79397feb0f34fae53a5e646ba7084b41a1668aee1b3a84c69638cc02bf3c6207","schema":"livegraph.boolean-propagation.v1","spec_hash":"3f0670daccd68a28799087161c97ccb1e90860f409d9b1bd91e59c4d0814e922","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v12\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-02efb0479f94a9929620","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --supports--> v0b\n- v0b --supports--> v03\n- v03 --supports--> v12\n- v12 --refutes--> v0c\n- v0c --supports--> v13\n- v13 --refutes--> v05\n- v05 --refutes--> v00\n- v00 --refutes--> v02\n- v02 --supports--> v0f\n- v0f --refutes--> v07\n- v07 --refutes--> v08\n- v08 --supports--> v0e\n- v0e --supports--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v0d\n- v04 --refutes--> v10\n- v07 --supports--> v06\n- v0e --supports--> v09\n- v0f --refutes--> v01\n- v13 --refutes--> v11\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":"60cfbe66b04051a2ef4b5178ceb891fc33ef2ed3c0e9720babb126583ff2213c","schema":"livegraph.boolean-propagation.v1","spec_hash":"02efb0479f94a99296200142103f9b00934623013780c12bbecd12b94e0901d9","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-fc87cf623b166f37d91e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v11 --supports--> v09\n- v09 --refutes--> v0e\n- v0e --refutes--> v02\n- v02 --supports--> v0c\n- v0c --refutes--> v0b\n- v0b --supports--> v12\n- v12 --supports--> v07\n- v07 --supports--> v08\n- v08 --supports--> v0a\n- v0a --refutes--> v13\n- v13 --refutes--> v05\n- v05 --refutes--> v10\n- v10 --supports--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v07 --refutes--> v0f\n- v08 --supports--> v06\n- v0e --refutes--> v00\n- v0e --refutes--> v04\n- v11 --supports--> v03\n- v13 --supports--> v01\n\n[VERIFIED ROOT]\nv11 = 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":"80e4cc22103a1bd7621e8bc04d7cfc6f620c331bb2aeef910c1487c4e49f1c9e","schema":"livegraph.boolean-propagation.v1","spec_hash":"fc87cf623b166f37d91e27e6c2d20974e3221e34e3edbe13cccecd329437358d","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-22fa261be3e746a6dfff","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v07\n- v07 --supports--> v00\n- v00 --supports--> v0a\n- v0a --refutes--> v05\n- v05 --refutes--> v06\n- v06 --refutes--> v09\n- v09 --refutes--> v0c\n- v0c --supports--> v11\n- v11 --supports--> v13\n- v13 --refutes--> v08\n- v08 --refutes--> v0b\n- v0b --supports--> v0e\n- v0e --supports--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0d\n- v00 --supports--> v0f\n- v09 --refutes--> v02\n- v09 --refutes--> v12\n- v0a --refutes--> v03\n- v0b --refutes--> v10\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":"64a4851765c86fdb15f88cf731d5b3070882d7d46b3508fca573cd26106b52b3","schema":"livegraph.boolean-propagation.v1","spec_hash":"22fa261be3e746a6dfff61bb5dc3c65594a80b3f434e343f80dd9674a45eb6d0","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-b5b7e714871e3f4b81d3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --refutes--> v12\n- v12 --refutes--> v08\n- v08 --refutes--> v10\n- v10 --supports--> v13\n- v13 --supports--> v02\n- v02 --refutes--> v11\n- v11 --refutes--> v07\n- v07 --supports--> v0b\n- v0b --supports--> v00\n- v00 --refutes--> v01\n- v01 --supports--> v0c\n- v0c --refutes--> v0f\n- v0f --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0a\n- v01 --supports--> v0d\n- v09 --refutes--> v04\n- v09 --supports--> v0e\n- v0c --refutes--> v05\n- v12 --refutes--> v06\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":"c7d4698bbcc90e83e94fd2ecfe5d726ee0521e3a74d5de1d5417aa59bdafbd84","schema":"livegraph.boolean-propagation.v1","spec_hash":"b5b7e714871e3f4b81d31a40ff2b002015a67021bb73615acf8d629fcef3dc6b","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-a2fa1b24b6e17b098a19","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0d --supports--> v01\n- v01 --supports--> v13\n- v13 --supports--> v09\n- v09 --refutes--> v0b\n- v0b --refutes--> v02\n- v02 --refutes--> v04\n- v04 --supports--> v0c\n- v0c --refutes--> v0f\n- v0f --refutes--> v06\n- v06 --refutes--> v12\n- v12 --refutes--> v11\n- v11 --supports--> v00\n- v00 --supports--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v0a\n- v06 --supports--> v10\n- v09 --supports--> v0e\n- v0b --supports--> v03\n- v0d --refutes--> v07\n- v11 --supports--> v08\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":"8ee12c3f4a8742c475489717f53a40c0eba355f5836ec377b94476db411146e7","schema":"livegraph.boolean-propagation.v1","spec_hash":"a2fa1b24b6e17b098a192ac4de30f382182bb834027e13ea732821505edf051d","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-336cdec15a1351e56f01","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --refutes--> v0d\n- v0d --refutes--> v04\n- v04 --supports--> v09\n- v09 --supports--> v0e\n- v0e --supports--> v0a\n- v0a --supports--> v07\n- v07 --supports--> v13\n- v13 --refutes--> v05\n- v05 --refutes--> v12\n- v12 --refutes--> v0b\n- v0b --supports--> v0f\n- v0f --refutes--> v0c\n- v0c --refutes--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v08\n- v0a --supports--> v00\n- v0c --refutes--> v03\n- v0f --refutes--> v10\n- v0f --supports--> v01\n- v0f --supports--> 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":"0598ec5e71811b667a7a9d8dd2d56ab6da429db94e6eaea10756d4e21474d80c","schema":"livegraph.boolean-propagation.v1","spec_hash":"336cdec15a1351e56f0194028bb1110fb224d5a2686c7531b92db9b42390a7fa","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-fd580264c554065ac0e3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --refutes--> v08\n- v08 --refutes--> v0e\n- v0e --supports--> v11\n- v11 --refutes--> v02\n- v02 --supports--> v10\n- v10 --refutes--> v07\n- v07 --supports--> v05\n- v05 --supports--> v0a\n- v0a --refutes--> v06\n- v06 --supports--> v00\n- v00 --refutes--> v03\n- v03 --supports--> v13\n- v13 --supports--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v0b\n- v07 --supports--> v0c\n- v08 --supports--> v0d\n- v0a --refutes--> v01\n- v0e --refutes--> v12\n- v13 --refutes--> v04\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":"951354e21f2beda5c4509a622ddb55a6c0125d7ae40e0dc332d6ae1f594117b5","schema":"livegraph.boolean-propagation.v1","spec_hash":"fd580264c554065ac0e3dc288401c11b37422e977dd71b80c469f148e181a3e2","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-ed689fbfdf6f4da5aa44","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --supports--> v0f\n- v0f --supports--> v0a\n- v0a --refutes--> v10\n- v10 --supports--> v00\n- v00 --supports--> v11\n- v11 --supports--> v03\n- v03 --supports--> v05\n- v05 --supports--> v0e\n- v0e --supports--> v09\n- v09 --refutes--> v0c\n- v0c --refutes--> v13\n- v13 --refutes--> v01\n- v01 --refutes--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v0e --supports--> v04\n- v0f --refutes--> v0d\n- v10 --refutes--> v0b\n- v10 --supports--> v12\n- v11 --supports--> v07\n- v13 --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":"027172095d16a14b421b1f338a3bb8f5e295ba07d25ff1f2150e66c47de621b3","schema":"livegraph.boolean-propagation.v1","spec_hash":"ed689fbfdf6f4da5aa44b581f4c84fd8f10a390f12c73fd1bb990c631d07af5d","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-551b489bc6b1302719f1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --refutes--> v09\n- v09 --refutes--> v11\n- v11 --refutes--> v0c\n- v0c --refutes--> v04\n- v04 --refutes--> v0a\n- v0a --refutes--> v0d\n- v0d --refutes--> v06\n- v06 --refutes--> v12\n- v12 --refutes--> v07\n- v07 --supports--> v0f\n- v0f --supports--> v00\n- v00 --refutes--> v10\n- v10 --supports--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v03\n- v04 --supports--> v13\n- v06 --refutes--> v08\n- v07 --refutes--> v0b\n- v07 --supports--> v0e\n- v11 --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":"558ece9083af9628e6147c30a68971eefacde2655984dfe88ec7451445084b2f","schema":"livegraph.boolean-propagation.v1","spec_hash":"551b489bc6b1302719f1f8faa1a2cc3a47d999bd9f4e33ec46aef2bbcab4e084","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"}],\"hop\":13,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-86ea66631873055be2e5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0b --supports--> v10\n- v10 --supports--> v00\n- v00 --supports--> v07\n- v07 --refutes--> v05\n- v05 --refutes--> v0e\n- v0e --refutes--> v0c\n- v0c --supports--> v11\n- v11 --supports--> v06\n- v06 --refutes--> v09\n- v09 --refutes--> v04\n- v04 --refutes--> v03\n- v03 --refutes--> v01\n- v01 --supports--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v02\n- v07 --refutes--> v12\n- v09 --supports--> v08\n- v0e --refutes--> v13\n- v10 --supports--> v0d\n- v11 --supports--> 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":"f3cfdd8f336799ebcf3cdef4770cfac7c089ed0af742c304e73876157e7e701b","schema":"livegraph.boolean-propagation.v1","spec_hash":"86ea66631873055be2e5ed2d8eb2092cb351532fb64730f3bcc09a5856bc2dd1","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":13,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":13,"item_id":"lggrpo1-cb1042137d9b75dd4f27","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --supports--> v0c\n- v0c --refutes--> v0d\n- v0d --refutes--> v0a\n- v0a --refutes--> v04\n- v04 --refutes--> v02\n- v02 --supports--> v08\n- v08 --supports--> v0e\n- v0e --supports--> v11\n- v11 --refutes--> v10\n- v10 --refutes--> v12\n- v12 --supports--> v06\n- v06 --refutes--> v13\n- v13 --supports--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v0b\n- v0a --refutes--> v05\n- v0c --refutes--> v01\n- v12 --supports--> v07\n- v13 --refutes--> v0f\n- v13 --supports--> v00\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":"08d1c2560c308af74089a19d9013bb92511e87d5b2819980f06d288307699ece","schema":"livegraph.boolean-propagation.v1","spec_hash":"cb1042137d9b75dd4f278a1afa540113153156fc2dc732da585eae18d3c081a4","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":13,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":13,"item_id":"lggrpo1-aa49e0201f022e6a5d5c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --refutes--> v09\n- v09 --refutes--> v02\n- v02 --supports--> v11\n- v11 --refutes--> v0c\n- v0c --refutes--> v0d\n- v0d --supports--> v10\n- v10 --supports--> v01\n- v01 --refutes--> v0f\n- v0f --refutes--> v03\n- v03 --refutes--> v05\n- v05 --refutes--> v07\n- v07 --supports--> v06\n- v06 --supports--> v0e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v04\n- v07 --refutes--> v0a\n- v09 --supports--> v13\n- v0f --refutes--> v12\n- v10 --supports--> v0b\n- v11 --supports--> 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":"6b6e5187025a8eab763cb70fa564cb0a0961cc671c55ece8eb0aff750a3c0060","schema":"livegraph.boolean-propagation.v1","spec_hash":"aa49e0201f022e6a5d5c37dddd3024d9d117b70c4fc2afefdb6788e6873e016f","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":13,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-b23b5876a66c5f4c6fba","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0f --refutes--> v0a\n- v0a --refutes--> v09\n- v09 --supports--> v07\n- v07 --refutes--> v04\n- v04 --refutes--> v06\n- v06 --refutes--> v15\n- v15 --refutes--> v14\n- v14 --supports--> v11\n- v11 --supports--> v12\n- v12 --refutes--> v0b\n- v0b --supports--> v00\n- v00 --refutes--> v0c\n- v0c --refutes--> v0d\n- v0d --supports--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v10\n- v06 --supports--> v03\n- v09 --refutes--> v13\n- v0a --supports--> v01\n- v0b --supports--> v02\n- v0d --refutes--> v08\n- v0d --refutes--> v0e\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":"c81f098d816587e0c92d221e3475044a641c61c8021161d09a17bd683fd4af64","schema":"livegraph.boolean-propagation.v1","spec_hash":"b23b5876a66c5f4c6fbacb61ebf68de4be92226f7cfe3a023b069481320766ab","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-42a3b6f3588675b2ddaa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --supports--> v08\n- v08 --refutes--> v03\n- v03 --supports--> v14\n- v14 --supports--> v09\n- v09 --refutes--> v0d\n- v0d --refutes--> v04\n- v04 --supports--> v06\n- v06 --refutes--> v13\n- v13 --supports--> v07\n- v07 --refutes--> v02\n- v02 --refutes--> v00\n- v00 --refutes--> v11\n- v11 --supports--> v0b\n- v0b --refutes--> v15\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v0e\n- v03 --supports--> v0f\n- v07 --refutes--> v0c\n- v08 --supports--> v01\n- v08 --supports--> v10\n- v0d --supports--> v05\n- v14 --supports--> v12\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":"ae6bfd6b4ef3be8dd0eafb8e2060377409c4378ef180108cdbedfd112f0a041f","schema":"livegraph.boolean-propagation.v1","spec_hash":"42a3b6f3588675b2ddaa038890e6822b471570dcc8cc424e5a91d93d44d9de00","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-b00c9da5ac0544e3fc7b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --supports--> v13\n- v13 --refutes--> v0f\n- v0f --supports--> v0e\n- v0e --supports--> v00\n- v00 --supports--> v10\n- v10 --refutes--> v07\n- v07 --supports--> v03\n- v03 --refutes--> v09\n- v09 --supports--> v04\n- v04 --refutes--> v0d\n- v0d --refutes--> v0c\n- v0c --supports--> v0b\n- v0b --supports--> v01\n- v01 --supports--> v11\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v14\n- v01 --refutes--> v08\n- v01 --supports--> v02\n- v03 --supports--> v0a\n- v04 --refutes--> v06\n- v0c --supports--> v12\n- v0f --refutes--> v15\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":"f850bd3aa29ec21dbb6fcfbcfd92a6eb63007cb88141e04e0c7adf0f94fa17d6","schema":"livegraph.boolean-propagation.v1","spec_hash":"b00c9da5ac0544e3fc7bec2aa1a90809beb33ae0da0dc3d03470300c5c08e4b2","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v13\"}],\"hop\":14,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-49ce2ce49ad000bde6fe","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --supports--> v09\n- v09 --refutes--> v15\n- v15 --refutes--> v11\n- v11 --supports--> v08\n- v08 --supports--> v01\n- v01 --refutes--> v06\n- v06 --refutes--> v0b\n- v0b --refutes--> v13\n- v13 --refutes--> v0e\n- v0e --supports--> v05\n- v05 --supports--> v10\n- v10 --refutes--> v0f\n- v0f --refutes--> v12\n- v12 --refutes--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v02\n- v01 --supports--> v00\n- v08 --refutes--> v07\n- v0e --supports--> v14\n- v0f --supports--> v03\n- v10 --refutes--> v0c\n- v11 --refutes--> v0d\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":"9a17b02edd7ff0a152408d15d633b3a8d309af12220aec4542127b8535ca3441","schema":"livegraph.boolean-propagation.v1","spec_hash":"49ce2ce49ad000bde6fe7f8e8c38996335c3331167b49f102ba35c3d0363a3e7","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-a20bf946c06484013cd4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v0f\n- v0f --supports--> v0a\n- v0a --refutes--> v12\n- v12 --supports--> v10\n- v10 --supports--> v05\n- v05 --supports--> v09\n- v09 --supports--> v0e\n- v0e --supports--> v11\n- v11 --supports--> v0d\n- v0d --supports--> v08\n- v08 --supports--> v13\n- v13 --supports--> v0c\n- v0c --refutes--> v15\n- v15 --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v04\n- v08 --refutes--> v06\n- v08 --refutes--> v0b\n- v09 --supports--> v01\n- v0c --supports--> v02\n- v12 --refutes--> v07\n- v15 --refutes--> v14\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":"dab16e52909d5aa81fa30d63b46687c7e610f5c0b66d8bbc38a802193558496f","schema":"livegraph.boolean-propagation.v1","spec_hash":"a20bf946c06484013cd4c6626b9c8f621180477ee0279cb9b27048404f6bd01e","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-e1c7e1db5b4978d647f6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v11 --supports--> v09\n- v09 --refutes--> v07\n- v07 --supports--> v0a\n- v0a --supports--> v03\n- v03 --refutes--> v0d\n- v0d --refutes--> v02\n- v02 --refutes--> v0e\n- v0e --supports--> v12\n- v12 --refutes--> v13\n- v13 --supports--> v08\n- v08 --supports--> v05\n- v05 --supports--> v06\n- v06 --refutes--> v0c\n- v0c --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v0f\n- v03 --supports--> v0b\n- v06 --supports--> v14\n- v09 --refutes--> v04\n- v09 --refutes--> v10\n- v0d --refutes--> v01\n- v13 --supports--> v15\n\n[VERIFIED ROOT]\nv11 = 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":"973ea5a4755614d31e2e4f1e60f6c4a75073b981505beed4d46e52b8b3850532","schema":"livegraph.boolean-propagation.v1","spec_hash":"e1c7e1db5b4978d647f6b319a5f1c41717585937545c6f0eafd761ad4ba945de","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":14,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-341fa5d6ba80b2ffbfcc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --refutes--> v02\n- v02 --supports--> v01\n- v01 --refutes--> v0c\n- v0c --refutes--> v07\n- v07 --refutes--> v0e\n- v0e --supports--> v08\n- v08 --refutes--> v04\n- v04 --refutes--> v0f\n- v0f --supports--> v13\n- v13 --refutes--> v15\n- v15 --refutes--> v05\n- v05 --refutes--> v03\n- v03 --refutes--> v0b\n- v0b --refutes--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v06\n- v08 --refutes--> v12\n- v08 --supports--> v11\n- v0c --supports--> v0d\n- v0c --supports--> v10\n- v13 --refutes--> v14\n- v13 --supports--> v0a\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":"b49dca11378d8ab5000d8e66bab65b508f104fc79b7a114c621519986fa933c7","schema":"livegraph.boolean-propagation.v1","spec_hash":"341fa5d6ba80b2ffbfcc9735c30369ffe8c6a1234d6f2eba2841da25ae62c347","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-ca7b52fa915a61b895fe","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --refutes--> v00\n- v00 --supports--> v14\n- v14 --supports--> v10\n- v10 --refutes--> v0d\n- v0d --refutes--> v05\n- v05 --refutes--> v0e\n- v0e --refutes--> v12\n- v12 --refutes--> v0a\n- v0a --supports--> v09\n- v09 --supports--> v04\n- v04 --refutes--> v03\n- v03 --refutes--> v06\n- v06 --supports--> v02\n- v02 --supports--> v0c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v15\n- v04 --refutes--> v07\n- v04 --refutes--> v0b\n- v09 --refutes--> v11\n- v10 --refutes--> v01\n- v10 --refutes--> v0f\n- v10 --supports--> v13\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":"c5426d346f8e63a9e4d3d7c150e019a2ab77c38d89e321084a4d77d43eaa5892","schema":"livegraph.boolean-propagation.v1","spec_hash":"ca7b52fa915a61b895fe2f472806297117b6d8de973f86957d1e00c53fab903b","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-b38a7b3681ec035ad7ad","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0f --refutes--> v13\n- v13 --supports--> v0c\n- v0c --supports--> v0e\n- v0e --supports--> v0a\n- v0a --refutes--> v03\n- v03 --supports--> v08\n- v08 --refutes--> v05\n- v05 --refutes--> v07\n- v07 --refutes--> v00\n- v00 --refutes--> v04\n- v04 --supports--> v12\n- v12 --supports--> v0b\n- v0b --refutes--> v01\n- v01 --supports--> v10\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v0d\n- v03 --supports--> v02\n- v05 --supports--> v09\n- v0c --refutes--> v14\n- v0c --supports--> v15\n- v12 --refutes--> v06\n- v13 --supports--> v11\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":"ff4d16bf12040b75bb2f0c1e203a15a7c7c6018e2a2819d0b6da1c3da41ca67e","schema":"livegraph.boolean-propagation.v1","spec_hash":"b38a7b3681ec035ad7adfb574ff5e612a8405d1c579e53697648097954cb29ee","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":14,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-182faae5bb9603988fa5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --refutes--> v04\n- v04 --refutes--> v00\n- v00 --refutes--> v14\n- v14 --supports--> v11\n- v11 --refutes--> v0f\n- v0f --supports--> v0e\n- v0e --supports--> v02\n- v02 --refutes--> v07\n- v07 --refutes--> v12\n- v12 --refutes--> v0b\n- v0b --supports--> v10\n- v10 --supports--> v13\n- v13 --refutes--> v0c\n- v0c --supports--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v06\n- v00 --supports--> v09\n- v02 --supports--> v0d\n- v04 --refutes--> v05\n- v0b --supports--> v15\n- v0c --supports--> v0a\n- v10 --refutes--> v01\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":"a2f662ade5c681992ffa3ff4d30c11f8fa6f9c269c8622e166c4f3e3d47cc9db","schema":"livegraph.boolean-propagation.v1","spec_hash":"182faae5bb9603988fa55a0e4d69b6226cb6256e7a0207d748ffee328f530692","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-9087da495eabde616bb3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --supports--> v01\n- v01 --supports--> v15\n- v15 --refutes--> v05\n- v05 --refutes--> v10\n- v10 --supports--> v13\n- v13 --refutes--> v08\n- v08 --refutes--> v0e\n- v0e --supports--> v0d\n- v0d --refutes--> v09\n- v09 --refutes--> v0f\n- v0f --supports--> v0a\n- v0a --refutes--> v0c\n- v0c --supports--> v0b\n- v0b --supports--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --supports--> v07\n- v08 --refutes--> v14\n- v08 --supports--> v03\n- v09 --refutes--> v02\n- v0d --supports--> v11\n- v0f --supports--> v04\n- v10 --supports--> v12\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":"370fdc8abb1adf66bf60d3cec0da12da5341095170d1aaeec1bd344b63ddbd46","schema":"livegraph.boolean-propagation.v1","spec_hash":"9087da495eabde616bb38db1c71443092e1d4b34a9352895c7bd4ceffbd4eaf8","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-bb9189b4edb8ab687bfa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v11 --supports--> v0d\n- v0d --refutes--> v15\n- v15 --supports--> v07\n- v07 --refutes--> v05\n- v05 --supports--> v04\n- v04 --supports--> v0b\n- v0b --refutes--> v0a\n- v0a --refutes--> v10\n- v10 --supports--> v12\n- v12 --refutes--> v09\n- v09 --refutes--> v0e\n- v0e --supports--> v14\n- v14 --refutes--> v0c\n- v0c --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v06\n- v09 --supports--> v0f\n- v0a --supports--> v13\n- v0b --refutes--> v02\n- v0c --supports--> v00\n- v0d --supports--> v01\n- v0d --supports--> v08\n\n[VERIFIED ROOT]\nv11 = 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":"805975c16e3bee1b9634d1c7f93e246c57f49588099dddf885777ced80970504","schema":"livegraph.boolean-propagation.v1","spec_hash":"bb9189b4edb8ab687bfaa9e5d31fe1eb62272e02bdded084734665f60a95e143","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-e6db2186671d88218981","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --refutes--> v12\n- v12 --supports--> v01\n- v01 --refutes--> v04\n- v04 --supports--> v07\n- v07 --supports--> v00\n- v00 --refutes--> v10\n- v10 --supports--> v06\n- v06 --supports--> v05\n- v05 --supports--> v13\n- v13 --refutes--> v0a\n- v0a --supports--> v0d\n- v0d --refutes--> v03\n- v03 --refutes--> v14\n- v14 --supports--> v11\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0e\n- v07 --refutes--> v15\n- v07 --supports--> v09\n- v0a --supports--> v0b\n- v0d --refutes--> v0c\n- v0d --supports--> v02\n- v0d --supports--> v0f\n\n[VERIFIED ROOT]\nv08 = 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":"fd75aae725f4862d0757396b4bc8993b6671c79e12bcb082f3e724766c05de4d","schema":"livegraph.boolean-propagation.v1","spec_hash":"e6db2186671d88218981004d3333a0d29d857ceaeb4c0f6bd5ca95b1c87b6bbe","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-4d4e1a00c71915ff1116","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v13 --supports--> v02\n- v02 --refutes--> v0c\n- v0c --refutes--> v10\n- v10 --refutes--> v06\n- v06 --refutes--> v0a\n- v0a --supports--> v09\n- v09 --refutes--> v15\n- v15 --refutes--> v01\n- v01 --refutes--> v12\n- v12 --refutes--> v0b\n- v0b --refutes--> v08\n- v08 --refutes--> v0d\n- v0d --supports--> v00\n- v00 --supports--> v14\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v04\n- v08 --supports--> v03\n- v08 --supports--> v0f\n- v10 --supports--> v11\n- v12 --supports--> v05\n- v12 --supports--> v0e\n- v15 --supports--> v07\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":"ac43742424c1c2ce9576c99cfcc32886b873530d685b572f9ac7ad84c7a4e534","schema":"livegraph.boolean-propagation.v1","spec_hash":"4d4e1a00c71915ff1116eba36e6262e2adae205f8b5172728ab0f393ea60fac1","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-2eda53e379b1210d464a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0f --supports--> v05\n- v05 --supports--> v12\n- v12 --refutes--> v11\n- v11 --refutes--> v09\n- v09 --supports--> v0e\n- v0e --supports--> v13\n- v13 --refutes--> v00\n- v00 --refutes--> v14\n- v14 --refutes--> v0d\n- v0d --refutes--> v03\n- v03 --refutes--> v04\n- v04 --refutes--> v0a\n- v0a --supports--> v01\n- v01 --supports--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v15\n- v0a --refutes--> v06\n- v11 --supports--> v07\n- v12 --refutes--> v0c\n- v12 --supports--> v0b\n- v14 --supports--> v02\n- v14 --supports--> v10\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":"717deb4ea64fa93bb42afafe95eb12e5f274dc719e302791800941e2cb8d0bf7","schema":"livegraph.boolean-propagation.v1","spec_hash":"2eda53e379b1210d464a45eac292834f1d25ffa6d8711d2b7d6e2669f4a519a9","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-225c876cdaae269fd292","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v09\n- v09 --refutes--> v0a\n- v0a --refutes--> v05\n- v05 --refutes--> v03\n- v03 --refutes--> v14\n- v14 --refutes--> v0f\n- v0f --supports--> v06\n- v06 --refutes--> v10\n- v10 --supports--> v01\n- v01 --refutes--> v08\n- v08 --refutes--> v11\n- v11 --refutes--> v0d\n- v0d --supports--> v07\n- v07 --refutes--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v02\n- v04 --supports--> v15\n- v06 --refutes--> v12\n- v06 --supports--> v0e\n- v08 --refutes--> v0c\n- v11 --refutes--> v13\n- v14 --refutes--> v00\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":"668aa7bd198006922c55ad134d19f4396df7ebb24329e1a4d265d68c5ecdfcc8","schema":"livegraph.boolean-propagation.v1","spec_hash":"225c876cdaae269fd292b3508b15f4edc9b3a6be0ec0bfa4f4d35f28ad0f41d7","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-338993fbca66c0b4c7dc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --refutes--> v11\n- v11 --refutes--> v05\n- v05 --refutes--> v0e\n- v0e --refutes--> v02\n- v02 --refutes--> v04\n- v04 --refutes--> v06\n- v06 --refutes--> v0b\n- v0b --refutes--> v12\n- v12 --supports--> v00\n- v00 --supports--> v13\n- v13 --supports--> v01\n- v01 --refutes--> v0c\n- v0c --refutes--> v03\n- v03 --supports--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v14\n- v03 --supports--> v15\n- v05 --supports--> v07\n- v0c --supports--> v0f\n- v0e --supports--> v10\n- v11 --refutes--> v0d\n- v11 --supports--> v0a\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":"b8f6a8fdd7789d836f2f3097b3a96d095b8b32df3f57fd22ebf4a623e3db85b4","schema":"livegraph.boolean-propagation.v1","spec_hash":"338993fbca66c0b4c7dc4aa43a88ec0441b2c19b342eef9a9aae116d8225225d","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":14,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-69e916bb288f73ead4c9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0e --refutes--> v14\n- v14 --refutes--> v0f\n- v0f --refutes--> v02\n- v02 --refutes--> v08\n- v08 --supports--> v04\n- v04 --refutes--> v03\n- v03 --refutes--> v0d\n- v0d --supports--> v01\n- v01 --refutes--> v13\n- v13 --refutes--> v11\n- v11 --refutes--> v10\n- v10 --refutes--> v06\n- v06 --refutes--> v0a\n- v0a --supports--> v0c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v05\n- v06 --supports--> v00\n- v08 --supports--> v09\n- v0d --supports--> v15\n- v0e --supports--> v07\n- v10 --refutes--> v0b\n- v13 --supports--> v12\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":"53f6ea1b26206b6b2f9583e40c1dbe292f815041f119707b5454e81e79720bdc","schema":"livegraph.boolean-propagation.v1","spec_hash":"69e916bb288f73ead4c9d9aad5d3580cc4498db44550284ded8a72a325354e7d","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-d8d68532159912c14cc8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v15 --supports--> v14\n- v14 --supports--> v08\n- v08 --refutes--> v0e\n- v0e --supports--> v06\n- v06 --supports--> v0d\n- v0d --supports--> v01\n- v01 --refutes--> v12\n- v12 --supports--> v02\n- v02 --refutes--> v0f\n- v0f --refutes--> v03\n- v03 --supports--> v05\n- v05 --supports--> v10\n- v10 --refutes--> v00\n- v00 --supports--> v0c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v0a\n- v06 --supports--> v13\n- v08 --refutes--> v04\n- v10 --supports--> v0b\n- v14 --refutes--> v09\n- v14 --supports--> v07\n- v15 --refutes--> v11\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":"6b1c161ba707710f907dac9f80a8880fe9228c24be8af7e133e3e65b9bee9621","schema":"livegraph.boolean-propagation.v1","spec_hash":"d8d68532159912c14cc87a3fcb42cc323a27f87a697300fc3f3a94899a604add","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v15\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-bde271360ff641f1f074","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v05\n- v05 --supports--> v09\n- v09 --supports--> v0b\n- v0b --supports--> v13\n- v13 --supports--> v00\n- v00 --supports--> v01\n- v01 --refutes--> v06\n- v06 --supports--> v03\n- v03 --supports--> v10\n- v10 --supports--> v0e\n- v0e --refutes--> v0f\n- v0f --supports--> v0c\n- v0c --supports--> v15\n- v15 --refutes--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v0a\n- v05 --refutes--> v08\n- v06 --supports--> v14\n- v09 --refutes--> v12\n- v13 --refutes--> v11\n- v15 --supports--> v02\n- v15 --supports--> v0d\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":"e9343b5e407f60f7af40aabf764345f60fb0b476c03eab80604942a0aa283bfc","schema":"livegraph.boolean-propagation.v1","spec_hash":"bde271360ff641f1f07495a880e9c34688e571fe3be3b2fa17b69c51e88acb3f","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-a2cfd1b2f4490a12aeb1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0e --refutes--> v0c\n- v0c --refutes--> v11\n- v11 --refutes--> v0f\n- v0f --supports--> v00\n- v00 --refutes--> v03\n- v03 --supports--> v06\n- v06 --refutes--> v02\n- v02 --refutes--> v13\n- v13 --refutes--> v05\n- v05 --supports--> v14\n- v14 --refutes--> v10\n- v10 --supports--> v0b\n- v0b --supports--> v0a\n- v0a --refutes--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v15\n- v05 --refutes--> v09\n- v05 --supports--> v04\n- v06 --refutes--> v07\n- v06 --supports--> v0d\n- v11 --refutes--> v12\n- v13 --supports--> v08\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":"859a97bb3f83b4462ca5aa60583eee7234e1821006ab2b7244d19819f7c8404b","schema":"livegraph.boolean-propagation.v1","spec_hash":"a2cfd1b2f4490a12aeb1962089212ed736b5a8e75dc7c431216dd2e7b0d008bd","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-98533221851a5f13bfa5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v13 --supports--> v02\n- v02 --refutes--> v04\n- v04 --refutes--> v0e\n- v0e --refutes--> v11\n- v11 --refutes--> v09\n- v09 --supports--> v06\n- v06 --supports--> v07\n- v07 --supports--> v0a\n- v0a --supports--> v0c\n- v0c --refutes--> v0d\n- v0d --supports--> v14\n- v14 --supports--> v10\n- v10 --refutes--> v01\n- v01 --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v05\n- v02 --supports--> v12\n- v09 --refutes--> v15\n- v0a --refutes--> v03\n- v0d --supports--> v0f\n- v0e --refutes--> v08\n- v13 --supports--> v0b\n\n[VERIFIED ROOT]\nv13 = 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":"029f3fcabd72e996ed3612881656679c7d29a88229dc23f75f7c1898fd9d3a3c","schema":"livegraph.boolean-propagation.v1","spec_hash":"98533221851a5f13bfa53d464d57ea53865700c0b13fc4722bb28ceebf93e4ec","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v13\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-40359f5b567be51cfb7e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --refutes--> v0b\n- v0b --refutes--> v14\n- v14 --refutes--> v02\n- v02 --refutes--> v15\n- v15 --supports--> v01\n- v01 --supports--> v0c\n- v0c --refutes--> v11\n- v11 --refutes--> v10\n- v10 --supports--> v08\n- v08 --refutes--> v07\n- v07 --supports--> v0e\n- v0e --supports--> v00\n- v00 --refutes--> v09\n- v09 --refutes--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v04\n- v01 --supports--> v03\n- v07 --supports--> v13\n- v0c --refutes--> v0a\n- v10 --supports--> v12\n- v11 --refutes--> v0f\n- v15 --supports--> v06\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":"42eaa0c1e6e72e8874589ff54b01cfb478a4f3f3592117477180098ab4b47b1b","schema":"livegraph.boolean-propagation.v1","spec_hash":"40359f5b567be51cfb7e1105cca8ed5cce22abc2883b02080adeb28841cdf54a","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-7f56f64ac39581d02215","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v15\n- v15 --refutes--> v10\n- v10 --supports--> v01\n- v01 --supports--> v0a\n- v0a --supports--> v02\n- v02 --supports--> v0e\n- v0e --supports--> v00\n- v00 --refutes--> v0b\n- v0b --refutes--> v0d\n- v0d --refutes--> v0f\n- v0f --refutes--> v08\n- v08 --supports--> v11\n- v11 --supports--> v12\n- v12 --supports--> v14\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v03\n- v02 --supports--> v13\n- v0a --supports--> v07\n- v0a --supports--> v0c\n- v0d --refutes--> v06\n- v10 --refutes--> v05\n- v10 --supports--> v09\n\n[VERIFIED ROOT]\nv04 = 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":"4d9feab334a1f60d6f2c51a248a74f1bef9ff60c183863c7d3c48e035ec5e49f","schema":"livegraph.boolean-propagation.v1","spec_hash":"7f56f64ac39581d02215b1cc7e8e39621d482a1f5b8592b12d6d0f3f05dabdb0","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-2eef1c0cfcea07ea5187","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v12 --supports--> v03\n- v03 --supports--> v09\n- v09 --supports--> v0b\n- v0b --supports--> v0c\n- v0c --refutes--> v06\n- v06 --supports--> v0e\n- v0e --refutes--> v02\n- v02 --refutes--> v0f\n- v0f --supports--> v0d\n- v0d --supports--> v00\n- v00 --refutes--> v11\n- v11 --supports--> v10\n- v10 --supports--> v08\n- v08 --refutes--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v15\n- v06 --refutes--> v07\n- v0b --refutes--> v04\n- v0c --refutes--> v0a\n- v0d --refutes--> v13\n- v0d --supports--> v14\n- v0f --refutes--> v01\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":"31c3b09aab17fe2425514e493eed7f1177448bff81365ad8fe8d8d4d8c246816","schema":"livegraph.boolean-propagation.v1","spec_hash":"2eef1c0cfcea07ea5187832720b4441bc0cb71a0cac09f0181c8d7dc0d15449d","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":14,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-cb6ad51ef5787079729f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v11 --refutes--> v00\n- v00 --supports--> v0c\n- v0c --refutes--> v10\n- v10 --supports--> v15\n- v15 --supports--> v0b\n- v0b --refutes--> v08\n- v08 --refutes--> v06\n- v06 --refutes--> v02\n- v02 --refutes--> v14\n- v14 --refutes--> v0a\n- v0a --refutes--> v13\n- v13 --refutes--> v12\n- v12 --refutes--> v03\n- v03 --supports--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v07\n- v02 --supports--> v0d\n- v06 --supports--> v04\n- v0a --supports--> v05\n- v0c --refutes--> v09\n- v12 --refutes--> v0e\n- v14 --supports--> v0f\n\n[VERIFIED ROOT]\nv11 = 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":"30ecdb1f138f506d0e379e965a948a9845d0b9880f539f8cbab5e3b01ff800b6","schema":"livegraph.boolean-propagation.v1","spec_hash":"cb6ad51ef5787079729f19e56a4b5395a1d6b15fee6fdbddedfe4de469112157","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-aae81f4773e60c83a102","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --refutes--> v0b\n- v0b --supports--> v15\n- v15 --refutes--> v13\n- v13 --refutes--> v0f\n- v0f --supports--> v07\n- v07 --refutes--> v0d\n- v0d --refutes--> v0c\n- v0c --refutes--> v10\n- v10 --supports--> v12\n- v12 --supports--> v01\n- v01 --refutes--> v0e\n- v0e --supports--> v11\n- v11 --refutes--> v04\n- v04 --supports--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v05\n- v0b --refutes--> v06\n- v0d --refutes--> v14\n- v0d --supports--> v08\n- v0f --refutes--> v00\n- v0f --supports--> v09\n- v12 --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":"2e0cf21fd676b399d38cdc2e9d3d28c11161765813fec84fc2ac7e4a184bd908","schema":"livegraph.boolean-propagation.v1","spec_hash":"aae81f4773e60c83a10216052139db73449264f1d7b1db318437afd3d2006f7d","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-e08dea23c4def66f5243","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v04\n- v04 --refutes--> v07\n- v07 --refutes--> v00\n- v00 --supports--> v09\n- v09 --refutes--> v0f\n- v0f --supports--> v05\n- v05 --refutes--> v0d\n- v0d --refutes--> v02\n- v02 --refutes--> v10\n- v10 --refutes--> v08\n- v08 --refutes--> v03\n- v03 --refutes--> v0b\n- v0b --supports--> v01\n- v01 --supports--> v14\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --supports--> v0c\n- v07 --supports--> v15\n- v09 --refutes--> v12\n- v0b --refutes--> v0e\n- v0b --supports--> v13\n- v0f --supports--> v0a\n- v0f --supports--> v11\n\n[VERIFIED ROOT]\nv06 = 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":"73a4f78df4ffeecbafc468560edc1eaa07aa886916dcaaa46bd679ec01b50d1d","schema":"livegraph.boolean-propagation.v1","spec_hash":"e08dea23c4def66f5243b9083973c4a68e716ffb20ca006bcae256e6606c393b","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"}],\"hop\":14,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-6cf27b75e796100b4790","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --refutes--> v14\n- v14 --supports--> v01\n- v01 --refutes--> v0c\n- v0c --refutes--> v06\n- v06 --refutes--> v05\n- v05 --refutes--> v10\n- v10 --refutes--> v13\n- v13 --refutes--> v07\n- v07 --refutes--> v11\n- v11 --refutes--> v00\n- v00 --refutes--> v0d\n- v0d --refutes--> v0a\n- v0a --supports--> v09\n- v09 --supports--> v0e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0f\n- v01 --supports--> v15\n- v05 --supports--> v12\n- v0d --supports--> v0b\n- v10 --supports--> v03\n- v14 --refutes--> v04\n- v14 --supports--> v02\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":"6e7e975888478e7a4fbbd4a9e512bd49a14514d1bfa9f748c9692dae5b56f19e","schema":"livegraph.boolean-propagation.v1","spec_hash":"6cf27b75e796100b4790404a846f3351cce292963750536d6f98cbbf0d92c47d","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-b2d698a78244003aaa3b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v13 --refutes--> v01\n- v01 --supports--> v0e\n- v0e --supports--> v04\n- v04 --supports--> v11\n- v11 --supports--> v0f\n- v0f --refutes--> v09\n- v09 --refutes--> v0d\n- v0d --refutes--> v05\n- v05 --supports--> v12\n- v12 --refutes--> v0b\n- v0b --refutes--> v07\n- v07 --refutes--> v15\n- v15 --supports--> v0c\n- v0c --supports--> v14\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v09 --refutes--> v06\n- v09 --refutes--> v08\n- v09 --supports--> v10\n- v0b --refutes--> v03\n- v12 --refutes--> v02\n- v15 --refutes--> v00\n- v15 --supports--> v0a\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":"95e4ed85bb3732a57fbdc73a19d503ccd58b55d240499208d115997f8ab7fc38","schema":"livegraph.boolean-propagation.v1","spec_hash":"b2d698a78244003aaa3b658ab9276e4bf649e8b7dfc893e22ceee00d5ceadefe","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v13\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-d3fcc0740331b67bba12","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --supports--> v01\n- v01 --supports--> v0e\n- v0e --refutes--> v07\n- v07 --refutes--> v15\n- v15 --supports--> v14\n- v14 --supports--> v06\n- v06 --supports--> v04\n- v04 --refutes--> v02\n- v02 --refutes--> v10\n- v10 --supports--> v12\n- v12 --supports--> v00\n- v00 --refutes--> v13\n- v13 --supports--> v0f\n- v0f --supports--> v11\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v09\n- v01 --refutes--> v0a\n- v01 --supports--> v05\n- v02 --refutes--> v0b\n- v03 --refutes--> v08\n- v12 --refutes--> v0c\n- v12 --refutes--> v0d\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":"447d3b6e9f357d6616a8712ade09398022e00b2fbb2956239cd9c1578dfecf40","schema":"livegraph.boolean-propagation.v1","spec_hash":"d3fcc0740331b67bba12c4a51786a4d6daffe65c8202feb01e60b1973987d351","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-5c5862441faf1c940b7c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --refutes--> v05\n- v05 --refutes--> v0e\n- v0e --supports--> v0d\n- v0d --supports--> v15\n- v15 --refutes--> v02\n- v02 --refutes--> v10\n- v10 --supports--> v0a\n- v0a --refutes--> v09\n- v09 --supports--> v08\n- v08 --refutes--> v06\n- v06 --supports--> v00\n- v00 --refutes--> v0b\n- v0b --supports--> v04\n- v04 --supports--> v11\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v12\n- v00 --supports--> v03\n- v09 --refutes--> v0c\n- v09 --refutes--> v0f\n- v09 --refutes--> v13\n- v0e --refutes--> v14\n- v15 --supports--> v01\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":"95acfbe7a73db60caf7995b08213402e7ba23ece32e19dfc7a8f95f015aa1f58","schema":"livegraph.boolean-propagation.v1","spec_hash":"5c5862441faf1c940b7ca803e58fb5dae18a463be603f4b710d71f5925ca2ebe","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-e446d9b2a0dbf1a43e60","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v12 --supports--> v0e\n- v0e --refutes--> v10\n- v10 --refutes--> v08\n- v08 --refutes--> v09\n- v09 --supports--> v06\n- v06 --refutes--> v05\n- v05 --refutes--> v13\n- v13 --supports--> v01\n- v01 --supports--> v00\n- v00 --refutes--> v0b\n- v0b --refutes--> v0d\n- v0d --supports--> v03\n- v03 --refutes--> v0f\n- v0f --supports--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v04\n- v06 --supports--> v15\n- v0b --refutes--> v0c\n- v0f --refutes--> v07\n- v12 --refutes--> v02\n- v12 --supports--> v14\n- v13 --supports--> v11\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":"a4e21b976bcd8c588b8f4fcd688b1cc7d5cc217ab89ce79fb172155e509b695c","schema":"livegraph.boolean-propagation.v1","spec_hash":"e446d9b2a0dbf1a43e6013ae63f5d3a8a0176cf1ba1f4ba9b6ce06bd5f5c05d5","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":14,\"root\":\"v12\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-c9ad3f15fc2e7e128351","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --refutes--> v04\n- v04 --supports--> v15\n- v15 --refutes--> v0c\n- v0c --refutes--> v0a\n- v0a --supports--> v09\n- v09 --refutes--> v06\n- v06 --supports--> v08\n- v08 --refutes--> v00\n- v00 --supports--> v02\n- v02 --supports--> v11\n- v11 --refutes--> v14\n- v14 --supports--> v03\n- v03 --supports--> v0f\n- v0f --supports--> v10\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v05\n- v02 --supports--> v0e\n- v04 --supports--> v12\n- v0a --refutes--> v0b\n- v0f --supports--> v07\n- v11 --refutes--> v0d\n- v15 --supports--> v13\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":"6ca30bd922dd261e6487a1f23f0c7656c097e91eec90b0ad8a2e9443adf6a9eb","schema":"livegraph.boolean-propagation.v1","spec_hash":"c9ad3f15fc2e7e1283516e7cbe612167eca4f0e39c75bdbca1288dd291b36db4","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-8c08b60b4b4d4b0696de","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --supports--> v0f\n- v0f --supports--> v00\n- v00 --supports--> v06\n- v06 --supports--> v0d\n- v0d --refutes--> v01\n- v01 --supports--> v0e\n- v0e --refutes--> v11\n- v11 --supports--> v0b\n- v0b --refutes--> v14\n- v14 --supports--> v05\n- v05 --refutes--> v0a\n- v0a --supports--> v02\n- v02 --supports--> v15\n- v15 --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v0c\n- v02 --refutes--> v13\n- v06 --refutes--> v09\n- v11 --refutes--> v04\n- v11 --refutes--> v12\n- v14 --refutes--> v10\n- v15 --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":"c7eed4c5151f51a8715739f30adf2eddaeb45d8285574fe0051b8cfcf1b08971","schema":"livegraph.boolean-propagation.v1","spec_hash":"8c08b60b4b4d4b0696ded9973b1531ed771aced3734020aeb1e927b9a2385456","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-37a72645ab07209a1651","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0b --refutes--> v0d\n- v0d --refutes--> v0a\n- v0a --supports--> v0c\n- v0c --supports--> v10\n- v10 --supports--> v11\n- v11 --supports--> v03\n- v03 --refutes--> v13\n- v13 --refutes--> v05\n- v05 --supports--> v04\n- v04 --refutes--> v0e\n- v0e --supports--> v12\n- v12 --supports--> v14\n- v14 --supports--> v01\n- v01 --refutes--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v08\n- v01 --refutes--> v09\n- v04 --supports--> v0f\n- v05 --refutes--> v06\n- v05 --supports--> v15\n- v10 --refutes--> v00\n- v13 --refutes--> v07\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":"5fff69e09e247017c741d26bf79eee12349fc0635b63339e9afe8339d3becf19","schema":"livegraph.boolean-propagation.v1","spec_hash":"37a72645ab07209a16517fb04d09ebefa8a4de193a4be5fa773f6af8fa357ab2","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-9442bf0888c322730bf0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --supports--> v02\n- v02 --refutes--> v0d\n- v0d --refutes--> v0f\n- v0f --supports--> v03\n- v03 --refutes--> v14\n- v14 --supports--> v15\n- v15 --supports--> v01\n- v01 --refutes--> v07\n- v07 --refutes--> v0b\n- v0b --supports--> v12\n- v12 --supports--> v0c\n- v0c --refutes--> v11\n- v11 --supports--> v0e\n- v0e --supports--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v00\n- v03 --supports--> v13\n- v08 --refutes--> v06\n- v08 --supports--> v04\n- v0b --refutes--> v10\n- v0c --refutes--> v05\n- v0d --refutes--> v09\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":"8dce42720577fbeb5febf8ffa61b0ecf7fed6c70ce3a6bedce75fc876af3ef85","schema":"livegraph.boolean-propagation.v1","spec_hash":"9442bf0888c322730bf0148b1e95f79ea23a7ce7b56eb11350e6cfb593a770b3","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-0776cc0f976f254a63a7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --supports--> v07\n- v07 --supports--> v0f\n- v0f --supports--> v11\n- v11 --supports--> v02\n- v02 --supports--> v12\n- v12 --supports--> v05\n- v05 --refutes--> v04\n- v04 --supports--> v09\n- v09 --supports--> v00\n- v00 --supports--> v0c\n- v0c --refutes--> v0d\n- v0d --supports--> v15\n- v15 --supports--> v01\n- v01 --refutes--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v13\n- v02 --refutes--> v10\n- v03 --refutes--> v0e\n- v09 --supports--> v0a\n- v0c --supports--> v0b\n- v11 --refutes--> v06\n- v15 --supports--> v14\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":"348e1fbd94e2cdcd76c2998acbccaf2a68af69ba8b9660696d349500329f821b","schema":"livegraph.boolean-propagation.v1","spec_hash":"0776cc0f976f254a63a70387e5c8c7519d740bd408d6fbcf1cd700620e46e084","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-2755d34cc1b9e1da5234","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v10 --refutes--> v03\n- v03 --supports--> v01\n- v01 --refutes--> v06\n- v06 --refutes--> v09\n- v09 --supports--> v0c\n- v0c --refutes--> v00\n- v00 --refutes--> v0a\n- v0a --refutes--> v08\n- v08 --refutes--> v11\n- v11 --supports--> v04\n- v04 --refutes--> v07\n- v07 --refutes--> v0b\n- v0b --supports--> v13\n- v13 --refutes--> v14\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v0e\n- v03 --refutes--> v0d\n- v06 --supports--> v15\n- v08 --refutes--> v05\n- v08 --supports--> v0f\n- v0b --refutes--> v02\n- v13 --supports--> v12\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":"e4dc68def2917966982b9781e9346a8c5ce44e49aeef123c7232880f41de0a9c","schema":"livegraph.boolean-propagation.v1","spec_hash":"2755d34cc1b9e1da5234fa7124bee846588120510ac6f9c483ebb3e2c529f42d","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":14,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-fa2d30873edab3a6cdfa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --supports--> v0d\n- v0d --supports--> v0b\n- v0b --refutes--> v06\n- v06 --supports--> v12\n- v12 --supports--> v0a\n- v0a --refutes--> v0e\n- v0e --refutes--> v11\n- v11 --supports--> v09\n- v09 --refutes--> v02\n- v02 --supports--> v0c\n- v0c --supports--> v10\n- v10 --supports--> v04\n- v04 --refutes--> v03\n- v03 --supports--> v15\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v05\n- v03 --supports--> v01\n- v08 --refutes--> v07\n- v0b --supports--> v13\n- v0d --refutes--> v14\n- v0e --supports--> v00\n- v12 --supports--> v0f\n\n[VERIFIED ROOT]\nv08 = 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":"46a58c055216ae9029769e17816e38d5cf0e2ca38d7105a3191a15ad6134efca","schema":"livegraph.boolean-propagation.v1","spec_hash":"fa2d30873edab3a6cdfabd4018720f2e51b9ee5b1373bc28c780100cef52cc64","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"}],\"hop\":14,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-fcca562a63a56fb3f6af","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --refutes--> v06\n- v06 --supports--> v03\n- v03 --supports--> v15\n- v15 --supports--> v11\n- v11 --supports--> v05\n- v05 --refutes--> v14\n- v14 --supports--> v13\n- v13 --refutes--> v0c\n- v0c --supports--> v00\n- v00 --refutes--> v01\n- v01 --supports--> v02\n- v02 --supports--> v07\n- v07 --supports--> v12\n- v12 --supports--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v10\n- v05 --supports--> v04\n- v06 --supports--> v0e\n- v12 --refutes--> v08\n- v13 --refutes--> v09\n- v13 --refutes--> v0d\n- v15 --refutes--> v0f\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":"6291d57c118cdd222f7e0b41a3cebbc361ef6e0191c552b7e096784c669507cc","schema":"livegraph.boolean-propagation.v1","spec_hash":"fcca562a63a56fb3f6afa049b498db644c434ed03fdc06e8a809fb6a3a1eb607","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-a23c7495987219e79f89","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v10 --refutes--> v02\n- v02 --refutes--> v14\n- v14 --refutes--> v13\n- v13 --refutes--> v0e\n- v0e --supports--> v00\n- v00 --refutes--> v0b\n- v0b --supports--> v08\n- v08 --refutes--> v05\n- v05 --supports--> v0d\n- v0d --supports--> v03\n- v03 --supports--> v07\n- v07 --refutes--> v12\n- v12 --supports--> v11\n- v11 --supports--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v09\n- v03 --refutes--> v0a\n- v03 --refutes--> v0f\n- v05 --supports--> v01\n- v07 --supports--> v15\n- v13 --supports--> v04\n- v14 --refutes--> v0c\n\n[VERIFIED ROOT]\nv10 = 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":"56f404e7ee017fc3b706fa817a4a132cd1eb00f9b8049240c90de5d2d013ad34","schema":"livegraph.boolean-propagation.v1","spec_hash":"a23c7495987219e79f89f7f1ad8903bdc8af8ff94ba45872c47cfdc678e7d90d","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-61f66930374a124b3531","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v11 --refutes--> v12\n- v12 --refutes--> v13\n- v13 --supports--> v09\n- v09 --supports--> v01\n- v01 --supports--> v06\n- v06 --refutes--> v15\n- v15 --refutes--> v0f\n- v0f --refutes--> v03\n- v03 --supports--> v10\n- v10 --supports--> v14\n- v14 --refutes--> v05\n- v05 --supports--> v00\n- v00 --supports--> v0a\n- v0a --supports--> v0e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v07\n- v05 --supports--> v0c\n- v05 --supports--> v0d\n- v0f --supports--> v0b\n- v11 --refutes--> v02\n- v12 --refutes--> v08\n- v13 --supports--> v04\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":"d1a3383d61239e1e7c790452add34787ad8adc46d09e312876210c4e76a234eb","schema":"livegraph.boolean-propagation.v1","spec_hash":"61f66930374a124b3531e1c808e3b44eca5a54d21de16d73f1fef4706158cacf","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-e9836c576310cf27b37c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0b --refutes--> v00\n- v00 --refutes--> v01\n- v01 --refutes--> v0c\n- v0c --supports--> v0e\n- v0e --refutes--> v08\n- v08 --refutes--> v11\n- v11 --supports--> v05\n- v05 --refutes--> v12\n- v12 --refutes--> v07\n- v07 --refutes--> v15\n- v15 --supports--> v02\n- v02 --supports--> v06\n- v06 --supports--> v0a\n- v0a --refutes--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v14\n- v02 --refutes--> v09\n- v05 --refutes--> v04\n- v08 --supports--> v03\n- v0c --supports--> v13\n- v11 --refutes--> v10\n- v12 --supports--> v0d\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":"189f90233033799f7735f7c797425b263e4643c5796fc9617bf2e147bfebb1ac","schema":"livegraph.boolean-propagation.v1","spec_hash":"e9836c576310cf27b37caebfa27fd196fcd3f193bd6f835d98a9c82a8fa72b82","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-5debd8e97a22493d769a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0f --refutes--> v0b\n- v0b --refutes--> v06\n- v06 --refutes--> v11\n- v11 --supports--> v0c\n- v0c --supports--> v00\n- v00 --refutes--> v01\n- v01 --refutes--> v0d\n- v0d --supports--> v0a\n- v0a --refutes--> v02\n- v02 --refutes--> v10\n- v10 --refutes--> v07\n- v07 --supports--> v13\n- v13 --refutes--> v05\n- v05 --refutes--> v12\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v08\n- v06 --refutes--> v0e\n- v06 --refutes--> v15\n- v0b --supports--> v04\n- v0f --supports--> v09\n- v11 --refutes--> v03\n- v13 --supports--> v14\n\n[VERIFIED ROOT]\nv0f = 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":"0209963115fed42f1b148d1433bbad73cf342b86fcf08afb4d2740627ee72e04","schema":"livegraph.boolean-propagation.v1","spec_hash":"5debd8e97a22493d769a22dd1b3f4e3a52ddce6873c28c044a0dfc9ccc2ecd07","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v13\"}],\"hop\":14,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-81460b6364262c56e9be","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0e --refutes--> v0a\n- v0a --refutes--> v11\n- v11 --supports--> v14\n- v14 --supports--> v00\n- v00 --supports--> v0d\n- v0d --refutes--> v0b\n- v0b --refutes--> v01\n- v01 --refutes--> v02\n- v02 --supports--> v09\n- v09 --supports--> v05\n- v05 --refutes--> v07\n- v07 --refutes--> v12\n- v12 --refutes--> v13\n- v13 --refutes--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v06\n- v05 --supports--> v15\n- v09 --refutes--> v10\n- v0a --supports--> v08\n- v12 --refutes--> v03\n- v13 --refutes--> v04\n- v13 --supports--> v0c\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":"da6609b33c8d97af6ba87a203fea15a4a6f14484be553d87a5e5e8a9ab47fff6","schema":"livegraph.boolean-propagation.v1","spec_hash":"81460b6364262c56e9be7e5cb2df864bc17b66f38a1ad90e113925125e67b8f5","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-f10dc390e4ae55d8e83e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --refutes--> v12\n- v12 --refutes--> v07\n- v07 --refutes--> v06\n- v06 --supports--> v00\n- v00 --refutes--> v11\n- v11 --refutes--> v0b\n- v0b --refutes--> v03\n- v03 --supports--> v0d\n- v0d --supports--> v05\n- v05 --refutes--> v0f\n- v0f --supports--> v14\n- v14 --refutes--> v15\n- v15 --refutes--> v13\n- v13 --supports--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v01\n- v06 --supports--> v0a\n- v0c --refutes--> v09\n- v11 --refutes--> v04\n- v12 --refutes--> v08\n- v13 --refutes--> v0e\n- v15 --supports--> v10\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":"10dc6431b42f09ab73ca1a045c138dc3b0dab993b1d41841071a1f205ecbad5e","schema":"livegraph.boolean-propagation.v1","spec_hash":"f10dc390e4ae55d8e83e5da580296369947e6686fc53812c41aad26f11b783d3","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-241e5f3623aa860a9ba4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --supports--> v06\n- v06 --supports--> v05\n- v05 --refutes--> v0d\n- v0d --refutes--> v14\n- v14 --refutes--> v15\n- v15 --refutes--> v07\n- v07 --refutes--> v0b\n- v0b --supports--> v13\n- v13 --refutes--> v12\n- v12 --supports--> v10\n- v10 --refutes--> v04\n- v04 --supports--> v02\n- v02 --supports--> v0c\n- v0c --refutes--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v00\n- v0b --supports--> v01\n- v0c --supports--> v0e\n- v12 --refutes--> v0f\n- v14 --refutes--> v09\n- v14 --refutes--> v11\n- v15 --supports--> v03\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":"728a9217b6179fea327e25f823dfcc39785a54a8bd9c70481349f64abe2b1443","schema":"livegraph.boolean-propagation.v1","spec_hash":"241e5f3623aa860a9ba43c0222bb996ac09fb05ef708d1ea67d28c74600686a3","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-504b7e2de951edbae191","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --supports--> v0d\n- v0d --supports--> v14\n- v14 --refutes--> v08\n- v08 --supports--> v13\n- v13 --supports--> v07\n- v07 --supports--> v0e\n- v0e --refutes--> v01\n- v01 --supports--> v05\n- v05 --supports--> v11\n- v11 --refutes--> v02\n- v02 --supports--> v00\n- v00 --supports--> v0f\n- v0f --refutes--> v0c\n- v0c --refutes--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v0a\n- v01 --supports--> v10\n- v03 --refutes--> v0b\n- v0c --refutes--> v12\n- v11 --refutes--> v15\n- v13 --refutes--> v04\n- v13 --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":"7fb4c4d80136480877610126cdc276199c58dd5f3c69169087003387ebcfdc3a","schema":"livegraph.boolean-propagation.v1","spec_hash":"504b7e2de951edbae1912f24bf41ad486c850d07563f57fe3cd5727da01f4913","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-68e4aa8dee50315ce248","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --refutes--> v14\n- v14 --supports--> v0c\n- v0c --refutes--> v12\n- v12 --refutes--> v04\n- v04 --refutes--> v08\n- v08 --refutes--> v11\n- v11 --supports--> v0a\n- v0a --refutes--> v02\n- v02 --refutes--> v0f\n- v0f --refutes--> v07\n- v07 --supports--> v06\n- v06 --supports--> v01\n- v01 --supports--> v0e\n- v0e --supports--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v10\n- v06 --supports--> v00\n- v0c --refutes--> v0d\n- v0c --supports--> v13\n- v0f --refutes--> v09\n- v0f --supports--> v15\n- v11 --refutes--> 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":"08f20a22f9bdcd669b87675c4f64dab9b0a28fd280c558b2024755516c2fb1a5","schema":"livegraph.boolean-propagation.v1","spec_hash":"68e4aa8dee50315ce24896b0a0027624fc642b31376cea662dafb70bdaaa6154","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-76235d4e715adc79342e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v10 --supports--> v02\n- v02 --supports--> v15\n- v15 --supports--> v09\n- v09 --supports--> v06\n- v06 --supports--> v12\n- v12 --refutes--> v13\n- v13 --supports--> v0d\n- v0d --supports--> v14\n- v14 --refutes--> v04\n- v04 --supports--> v0e\n- v0e --supports--> v05\n- v05 --refutes--> v0a\n- v0a --refutes--> v07\n- v07 --supports--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v00\n- v09 --refutes--> v03\n- v0e --supports--> v01\n- v10 --refutes--> v11\n- v10 --supports--> v0c\n- v15 --refutes--> v08\n- v15 --refutes--> v0b\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":"ebf1d0599680cc6c9a14faf1808c53a797978a6162307e2f35b16476e9bbca58","schema":"livegraph.boolean-propagation.v1","spec_hash":"76235d4e715adc79342efd3d9ad41da59b27174a8a726cca9ea19c006f724ce8","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-37b507b0a15de228dc77","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --refutes--> v0a\n- v0a --refutes--> v03\n- v03 --supports--> v0e\n- v0e --supports--> v05\n- v05 --supports--> v04\n- v04 --refutes--> v11\n- v11 --supports--> v02\n- v02 --refutes--> v08\n- v08 --supports--> v06\n- v06 --supports--> v0c\n- v0c --refutes--> v0b\n- v0b --refutes--> v10\n- v10 --refutes--> v0f\n- v0f --refutes--> v13\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v09\n- v02 --supports--> v15\n- v07 --supports--> v12\n- v0a --refutes--> v0d\n- v0a --supports--> v01\n- v0c --refutes--> v00\n- v0e --supports--> v14\n\n[VERIFIED ROOT]\nv07 = 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":"156bc427972dfa6e0408595f3ef570e8a9ed2aa6c4257ae9b45a95333a41d744","schema":"livegraph.boolean-propagation.v1","spec_hash":"37b507b0a15de228dc77b106f2392feb4b81393e8d4cdc5bbe3e01abe725154f","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v11\"}],\"hop\":14,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-0aa101ba4a803aced5af","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --supports--> v07\n- v07 --refutes--> v08\n- v08 --refutes--> v14\n- v14 --refutes--> v06\n- v06 --refutes--> v0b\n- v0b --supports--> v0e\n- v0e --supports--> v02\n- v02 --supports--> v05\n- v05 --refutes--> v12\n- v12 --refutes--> v0a\n- v0a --supports--> v00\n- v00 --refutes--> v04\n- v04 --refutes--> v0d\n- v0d --supports--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0c\n- v00 --supports--> v10\n- v08 --supports--> v01\n- v0b --supports--> v13\n- v0d --refutes--> v11\n- v14 --supports--> v03\n- v14 --supports--> v15\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":"4adea4fdfdf185d72c93383bd801bd1712f256ed97b0b08581ec682dede4f680","schema":"livegraph.boolean-propagation.v1","spec_hash":"0aa101ba4a803aced5afe1c2406427b2ab5d4fed6ff14193f0684f6846d0ee5c","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-a9b9e45227fe0cf1ee3e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0f --supports--> v0c\n- v0c --supports--> v07\n- v07 --refutes--> v0d\n- v0d --refutes--> v02\n- v02 --supports--> v06\n- v06 --refutes--> v03\n- v03 --refutes--> v04\n- v04 --supports--> v0e\n- v0e --supports--> v15\n- v15 --supports--> v12\n- v12 --supports--> v11\n- v11 --supports--> v01\n- v01 --supports--> v00\n- v00 --supports--> v13\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v08\n- v03 --supports--> v0b\n- v06 --refutes--> v10\n- v0c --refutes--> v05\n- v0d --refutes--> v0a\n- v12 --refutes--> v14\n- v15 --supports--> v09\n\n[VERIFIED ROOT]\nv0f = 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":"1a5169b964cdba54e3247b3dd6d37243bfef8fb81931ee68362b12091bc3fafc","schema":"livegraph.boolean-propagation.v1","spec_hash":"a9b9e45227fe0cf1ee3e5d1dd7787742e88f2c244df34ced8eb54607b2097b51","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-591a8a31cd13d2ca5efe","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v15 --refutes--> v07\n- v07 --refutes--> v03\n- v03 --supports--> v06\n- v06 --refutes--> v00\n- v00 --supports--> v11\n- v11 --refutes--> v0f\n- v0f --supports--> v0c\n- v0c --refutes--> v14\n- v14 --refutes--> v05\n- v05 --supports--> v10\n- v10 --refutes--> v01\n- v01 --supports--> v0e\n- v0e --supports--> v13\n- v13 --refutes--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v08\n- v06 --supports--> v02\n- v0c --supports--> v0d\n- v0f --refutes--> v04\n- v0f --refutes--> v0a\n- v14 --refutes--> v12\n- v15 --refutes--> v09\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":"c3ca7bec1c88d488ea13f1dd5d2f357d2d729d2e5e624b0c203a04fea52cc1fa","schema":"livegraph.boolean-propagation.v1","spec_hash":"591a8a31cd13d2ca5efe16c43a9643ffd463c5854f44d7eb39f55a5a7b56dfbe","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v15\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-fca94add9904d80b0ccd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --refutes--> v0a\n- v0a --supports--> v0f\n- v0f --supports--> v0c\n- v0c --refutes--> v0e\n- v0e --refutes--> v15\n- v15 --supports--> v05\n- v05 --supports--> v10\n- v10 --supports--> v0d\n- v0d --supports--> v06\n- v06 --supports--> v02\n- v02 --supports--> v08\n- v08 --refutes--> v09\n- v09 --refutes--> v11\n- v11 --supports--> v12\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v13\n- v08 --supports--> v04\n- v09 --supports--> v00\n- v09 --supports--> v01\n- v09 --supports--> v14\n- v0d --refutes--> v0b\n- v10 --refutes--> v07\n\n[VERIFIED ROOT]\nv03 = 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":"e3549a1a98c7be55af8802f2136bda5aa27f3c18ddef8f4682718aedeb5b7eac","schema":"livegraph.boolean-propagation.v1","spec_hash":"fca94add9904d80b0ccd9fd8062960b120bd6bd9be1b8b700a88ddfcb5be2d4b","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-1d2fbdd45ec1c0e74ce7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --refutes--> v0e\n- v0e --supports--> v15\n- v15 --supports--> v14\n- v14 --supports--> v01\n- v01 --refutes--> v05\n- v05 --refutes--> v08\n- v08 --refutes--> v09\n- v09 --refutes--> v11\n- v11 --supports--> v0c\n- v0c --supports--> v0a\n- v0a --refutes--> v06\n- v06 --refutes--> v0b\n- v0b --supports--> v13\n- v13 --supports--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v10\n- v08 --refutes--> v07\n- v08 --supports--> v00\n- v0c --refutes--> v04\n- v13 --supports--> v0d\n- v14 --refutes--> v0f\n- v14 --refutes--> v12\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":"dfe7cf94920eec34d0a5ddde625e8fdeef47ed3bd5b462fc9947ecd2132eb9f4","schema":"livegraph.boolean-propagation.v1","spec_hash":"1d2fbdd45ec1c0e74ce7d3b98191c3887cf412f418b71eb91034947202f1d1a3","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-2dd8fc83a4870ad91966","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --supports--> v02\n- v02 --refutes--> v06\n- v06 --supports--> v07\n- v07 --supports--> v14\n- v14 --refutes--> v04\n- v04 --supports--> v0e\n- v0e --supports--> v0b\n- v0b --refutes--> v0c\n- v0c --supports--> v12\n- v12 --supports--> v11\n- v11 --refutes--> v08\n- v08 --supports--> v15\n- v15 --supports--> v13\n- v13 --supports--> v10\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v03\n- v0b --supports--> v0f\n- v0c --supports--> v0d\n- v0e --supports--> v00\n- v11 --refutes--> v09\n- v12 --refutes--> v0a\n- v12 --supports--> v01\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":"81fa7f0c5387dc7733ab0df67172a4eb69903d2015088a68342671eaaf65c9b6","schema":"livegraph.boolean-propagation.v1","spec_hash":"2dd8fc83a4870ad919660e5dd48ebe7df3c50429b4161a91af8f21452cfbb3c7","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-4fdbfc39dab0ad0c3f9b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v12 --supports--> v04\n- v04 --refutes--> v14\n- v14 --supports--> v07\n- v07 --supports--> v09\n- v09 --refutes--> v0e\n- v0e --refutes--> v00\n- v00 --refutes--> v0f\n- v0f --refutes--> v0c\n- v0c --refutes--> v01\n- v01 --refutes--> v0b\n- v0b --supports--> v11\n- v11 --supports--> v05\n- v05 --supports--> v02\n- v02 --refutes--> v13\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v06\n- v05 --refutes--> v10\n- v07 --refutes--> v03\n- v07 --supports--> v08\n- v09 --refutes--> v0a\n- v0c --refutes--> v15\n- v12 --supports--> v0d\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":"92d4aa1746904fb6f2b17214e336ce45942dd63386a2443f057f136069ffb027","schema":"livegraph.boolean-propagation.v1","spec_hash":"4fdbfc39dab0ad0c3f9bcca097b14a230cadbec0443d4c8d0baa55c0731f4343","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v12\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-1a124459b9a132b6ed47","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v0a\n- v0a --refutes--> v03\n- v03 --refutes--> v10\n- v10 --refutes--> v11\n- v11 --refutes--> v01\n- v01 --supports--> v04\n- v04 --supports--> v08\n- v08 --refutes--> v12\n- v12 --supports--> v14\n- v14 --supports--> v07\n- v07 --supports--> v00\n- v00 --refutes--> v0c\n- v0c --supports--> v15\n- v15 --supports--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v0b\n- v04 --supports--> v0e\n- v06 --supports--> v09\n- v07 --supports--> v13\n- v0a --refutes--> v0d\n- v10 --refutes--> v05\n- v14 --refutes--> v02\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":"5aefb2141c110f8ca8671b6fe9a0209942a52458e4c78b54315ce4681a07ac80","schema":"livegraph.boolean-propagation.v1","spec_hash":"1a124459b9a132b6ed475b2354dff76dc7349c467804443865b6383d7adc667f","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-686fe19d383e745b33df","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v15 --supports--> v13\n- v13 --refutes--> v05\n- v05 --refutes--> v03\n- v03 --refutes--> v10\n- v10 --supports--> v01\n- v01 --refutes--> v02\n- v02 --refutes--> v06\n- v06 --refutes--> v12\n- v12 --supports--> v0a\n- v0a --refutes--> v07\n- v07 --supports--> v0c\n- v0c --refutes--> v11\n- v11 --refutes--> v04\n- v04 --refutes--> v14\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v09\n- v07 --refutes--> v00\n- v0c --refutes--> v08\n- v11 --refutes--> v0b\n- v12 --refutes--> v0f\n- v13 --refutes--> v0d\n- v13 --refutes--> v0e\n\n[VERIFIED ROOT]\nv15 = 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":"3e58f36d36ff00f4bcd6da58858579831b68bcfa35325eca9a8058712519bffe","schema":"livegraph.boolean-propagation.v1","spec_hash":"686fe19d383e745b33df4f3ae463b7de017555cbcb01db0660c7377a848afba9","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v15\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-2d0e480afdc22d84f7fe","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --supports--> v0d\n- v0d --refutes--> v02\n- v02 --refutes--> v01\n- v01 --refutes--> v06\n- v06 --refutes--> v0c\n- v0c --supports--> v10\n- v10 --supports--> v05\n- v05 --refutes--> v07\n- v07 --refutes--> v14\n- v14 --refutes--> v0f\n- v0f --supports--> v03\n- v03 --supports--> v12\n- v12 --refutes--> v0b\n- v0b --refutes--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v0e\n- v03 --refutes--> v13\n- v05 --refutes--> v11\n- v07 --refutes--> v15\n- v09 --refutes--> v04\n- v09 --supports--> v08\n- v10 --refutes--> v00\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":"4daec25574a6fbc35a548506509f583580f386d388b268cd1e25b1592cb93934","schema":"livegraph.boolean-propagation.v1","spec_hash":"2d0e480afdc22d84f7fe9952e7a8e5ab27fd49eb7220b2580e3bb2ae1f92aa65","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":14,"item_id":"lggrpo1-2d5d17832d148157b3a0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v13 --refutes--> v05\n- v05 --refutes--> v14\n- v14 --supports--> v0c\n- v0c --supports--> v04\n- v04 --supports--> v15\n- v15 --refutes--> v10\n- v10 --refutes--> v0f\n- v0f --refutes--> v01\n- v01 --supports--> v0a\n- v0a --supports--> v00\n- v00 --supports--> v09\n- v09 --refutes--> v03\n- v03 --supports--> v08\n- v08 --refutes--> v0e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v0d\n- v01 --refutes--> v11\n- v01 --supports--> v06\n- v04 --supports--> v12\n- v0f --refutes--> v02\n- v0f --supports--> v07\n- v0f --supports--> v0b\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":"818ade73b8cc63322dfc459f9dc3019c08510a5ba2e1a4fb857a2111c7060da8","schema":"livegraph.boolean-propagation.v1","spec_hash":"2d5d17832d148157b3a0b2c799478b007a229b627633add8ced2a00d773423e7","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":14,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":14,"item_id":"lggrpo1-d146304ec7fa329d10a1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --supports--> v0a\n- v0a --supports--> v0c\n- v0c --refutes--> v0f\n- v0f --refutes--> v05\n- v05 --supports--> v08\n- v08 --refutes--> v03\n- v03 --refutes--> v11\n- v11 --refutes--> v10\n- v10 --supports--> v09\n- v09 --supports--> v01\n- v01 --refutes--> v14\n- v14 --refutes--> v0e\n- v0e --refutes--> v06\n- v06 --supports--> v13\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v15\n- v03 --refutes--> v0d\n- v08 --supports--> v00\n- v0a --refutes--> v0b\n- v0c --supports--> v07\n- v0e --refutes--> v02\n- v0e --refutes--> v12\n\n[VERIFIED ROOT]\nv04 = 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":"43bcfead73cdc847d8f2735bb488a70e6102523ab3df6f4aa14778b12a045069","schema":"livegraph.boolean-propagation.v1","spec_hash":"d146304ec7fa329d10a1944369dcd5dbb1be046eaf982e47f7e393bb4e8a168d","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":14,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-bfe8ead9391c133bed3a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --refutes--> v0c\n- v0c --refutes--> v08\n- v08 --supports--> v14\n- v14 --refutes--> v0a\n- v0a --refutes--> v12\n- v12 --supports--> v0b\n- v0b --refutes--> v11\n- v11 --refutes--> v13\n- v13 --supports--> v15\n- v15 --supports--> v02\n- v02 --refutes--> v10\n- v10 --refutes--> v05\n- v05 --supports--> v0e\n- v0e --refutes--> v07\n- v07 --refutes--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v06\n- v02 --supports--> v03\n- v02 --supports--> v04\n- v07 --supports--> v0d\n- v0a --supports--> v00\n- v12 --refutes--> v16\n- v15 --refutes--> v09\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":"979b67ac7cf40761f7c2a8ac3206b30def609ec2fd5cd7a4cb5cea9340d6df52","schema":"livegraph.boolean-propagation.v1","spec_hash":"bfe8ead9391c133bed3a4a4b49ac88ad7da6802087a7a3e5733580fd9824fb44","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-242838061cb9a27274e3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --supports--> v0e\n- v0e --supports--> v14\n- v14 --refutes--> v12\n- v12 --refutes--> v01\n- v01 --refutes--> v0b\n- v0b --refutes--> v0c\n- v0c --supports--> v0a\n- v0a --refutes--> v13\n- v13 --refutes--> v00\n- v00 --supports--> v10\n- v10 --supports--> v02\n- v02 --refutes--> v03\n- v03 --supports--> v08\n- v08 --supports--> v16\n- v16 --refutes--> v11\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v04\n- v08 --refutes--> v05\n- v0c --refutes--> v15\n- v10 --refutes--> v0d\n- v12 --supports--> v06\n- v13 --refutes--> v07\n- v16 --supports--> v0f\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":"1c655df1012549d551979e243c691fa4aec20d8aca997c7fb3d485170617d57a","schema":"livegraph.boolean-propagation.v1","spec_hash":"242838061cb9a27274e312746ce965522e96708044d1bdcd25741bf12931ae19","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-db7165645a5b2410a900","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v13\n- v13 --refutes--> v0d\n- v0d --refutes--> v02\n- v02 --supports--> v07\n- v07 --refutes--> v09\n- v09 --refutes--> v14\n- v14 --supports--> v0f\n- v0f --supports--> v12\n- v12 --supports--> v01\n- v01 --supports--> v15\n- v15 --refutes--> v0c\n- v0c --refutes--> v0a\n- v0a --supports--> v06\n- v06 --refutes--> v08\n- v08 --supports--> v10\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v07 --refutes--> v16\n- v0a --supports--> v03\n- v0f --supports--> v00\n- v0f --supports--> v05\n- v12 --supports--> v0e\n- v14 --refutes--> v11\n- v14 --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":"5b0c4d1b336413af2dc1af1cdcf88fc28091c2ac3cf5b12646d87bee7fe62cea","schema":"livegraph.boolean-propagation.v1","spec_hash":"db7165645a5b2410a900f2ac7603ae110390b6dfb3d3028d99ae202af3e811e8","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-e8d75e388b46eeb20873","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v16 --refutes--> v0b\n- v0b --supports--> v02\n- v02 --refutes--> v14\n- v14 --refutes--> v0f\n- v0f --refutes--> v10\n- v10 --supports--> v15\n- v15 --supports--> v08\n- v08 --supports--> v13\n- v13 --supports--> v07\n- v07 --refutes--> v05\n- v05 --supports--> v06\n- v06 --supports--> v01\n- v01 --supports--> v0c\n- v0c --refutes--> v09\n- v09 --refutes--> v12\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v0e\n- v06 --supports--> v11\n- v07 --refutes--> v0a\n- v10 --refutes--> v00\n- v13 --refutes--> v04\n- v14 --supports--> v0d\n- v16 --refutes--> v03\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":"82ca0a243c32a643801794325154731ce04d14bca4f5da76cc98f271f29c9906","schema":"livegraph.boolean-propagation.v1","spec_hash":"e8d75e388b46eeb20873917978f9e54956f433467208279a5694065ed1bac501","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v16\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-ce557799a34f53a6695e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v13 --supports--> v16\n- v16 --refutes--> v00\n- v00 --refutes--> v0e\n- v0e --supports--> v0a\n- v0a --refutes--> v04\n- v04 --refutes--> v0d\n- v0d --supports--> v09\n- v09 --refutes--> v0b\n- v0b --refutes--> v15\n- v15 --supports--> v07\n- v07 --supports--> v02\n- v02 --refutes--> v11\n- v11 --refutes--> v14\n- v14 --supports--> v03\n- v03 --refutes--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v0f\n- v03 --refutes--> v06\n- v03 --refutes--> v0c\n- v07 --refutes--> v12\n- v0e --supports--> v08\n- v13 --refutes--> v01\n- v14 --refutes--> v10\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":"4fafa11d6764befcf9b124b33f8453ae102dc6a31915ae38dd4c89da891e5f72","schema":"livegraph.boolean-propagation.v1","spec_hash":"ce557799a34f53a6695e47efb83bdedc0438b89df9d1ab374088f07b608179a7","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-2145e3c8e958b32fdce7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --supports--> v06\n- v06 --supports--> v09\n- v09 --supports--> v08\n- v08 --refutes--> v04\n- v04 --supports--> v13\n- v13 --refutes--> v16\n- v16 --refutes--> v14\n- v14 --refutes--> v0b\n- v0b --supports--> v11\n- v11 --supports--> v0c\n- v0c --refutes--> v12\n- v12 --supports--> v07\n- v07 --refutes--> v00\n- v00 --supports--> v0a\n- v0a --refutes--> v10\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v0f\n- v05 --supports--> v03\n- v07 --refutes--> v15\n- v07 --supports--> v0d\n- v0a --refutes--> v01\n- v11 --supports--> v02\n- v16 --supports--> v0e\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":"3b5ec056acf2be347be20f57fc040d513e16beeff83442629cbe1338ec82a477","schema":"livegraph.boolean-propagation.v1","spec_hash":"2145e3c8e958b32fdce703fadf98544d2f2a7cd4d3918734cbf7ce00a6056885","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-eba8f4c59fc692452239","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v11 --supports--> v14\n- v14 --supports--> v05\n- v05 --refutes--> v10\n- v10 --supports--> v00\n- v00 --refutes--> v0a\n- v0a --supports--> v03\n- v03 --supports--> v07\n- v07 --refutes--> v0e\n- v0e --refutes--> v0c\n- v0c --refutes--> v08\n- v08 --supports--> v04\n- v04 --refutes--> v0d\n- v0d --supports--> v0b\n- v0b --supports--> v06\n- v06 --supports--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v02\n- v03 --refutes--> v13\n- v03 --refutes--> v15\n- v07 --refutes--> v01\n- v07 --refutes--> v0f\n- v08 --refutes--> v12\n- v10 --refutes--> v16\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":"8aea5b4374b936cf292305edd765f97b544c69bc2934f2efa355e6ce1690625d","schema":"livegraph.boolean-propagation.v1","spec_hash":"eba8f4c59fc6924522390e1ad779d563fdf4277765520ec8b87aacf048a9d844","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":15,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-1d24afabb7048e173bdb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --refutes--> v15\n- v15 --supports--> v0d\n- v0d --refutes--> v04\n- v04 --supports--> v03\n- v03 --supports--> v0e\n- v0e --refutes--> v12\n- v12 --refutes--> v13\n- v13 --refutes--> v0a\n- v0a --supports--> v08\n- v08 --refutes--> v16\n- v16 --supports--> v0b\n- v0b --refutes--> v10\n- v10 --refutes--> v00\n- v00 --refutes--> v14\n- v14 --refutes--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v08 --refutes--> v06\n- v09 --supports--> v02\n- v09 --supports--> v05\n- v0b --refutes--> v0c\n- v0b --supports--> v11\n- v13 --refutes--> v07\n- v15 --refutes--> v0f\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":"aacee0da6ddd34e2c98266e80e894f5f32c3a4452da8af04e7561b4de6344460","schema":"livegraph.boolean-propagation.v1","spec_hash":"1d24afabb7048e173bdbdf4505b176b3fead76253e9e36f12f6cabdf879e5012","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-c66b27f31cf1b4e0e5a6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v16 --supports--> v07\n- v07 --refutes--> v0c\n- v0c --supports--> v0a\n- v0a --supports--> v15\n- v15 --refutes--> v02\n- v02 --refutes--> v03\n- v03 --supports--> v0e\n- v0e --supports--> v06\n- v06 --refutes--> v11\n- v11 --supports--> v01\n- v01 --supports--> v13\n- v13 --supports--> v12\n- v12 --supports--> v00\n- v00 --supports--> v04\n- v04 --supports--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0d\n- v02 --supports--> v05\n- v03 --supports--> v0b\n- v07 --supports--> v08\n- v0a --supports--> v10\n- v0e --supports--> v09\n- v12 --supports--> v14\n\n[VERIFIED ROOT]\nv16 = 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":"d18b12bef4a7d986b52e13858e76e1deff80e5f106159693fced14bdc96b62fe","schema":"livegraph.boolean-propagation.v1","spec_hash":"c66b27f31cf1b4e0e5a6f03e686576c6c399dc75c2b923909e4ffd87c30fed70","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v16\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-aa803e642b69490869d8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --refutes--> v0b\n- v0b --refutes--> v0c\n- v0c --supports--> v10\n- v10 --supports--> v00\n- v00 --refutes--> v01\n- v01 --refutes--> v06\n- v06 --refutes--> v0d\n- v0d --refutes--> v02\n- v02 --supports--> v09\n- v09 --refutes--> v0f\n- v0f --supports--> v0e\n- v0e --supports--> v16\n- v16 --refutes--> v12\n- v12 --supports--> v07\n- v07 --refutes--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v03\n- v07 --supports--> v11\n- v09 --refutes--> v0a\n- v0c --refutes--> v14\n- v0e --supports--> v08\n- v10 --supports--> v15\n- v16 --refutes--> v13\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":"acd4bf37d3797ee0a29c031cf4cd6285f92bb239bbd3ab7a1b8946a8d830021f","schema":"livegraph.boolean-propagation.v1","spec_hash":"aa803e642b69490869d8ca0af2e235d13dcf110c36e36b7f5d911c8f834a7cdf","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-2d293907845ed3a2205d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --supports--> v09\n- v09 --supports--> v0b\n- v0b --supports--> v0a\n- v0a --refutes--> v16\n- v16 --refutes--> v0d\n- v0d --supports--> v00\n- v00 --refutes--> v0e\n- v0e --refutes--> v0f\n- v0f --refutes--> v12\n- v12 --refutes--> v08\n- v08 --refutes--> v0c\n- v0c --supports--> v06\n- v06 --refutes--> v10\n- v10 --refutes--> v01\n- v01 --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v14\n- v01 --supports--> v02\n- v01 --supports--> v11\n- v07 --supports--> v13\n- v0a --supports--> v15\n- v0c --refutes--> v04\n- v16 --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":"4766713ab1903af33b1aa96f8ec5b77b27fc33e38c0fc3c711c9ca5fcf353c83","schema":"livegraph.boolean-propagation.v1","spec_hash":"2d293907845ed3a2205d30803fc57744b60545382ce8803d21d90be8beb913ed","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-f70fbb02a7157ed44a3f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --supports--> v04\n- v04 --refutes--> v0e\n- v0e --supports--> v05\n- v05 --refutes--> v16\n- v16 --refutes--> v07\n- v07 --refutes--> v03\n- v03 --refutes--> v14\n- v14 --refutes--> v10\n- v10 --refutes--> v09\n- v09 --refutes--> v01\n- v01 --refutes--> v11\n- v11 --refutes--> v00\n- v00 --refutes--> v0d\n- v0d --supports--> v0c\n- v0c --refutes--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v06\n- v05 --supports--> v0a\n- v09 --supports--> v13\n- v0c --refutes--> v08\n- v0d --refutes--> v0f\n- v0e --supports--> v15\n- v11 --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":"97e3881f3d040ca1c6645e592ea2709828cec9c7dbcd6ae28c699aa02cfba919","schema":"livegraph.boolean-propagation.v1","spec_hash":"f70fbb02a7157ed44a3f97e76c7d1406e09ff4f43fec5c5917944bbad0609454","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-3baed158b62736e8d3c9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --supports--> v09\n- v09 --refutes--> v08\n- v08 --supports--> v14\n- v14 --supports--> v0d\n- v0d --refutes--> v0e\n- v0e --refutes--> v13\n- v13 --refutes--> v05\n- v05 --supports--> v04\n- v04 --refutes--> v07\n- v07 --refutes--> v01\n- v01 --supports--> v11\n- v11 --supports--> v15\n- v15 --supports--> v03\n- v03 --refutes--> v16\n- v16 --refutes--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v0b\n- v04 --refutes--> v06\n- v04 --supports--> v12\n- v11 --refutes--> v00\n- v11 --refutes--> v10\n- v11 --supports--> v0c\n- v13 --refutes--> v0f\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":"c1d1c2c50ccfbec1cbc13891dd25b0888812e281c8ee216b34497670d0c52005","schema":"livegraph.boolean-propagation.v1","spec_hash":"3baed158b62736e8d3c93c10bbdf23d0cee6ce3f5bcc7445ef2502d5a434e53a","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-fa2d4c1e62a71a37b46e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --refutes--> v00\n- v00 --refutes--> v0f\n- v0f --refutes--> v14\n- v14 --refutes--> v13\n- v13 --refutes--> v05\n- v05 --supports--> v0d\n- v0d --refutes--> v06\n- v06 --supports--> v0b\n- v0b --refutes--> v07\n- v07 --refutes--> v08\n- v08 --refutes--> v16\n- v16 --refutes--> v11\n- v11 --supports--> v0e\n- v0e --supports--> v03\n- v03 --refutes--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v10\n- v05 --refutes--> v0a\n- v08 --supports--> v12\n- v0e --refutes--> v15\n- v0f --supports--> v0c\n- v13 --refutes--> v01\n- v14 --refutes--> 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":"d53958f174853c9fd312dfd7dd7fac619f2389cf43578ef78aa82accb8f16d85","schema":"livegraph.boolean-propagation.v1","spec_hash":"fa2d4c1e62a71a37b46ebe93b8e9a0025f52cf5ff303df83c32c3078ec5e55c7","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"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\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-470ec6f2fe96817c0043","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v12 --supports--> v14\n- v14 --refutes--> v0c\n- v0c --supports--> v13\n- v13 --refutes--> v06\n- v06 --refutes--> v02\n- v02 --supports--> v00\n- v00 --supports--> v05\n- v05 --refutes--> v0b\n- v0b --refutes--> v11\n- v11 --supports--> v0d\n- v0d --refutes--> v04\n- v04 --supports--> v15\n- v15 --supports--> v0e\n- v0e --supports--> v0a\n- v0a --refutes--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v10\n- v00 --supports--> v03\n- v02 --refutes--> v16\n- v05 --refutes--> v08\n- v05 --supports--> v01\n- v11 --refutes--> v09\n- v13 --refutes--> v07\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":"b635d0c7b8b72b72a84b97c0cad49096813629b764ab3280f0ba1b504483a212","schema":"livegraph.boolean-propagation.v1","spec_hash":"470ec6f2fe96817c00430879361f620c3d10a56341c85bc37c67bc7a98f0dfd9","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-f412625ef2ef46f7a0ff","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0b --refutes--> v0d\n- v0d --refutes--> v02\n- v02 --refutes--> v10\n- v10 --supports--> v00\n- v00 --supports--> v12\n- v12 --refutes--> v0f\n- v0f --supports--> v0e\n- v0e --supports--> v03\n- v03 --refutes--> v0a\n- v0a --supports--> v13\n- v13 --supports--> v08\n- v08 --supports--> v06\n- v06 --supports--> v15\n- v15 --refutes--> v0c\n- v0c --refutes--> v11\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v05\n- v02 --refutes--> v09\n- v03 --refutes--> v01\n- v03 --refutes--> v14\n- v03 --supports--> v07\n- v13 --supports--> v16\n- v15 --supports--> v04\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":"a2b3315708fd37486d26aee3818525a95d55a52966c85ac1acc5afb957f04940","schema":"livegraph.boolean-propagation.v1","spec_hash":"f412625ef2ef46f7a0ffbd077dfe4bb52842ac51ec1c74120a129297a33e1991","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-21524e0a47c64568362d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --supports--> v0f\n- v0f --supports--> v00\n- v00 --refutes--> v01\n- v01 --refutes--> v04\n- v04 --refutes--> v0b\n- v0b --supports--> v16\n- v16 --supports--> v12\n- v12 --supports--> v0a\n- v0a --supports--> v0e\n- v0e --supports--> v02\n- v02 --supports--> v07\n- v07 --refutes--> v0d\n- v0d --supports--> v11\n- v11 --supports--> v13\n- v13 --refutes--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v08\n- v02 --supports--> v06\n- v0d --refutes--> v14\n- v0e --refutes--> v03\n- v0f --refutes--> v09\n- v12 --refutes--> v10\n- v12 --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":"6d045a17b5c080c79f37214024ae8639791774586b922fb9b778cd81b4a7233a","schema":"livegraph.boolean-propagation.v1","spec_hash":"21524e0a47c64568362d579d1fd63e91b48ebf1a04dc42934103d97015fdd475","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-23e788c63fa8223be977","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --supports--> v12\n- v12 --supports--> v0f\n- v0f --supports--> v0b\n- v0b --supports--> v0a\n- v0a --refutes--> v02\n- v02 --refutes--> v0e\n- v0e --supports--> v00\n- v00 --supports--> v0d\n- v0d --supports--> v15\n- v15 --supports--> v13\n- v13 --refutes--> v0c\n- v0c --supports--> v07\n- v07 --refutes--> v09\n- v09 --supports--> v08\n- v08 --refutes--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v01\n- v02 --refutes--> v10\n- v06 --refutes--> v03\n- v07 --refutes--> v11\n- v0d --refutes--> v16\n- v0e --refutes--> v14\n- v0e --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":"f6d32d915c40e35808da11ec2118ea79507576fa7baee00491a8747d3acf2f07","schema":"livegraph.boolean-propagation.v1","spec_hash":"23e788c63fa8223be977c53fae2f5514813d28506c46fd93e15e6455872593ea","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-6ffb32e582428246926f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v13 --refutes--> v0d\n- v0d --supports--> v10\n- v10 --supports--> v05\n- v05 --refutes--> v12\n- v12 --refutes--> v15\n- v15 --supports--> v04\n- v04 --supports--> v14\n- v14 --supports--> v0e\n- v0e --supports--> v0a\n- v0a --refutes--> v01\n- v01 --refutes--> v09\n- v09 --supports--> v16\n- v16 --refutes--> v0c\n- v0c --refutes--> v08\n- v08 --supports--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v0f\n- v05 --supports--> v03\n- v05 --supports--> v11\n- v08 --supports--> v02\n- v0d --supports--> v07\n- v10 --refutes--> v00\n- v13 --refutes--> v0b\n\n[VERIFIED ROOT]\nv13 = 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":"1750487efa5cec970c4b71eabb36bd4067e0d6cec30fd36fe6101a5e2a856fc1","schema":"livegraph.boolean-propagation.v1","spec_hash":"6ffb32e582428246926f97fc6800d0e7658808a98ee64707a50f704fe68e0222","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-f0e096ea50b837c099ef","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --supports--> v08\n- v08 --refutes--> v01\n- v01 --supports--> v05\n- v05 --supports--> v0e\n- v0e --refutes--> v14\n- v14 --supports--> v0f\n- v0f --refutes--> v13\n- v13 --refutes--> v12\n- v12 --supports--> v0d\n- v0d --supports--> v15\n- v15 --refutes--> v11\n- v11 --refutes--> v16\n- v16 --supports--> v0a\n- v0a --supports--> v0b\n- v0b --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v0c\n- v08 --refutes--> v07\n- v0a --supports--> v02\n- v0d --supports--> v09\n- v0e --refutes--> v06\n- v15 --refutes--> v04\n- v16 --supports--> v10\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":"b656b4097d23055714e8653183aee25736afeee5a67140e882268f63f20c095a","schema":"livegraph.boolean-propagation.v1","spec_hash":"f0e096ea50b837c099ef4a5f969daf698b31cf2e3cd29d17468a17585ce51595","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-b3e9347690da37a3bd72","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --supports--> v0c\n- v0c --supports--> v0a\n- v0a --supports--> v03\n- v03 --refutes--> v08\n- v08 --refutes--> v13\n- v13 --supports--> v16\n- v16 --supports--> v01\n- v01 --supports--> v14\n- v14 --supports--> v0d\n- v0d --refutes--> v15\n- v15 --supports--> v0e\n- v0e --supports--> v11\n- v11 --supports--> v09\n- v09 --refutes--> v10\n- v10 --supports--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v07 --supports--> v02\n- v07 --supports--> v06\n- v09 --supports--> v04\n- v0c --supports--> v05\n- v0e --supports--> v0b\n- v13 --refutes--> v12\n- v14 --supports--> 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":"9f8a5970ea0d40f93e8a9d3ebc4c23b6de0ff0b3c84fc7239375246487bf139b","schema":"livegraph.boolean-propagation.v1","spec_hash":"b3e9347690da37a3bd7296c9816493796764d04710e0f80be861b6502ed6c6e2","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-745d4ad6ea7eeb923497","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --refutes--> v02\n- v02 --supports--> v0e\n- v0e --supports--> v0a\n- v0a --supports--> v12\n- v12 --refutes--> v0b\n- v0b --refutes--> v09\n- v09 --refutes--> v0f\n- v0f --refutes--> v14\n- v14 --supports--> v10\n- v10 --supports--> v13\n- v13 --supports--> v04\n- v04 --supports--> v01\n- v01 --supports--> v00\n- v00 --refutes--> v0c\n- v0c --supports--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v11\n- v01 --refutes--> v05\n- v04 --refutes--> v0d\n- v08 --supports--> v03\n- v09 --refutes--> v15\n- v10 --supports--> v16\n- v13 --refutes--> v06\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":"020afe7c765370e402a253abd5f6e8015ce7720c7e23d1248fdca4efd993c1b5","schema":"livegraph.boolean-propagation.v1","spec_hash":"745d4ad6ea7eeb923497820a4d001036300d775f9d4c80f6fc6de170f8c0462c","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":15,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-3dc4da11027ac54699f6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --supports--> v06\n- v06 --supports--> v07\n- v07 --refutes--> v12\n- v12 --refutes--> v0a\n- v0a --supports--> v16\n- v16 --supports--> v0c\n- v0c --supports--> v0b\n- v0b --refutes--> v03\n- v03 --supports--> v11\n- v11 --refutes--> v13\n- v13 --supports--> v04\n- v04 --supports--> v00\n- v00 --refutes--> v15\n- v15 --supports--> v0d\n- v0d --supports--> v0e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v0f\n- v04 --refutes--> v14\n- v05 --supports--> v09\n- v0b --supports--> v01\n- v12 --supports--> v02\n- v13 --supports--> v08\n- v13 --supports--> v10\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":"daceb9fe8dc73e794b05bcceac2042e7b4bc5066c0e92cc9abb3ec9088c7088f","schema":"livegraph.boolean-propagation.v1","spec_hash":"3dc4da11027ac54699f6e2365aaa740940af48fc29843a82e9b93e2e034ca888","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-8e05c5f6f5a1e8c7d584","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v12 --refutes--> v14\n- v14 --supports--> v10\n- v10 --supports--> v01\n- v01 --refutes--> v02\n- v02 --supports--> v05\n- v05 --refutes--> v00\n- v00 --supports--> v07\n- v07 --refutes--> v0e\n- v0e --refutes--> v06\n- v06 --refutes--> v0d\n- v0d --supports--> v03\n- v03 --supports--> v15\n- v15 --refutes--> v0b\n- v0b --refutes--> v11\n- v11 --supports--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v07 --supports--> v04\n- v0b --refutes--> v09\n- v0b --refutes--> v0c\n- v10 --refutes--> v13\n- v10 --supports--> v16\n- v11 --refutes--> v0a\n- v14 --refutes--> v08\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":"1ca09d1a7d6a768ad45e274fd1e5a414da76529fe7179636be99b9341dcaad89","schema":"livegraph.boolean-propagation.v1","spec_hash":"8e05c5f6f5a1e8c7d5846a51b99bf399bae49595b953257cfea1bf143c45eb33","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-feeb28c528a9642873c9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --refutes--> v0e\n- v0e --refutes--> v06\n- v06 --supports--> v07\n- v07 --supports--> v01\n- v01 --refutes--> v11\n- v11 --supports--> v03\n- v03 --supports--> v10\n- v10 --refutes--> v15\n- v15 --refutes--> v02\n- v02 --supports--> v0a\n- v0a --supports--> v14\n- v14 --supports--> v0d\n- v0d --supports--> v05\n- v05 --supports--> v04\n- v04 --supports--> v16\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v09\n- v06 --refutes--> v08\n- v07 --refutes--> v12\n- v07 --refutes--> v13\n- v0a --refutes--> v0f\n- v0d --refutes--> v0c\n- v0d --supports--> v0b\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":"b246edb30c7b03d63923f8dd5c932ac52c17f5748e3124d6ba0d5bbf2228603d","schema":"livegraph.boolean-propagation.v1","spec_hash":"feeb28c528a9642873c9aededa183028d58341edc7e0201ab0f00e14dbc3db4b","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v16\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-c13f79d9a1f5e6410851","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v03\n- v03 --refutes--> v11\n- v11 --supports--> v00\n- v00 --refutes--> v16\n- v16 --supports--> v14\n- v14 --refutes--> v0c\n- v0c --supports--> v09\n- v09 --supports--> v15\n- v15 --refutes--> v0d\n- v0d --refutes--> v05\n- v05 --supports--> v0e\n- v0e --supports--> v07\n- v07 --supports--> v12\n- v12 --refutes--> v13\n- v13 --supports--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v10\n- v04 --refutes--> v0a\n- v07 --supports--> v0b\n- v11 --refutes--> v06\n- v12 --refutes--> v02\n- v12 --supports--> v0f\n- v16 --supports--> 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":"02ebfb0d8122760202ff2ca60ac676d301c563debc1bd80157d20c1ee137c925","schema":"livegraph.boolean-propagation.v1","spec_hash":"c13f79d9a1f5e641085193e895e8aed6d67ad633df381f4b8ed61137a9462282","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-0e326ea2ecce1cf45265","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0d --refutes--> v02\n- v02 --supports--> v15\n- v15 --refutes--> v12\n- v12 --refutes--> v0f\n- v0f --refutes--> v16\n- v16 --supports--> v05\n- v05 --supports--> v0c\n- v0c --supports--> v04\n- v04 --refutes--> v11\n- v11 --supports--> v06\n- v06 --supports--> v09\n- v09 --supports--> v0e\n- v0e --refutes--> v14\n- v14 --supports--> v0a\n- v0a --supports--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v0b\n- v05 --supports--> v00\n- v0c --refutes--> v07\n- v0d --supports--> v10\n- v11 --refutes--> v08\n- v14 --supports--> v13\n- v15 --supports--> v03\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":"9b06fe952dfe398fbea3e59bec8e7ece0ef7fec98e527201ab8ad1e57b4ad72c","schema":"livegraph.boolean-propagation.v1","spec_hash":"0e326ea2ecce1cf45265d0b2c301325e28cab7cf41cee853f531b538c9fab695","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-81b2867999bdd549e10b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --refutes--> v0c\n- v0c --refutes--> v10\n- v10 --refutes--> v00\n- v00 --supports--> v14\n- v14 --supports--> v04\n- v04 --refutes--> v16\n- v16 --refutes--> v03\n- v03 --refutes--> v0b\n- v0b --refutes--> v15\n- v15 --refutes--> v07\n- v07 --supports--> v11\n- v11 --supports--> v02\n- v02 --supports--> v0d\n- v0d --refutes--> v05\n- v05 --refutes--> v13\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v0e\n- v0b --supports--> v12\n- v0c --supports--> v06\n- v0c --supports--> v08\n- v11 --refutes--> v09\n- v11 --supports--> v0a\n- v14 --refutes--> v0f\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":"16733bd8b5bd1c2db63698736c4b5164ec3d8143f4a6d77b0107189ae4cf2585","schema":"livegraph.boolean-propagation.v1","spec_hash":"81b2867999bdd549e10b8cada52c2a2d5fb917b5e60a0e2d36d2be1c6c9e9fed","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-0d15e4b7944ce3ccc274","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v10 --refutes--> v12\n- v12 --supports--> v05\n- v05 --refutes--> v0e\n- v0e --refutes--> v0d\n- v0d --supports--> v01\n- v01 --supports--> v13\n- v13 --refutes--> v0f\n- v0f --refutes--> v16\n- v16 --refutes--> v06\n- v06 --supports--> v04\n- v04 --refutes--> v03\n- v03 --supports--> v0c\n- v0c --supports--> v09\n- v09 --refutes--> v07\n- v07 --refutes--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v14\n- v06 --refutes--> v0b\n- v06 --supports--> v00\n- v07 --supports--> v15\n- v0e --refutes--> v0a\n- v10 --supports--> v08\n- v10 --supports--> v11\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":"5570783b2663c48679bc3a129f421b7bc3df29c76c78ababb16ed25a9cc7bf19","schema":"livegraph.boolean-propagation.v1","spec_hash":"0d15e4b7944ce3ccc2742795f681f2585b71f3f6fbb51cfe82932915abed4151","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-8e1d52ba9d0d75d776c5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --refutes--> v0f\n- v0f --supports--> v15\n- v15 --refutes--> v0d\n- v0d --supports--> v07\n- v07 --refutes--> v13\n- v13 --supports--> v16\n- v16 --refutes--> v04\n- v04 --refutes--> v0b\n- v0b --supports--> v05\n- v05 --supports--> v0a\n- v0a --refutes--> v08\n- v08 --supports--> v03\n- v03 --refutes--> v14\n- v14 --refutes--> v06\n- v06 --refutes--> v10\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v00\n- v06 --supports--> v02\n- v06 --supports--> v09\n- v06 --supports--> v11\n- v08 --refutes--> v12\n- v0b --refutes--> v0c\n- v0f --supports--> v0e\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":"2f4311498dac94aeccbd22565e07503377ca0ef379379c6e314525e8623cb51d","schema":"livegraph.boolean-propagation.v1","spec_hash":"8e1d52ba9d0d75d776c5d66e8b4a49cbc5efe6ecfcee6878c7cb4a277a1351bd","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-d5ea847bcf263b375ae0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v02\n- v02 --refutes--> v0e\n- v0e --supports--> v03\n- v03 --refutes--> v12\n- v12 --supports--> v16\n- v16 --refutes--> v0f\n- v0f --supports--> v13\n- v13 --supports--> v14\n- v14 --supports--> v11\n- v11 --refutes--> v10\n- v10 --supports--> v08\n- v08 --supports--> v07\n- v07 --refutes--> v00\n- v00 --supports--> v06\n- v06 --supports--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v0c\n- v03 --supports--> v05\n- v04 --refutes--> v0b\n- v06 --supports--> v0a\n- v08 --supports--> v01\n- v0f --supports--> v09\n- v10 --supports--> v15\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":"e54daefc57c9dd9c439917cdd5b3fdf3d5e5c248466249a3e312685784b82db1","schema":"livegraph.boolean-propagation.v1","spec_hash":"d5ea847bcf263b375ae0df6bfe62da971a449c97621a0730fb0b0316a24933d0","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-69a42b1d6491bfa98426","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0f --refutes--> v09\n- v09 --refutes--> v0d\n- v0d --refutes--> v08\n- v08 --refutes--> v00\n- v00 --refutes--> v06\n- v06 --refutes--> v16\n- v16 --supports--> v15\n- v15 --refutes--> v05\n- v05 --supports--> v03\n- v03 --supports--> v11\n- v11 --refutes--> v13\n- v13 --supports--> v01\n- v01 --supports--> v04\n- v04 --refutes--> v02\n- v02 --supports--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v0b\n- v03 --supports--> v0e\n- v03 --supports--> v14\n- v09 --refutes--> v07\n- v0f --supports--> v0c\n- v11 --supports--> v12\n- v15 --supports--> v10\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":"cca6128974423fd17041de262cffc87409ff5e367d0d1811d1d5a9383615dd59","schema":"livegraph.boolean-propagation.v1","spec_hash":"69a42b1d6491bfa98426460d09db4e3290a5f8e4edfa50ba26b35bb82a3fe5cd","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-befeb394bdc89ef4597a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --refutes--> v08\n- v08 --supports--> v05\n- v05 --supports--> v0b\n- v0b --supports--> v13\n- v13 --refutes--> v16\n- v16 --refutes--> v0e\n- v0e --refutes--> v03\n- v03 --supports--> v01\n- v01 --supports--> v06\n- v06 --supports--> v00\n- v00 --refutes--> v11\n- v11 --refutes--> v04\n- v04 --refutes--> v0c\n- v0c --supports--> v02\n- v02 --refutes--> v12\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v15\n- v01 --supports--> v10\n- v02 --refutes--> v07\n- v03 --supports--> v0f\n- v0b --supports--> v0a\n- v13 --refutes--> v14\n- v13 --supports--> v0d\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":"208d27a52350498b97d2378eb40f427e2ef896e44a938a42555c0682b3e9edcc","schema":"livegraph.boolean-propagation.v1","spec_hash":"befeb394bdc89ef4597a6228a5b0b2e16d75f51a3160edf782c54c69bb379a90","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-e961dfa1a1bfadd948cc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v13 --supports--> v0a\n- v0a --supports--> v06\n- v06 --refutes--> v0d\n- v0d --refutes--> v07\n- v07 --supports--> v14\n- v14 --supports--> v05\n- v05 --supports--> v0f\n- v0f --supports--> v09\n- v09 --refutes--> v00\n- v00 --refutes--> v11\n- v11 --supports--> v01\n- v01 --refutes--> v0b\n- v0b --refutes--> v0e\n- v0e --supports--> v0c\n- v0c --supports--> v12\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v08\n- v06 --supports--> v03\n- v0a --refutes--> v04\n- v0a --refutes--> v10\n- v0d --supports--> v15\n- v0e --supports--> v02\n- v13 --refutes--> v16\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":"00508a18ef864779024394771a762a61bdfff6912e54ec147e3ac6141cc40eaf","schema":"livegraph.boolean-propagation.v1","spec_hash":"e961dfa1a1bfadd948cce5ea611d600475b01a27501a29b4fbed80a550b79faf","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":15,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-d84d1b55219946940285","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --supports--> v0a\n- v0a --refutes--> v0f\n- v0f --refutes--> v12\n- v12 --supports--> v06\n- v06 --refutes--> v16\n- v16 --refutes--> v13\n- v13 --supports--> v14\n- v14 --refutes--> v05\n- v05 --supports--> v00\n- v00 --refutes--> v10\n- v10 --refutes--> v09\n- v09 --supports--> v0d\n- v0d --refutes--> v03\n- v03 --supports--> v0b\n- v0b --supports--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v01\n- v09 --supports--> v0e\n- v0b --refutes--> v15\n- v0c --refutes--> v02\n- v0f --refutes--> v08\n- v13 --supports--> v04\n- v16 --supports--> v11\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":"0811b37c02893dbeb7190ccbb7350c702d8b3d78944c5c822938cafaee7ae78b","schema":"livegraph.boolean-propagation.v1","spec_hash":"d84d1b552199469402853b4ac3a1afb49bdb017b4515c326a51536d4c77ce25c","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-29f00e8305864f88d177","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v10 --supports--> v0f\n- v0f --supports--> v04\n- v04 --refutes--> v14\n- v14 --supports--> v07\n- v07 --refutes--> v03\n- v03 --supports--> v05\n- v05 --supports--> v09\n- v09 --supports--> v0c\n- v0c --supports--> v0e\n- v0e --supports--> v16\n- v16 --refutes--> v06\n- v06 --supports--> v13\n- v13 --supports--> v15\n- v15 --refutes--> v02\n- v02 --supports--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --supports--> v00\n- v07 --refutes--> v12\n- v09 --refutes--> v0d\n- v0c --supports--> v01\n- v0e --refutes--> v0a\n- v10 --supports--> v11\n- v14 --supports--> v0b\n\n[VERIFIED ROOT]\nv10 = 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":"98639e5acc92bfad55cf3af4861cb462e8fda7e7c3653c9c1f9ef45e16c65eb1","schema":"livegraph.boolean-propagation.v1","spec_hash":"29f00e8305864f88d177aceb579bd648fd6828fd678600e85ef7c22a9af7786f","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-8823b4c0a53c197b7cc8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --supports--> v01\n- v01 --refutes--> v0e\n- v0e --refutes--> v0f\n- v0f --supports--> v07\n- v07 --supports--> v02\n- v02 --refutes--> v00\n- v00 --supports--> v06\n- v06 --refutes--> v11\n- v11 --refutes--> v0b\n- v0b --supports--> v16\n- v16 --supports--> v0a\n- v0a --supports--> v10\n- v10 --supports--> v05\n- v05 --supports--> v12\n- v12 --refutes--> v14\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v15\n- v02 --refutes--> v08\n- v02 --supports--> v04\n- v0a --refutes--> v0d\n- v0c --supports--> v13\n- v0f --supports--> v09\n- v16 --refutes--> v03\n\n[VERIFIED ROOT]\nv0c = 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":"4386487090bd14058ac3f5f046b94bf9975d7c4a32cfe17f2e62e4286272160f","schema":"livegraph.boolean-propagation.v1","spec_hash":"8823b4c0a53c197b7cc80e4117e194cceeed05a27395d2bb82365a846abeed77","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-fcc04ab80a922d1c86f7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v05\n- v05 --supports--> v11\n- v11 --supports--> v15\n- v15 --supports--> v0b\n- v0b --supports--> v07\n- v07 --refutes--> v14\n- v14 --refutes--> v0e\n- v0e --supports--> v13\n- v13 --refutes--> v01\n- v01 --supports--> v08\n- v08 --supports--> v00\n- v00 --refutes--> v0a\n- v0a --supports--> v12\n- v12 --refutes--> v0f\n- v0f --supports--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v08 --refutes--> v02\n- v0a --refutes--> v16\n- v0e --refutes--> v06\n- v0f --refutes--> v03\n- v14 --supports--> v0c\n- v15 --supports--> v0d\n- v15 --supports--> v10\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":"590dfe62792ade17ce247863fb8616580ae89de849af0a0edfdf327d293044dd","schema":"livegraph.boolean-propagation.v1","spec_hash":"fcc04ab80a922d1c86f71385c7baa9eee52cb96e6385b1eef06b93e1116f29e0","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-bb645b9be4d4b74a7a82","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --supports--> v0e\n- v0e --refutes--> v07\n- v07 --refutes--> v01\n- v01 --supports--> v09\n- v09 --supports--> v16\n- v16 --supports--> v0d\n- v0d --refutes--> v0b\n- v0b --refutes--> v12\n- v12 --refutes--> v03\n- v03 --refutes--> v15\n- v15 --supports--> v08\n- v08 --supports--> v05\n- v05 --supports--> v14\n- v14 --supports--> v13\n- v13 --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v11\n- v03 --refutes--> v04\n- v05 --supports--> v02\n- v08 --refutes--> v0c\n- v0b --supports--> v10\n- v0d --refutes--> v06\n- v0d --supports--> v0f\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":"2912b3b34941e67c12ebfec677a4b4772366346fa523232a6543307bce3d7c49","schema":"livegraph.boolean-propagation.v1","spec_hash":"bb645b9be4d4b74a7a82970646ac022ba166a1c0e59ff079d328b02f7b4f9e40","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-a0acb8c02cb22211dc94","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --supports--> v06\n- v06 --refutes--> v09\n- v09 --refutes--> v0c\n- v0c --supports--> v0b\n- v0b --supports--> v05\n- v05 --refutes--> v0f\n- v0f --supports--> v13\n- v13 --refutes--> v04\n- v04 --refutes--> v02\n- v02 --supports--> v08\n- v08 --supports--> v03\n- v03 --refutes--> v0e\n- v0e --supports--> v0a\n- v0a --refutes--> v14\n- v14 --supports--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v00\n- v08 --refutes--> v07\n- v08 --supports--> v15\n- v0b --supports--> v10\n- v0c --supports--> v12\n- v0f --refutes--> v11\n- v0f --supports--> v16\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":"83933e436aaeb455a4b4888dc14b05026abe929fce3efe347cfa3b6eb4a63848","schema":"livegraph.boolean-propagation.v1","spec_hash":"a0acb8c02cb22211dc94e6b9447bb769c53a0346fa0e1ab5b7b1596b077bfbf2","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":15,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-61ed78b32e636d196a47","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v06\n- v06 --refutes--> v09\n- v09 --refutes--> v0f\n- v0f --refutes--> v0b\n- v0b --supports--> v04\n- v04 --refutes--> v0c\n- v0c --refutes--> v01\n- v01 --refutes--> v10\n- v10 --supports--> v14\n- v14 --refutes--> v13\n- v13 --refutes--> v07\n- v07 --refutes--> v03\n- v03 --supports--> v08\n- v08 --refutes--> v15\n- v15 --supports--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v12\n- v03 --supports--> v05\n- v06 --refutes--> v02\n- v07 --refutes--> v0e\n- v08 --refutes--> v16\n- v08 --supports--> v0a\n- v0c --refutes--> v11\n\n[VERIFIED ROOT]\nv00 = 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":"dbd23e05bae44d2a12a53dc520fc87e7192302b3711d0f6d113db4e03a7fb33e","schema":"livegraph.boolean-propagation.v1","spec_hash":"61ed78b32e636d196a47fca91f697345210f2081e7aaaab7306399f44cd717fe","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-ff23fcca9db57e2053d6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --supports--> v03\n- v03 --supports--> v0a\n- v0a --supports--> v08\n- v08 --supports--> v00\n- v00 --supports--> v12\n- v12 --refutes--> v0e\n- v0e --refutes--> v15\n- v15 --refutes--> v0b\n- v0b --supports--> v0f\n- v0f --refutes--> v0c\n- v0c --supports--> v16\n- v16 --supports--> v0d\n- v0d --refutes--> v01\n- v01 --supports--> v05\n- v05 --supports--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v07\n- v09 --refutes--> v14\n- v09 --supports--> v02\n- v0c --refutes--> v11\n- v0d --refutes--> v13\n- v0e --supports--> v10\n- v16 --refutes--> v04\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":"99b2f0f9b6e9302e0461143d097f3a5df7544d774fa5893905fe0d62e430bee8","schema":"livegraph.boolean-propagation.v1","spec_hash":"ff23fcca9db57e2053d61bc5700464581ccd23e8933d369dc5eb0ac47f11c454","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-4766557e6a4b57d1a0fd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v0e\n- v0e --refutes--> v0b\n- v0b --refutes--> v0c\n- v0c --supports--> v13\n- v13 --refutes--> v09\n- v09 --supports--> v12\n- v12 --refutes--> v07\n- v07 --supports--> v10\n- v10 --refutes--> v01\n- v01 --refutes--> v03\n- v03 --refutes--> v0f\n- v0f --supports--> v14\n- v14 --refutes--> v08\n- v08 --refutes--> v04\n- v04 --supports--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v02\n- v04 --refutes--> v0d\n- v04 --supports--> v15\n- v09 --refutes--> v00\n- v0c --refutes--> v05\n- v10 --refutes--> v11\n- v12 --refutes--> v16\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":"47eba2bdace1921eaf68d92bb695880b658b1359572e2b52c85865d88eb7ecbd","schema":"livegraph.boolean-propagation.v1","spec_hash":"4766557e6a4b57d1a0fd9e00835ca5f84dbc7140652cdf70e181d40b9346660e","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v14\"}],\"hop\":15,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-255c7df51613daf3dd02","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --supports--> v14\n- v14 --refutes--> v16\n- v16 --supports--> v12\n- v12 --supports--> v05\n- v05 --refutes--> v10\n- v10 --refutes--> v08\n- v08 --supports--> v0b\n- v0b --refutes--> v0e\n- v0e --refutes--> v09\n- v09 --supports--> v01\n- v01 --refutes--> v0f\n- v0f --refutes--> v11\n- v11 --supports--> v03\n- v03 --refutes--> v00\n- v00 --supports--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0a\n- v03 --supports--> v15\n- v0b --supports--> v07\n- v0c --refutes--> v04\n- v0c --refutes--> v06\n- v0c --refutes--> v0d\n- v11 --refutes--> v13\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":"dbcb75b1253c6b4d839372c947f19917d92dd350fb229429582a6f21274e79da","schema":"livegraph.boolean-propagation.v1","spec_hash":"255c7df51613daf3dd029b560a5e73deff3fae3616499e63cb331385c1ba72be","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-a228ac1423bf6c4caeb1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v11 --refutes--> v14\n- v14 --supports--> v06\n- v06 --refutes--> v16\n- v16 --supports--> v0e\n- v0e --supports--> v07\n- v07 --supports--> v0a\n- v0a --supports--> v00\n- v00 --supports--> v04\n- v04 --refutes--> v15\n- v15 --supports--> v01\n- v01 --supports--> v02\n- v02 --refutes--> v0c\n- v0c --supports--> v0b\n- v0b --supports--> v03\n- v03 --supports--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v09\n- v01 --refutes--> v08\n- v01 --supports--> v13\n- v02 --refutes--> v12\n- v03 --supports--> v05\n- v06 --refutes--> v10\n- v14 --refutes--> v0f\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":"6c39dfdd499f95672de0da780f02546a28263a7fb65c5e2ac192ff76f837cbfc","schema":"livegraph.boolean-propagation.v1","spec_hash":"a228ac1423bf6c4caeb1baaa17a51dace977373c5deb924ab1535dfc004732ad","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-7f366ba32506530d6952","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v16 --refutes--> v03\n- v03 --supports--> v13\n- v13 --supports--> v08\n- v08 --refutes--> v01\n- v01 --refutes--> v0a\n- v0a --refutes--> v0b\n- v0b --refutes--> v06\n- v06 --supports--> v0e\n- v0e --refutes--> v14\n- v14 --supports--> v04\n- v04 --refutes--> v15\n- v15 --supports--> v07\n- v07 --supports--> v09\n- v09 --refutes--> v10\n- v10 --refutes--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v12\n- v0a --refutes--> v0f\n- v0e --supports--> v00\n- v0e --supports--> v0d\n- v14 --supports--> v02\n- v16 --refutes--> v0c\n- v16 --refutes--> v11\n\n[VERIFIED ROOT]\nv16 = 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":"9924a7197d6875bae5570b646277cb18cd920639ec55e22c844e66c35fd34a4e","schema":"livegraph.boolean-propagation.v1","spec_hash":"7f366ba32506530d6952e0294ce12f9942ecb4b306553a7bdec93b896e7443be","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v16\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-3b793a0ad716eecfe0bb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v10 --refutes--> v0c\n- v0c --refutes--> v06\n- v06 --refutes--> v03\n- v03 --supports--> v05\n- v05 --supports--> v01\n- v01 --refutes--> v07\n- v07 --supports--> v0f\n- v0f --supports--> v0e\n- v0e --refutes--> v16\n- v16 --supports--> v0a\n- v0a --refutes--> v00\n- v00 --supports--> v04\n- v04 --supports--> v08\n- v08 --supports--> v02\n- v02 --refutes--> v15\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v13\n- v01 --supports--> v11\n- v01 --supports--> v12\n- v07 --refutes--> v14\n- v0a --refutes--> v0d\n- v0f --supports--> v09\n- v16 --refutes--> v0b\n\n[VERIFIED ROOT]\nv10 = 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":"ce8b63a4f88f3bce3c7c209c47bcd548ead5e7f7c8120e02f598f28e42215092","schema":"livegraph.boolean-propagation.v1","spec_hash":"3b793a0ad716eecfe0bb698a280927e9aee803c4f132992724c5aa40bf515f37","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-d9836cf00578022d6d6a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --supports--> v11\n- v11 --supports--> v00\n- v00 --supports--> v04\n- v04 --refutes--> v0b\n- v0b --refutes--> v02\n- v02 --refutes--> v10\n- v10 --supports--> v15\n- v15 --supports--> v05\n- v05 --refutes--> v09\n- v09 --refutes--> v14\n- v14 --refutes--> v0a\n- v0a --supports--> v0f\n- v0f --supports--> v01\n- v01 --refutes--> v0d\n- v0d --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v0a --supports--> v08\n- v0a --supports--> v12\n- v0b --refutes--> v0e\n- v0b --supports--> v0c\n- v0d --refutes--> v16\n- v10 --supports--> v06\n- v11 --refutes--> v13\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":"d055482d6aeebe96697b916363642656c2c36fb63576692e1b7760aa5e586ad2","schema":"livegraph.boolean-propagation.v1","spec_hash":"d9836cf00578022d6d6af6d9d1e6a1483c72de50f7ec836dc0949e3b64318dc0","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-e918a7da6fae6c7a3e9e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0e --supports--> v07\n- v07 --refutes--> v16\n- v16 --supports--> v11\n- v11 --refutes--> v09\n- v09 --supports--> v0d\n- v0d --refutes--> v06\n- v06 --supports--> v01\n- v01 --refutes--> v14\n- v14 --supports--> v13\n- v13 --refutes--> v03\n- v03 --supports--> v0b\n- v0b --supports--> v00\n- v00 --supports--> v0c\n- v0c --supports--> v12\n- v12 --refutes--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v07 --refutes--> v0f\n- v09 --refutes--> v02\n- v09 --refutes--> v15\n- v0e --refutes--> v10\n- v11 --supports--> v04\n- v12 --refutes--> v0a\n- v12 --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":"bdec42b1d3b9754d69601bb86e16e27904b248ea5ea1d7829c86af62cc66cfdd","schema":"livegraph.boolean-propagation.v1","spec_hash":"e918a7da6fae6c7a3e9ecfb242ff7cc154f1d3ffd1e50e1aeb595d9d8e55f840","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-936b954026bf59a52a1d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --refutes--> v0c\n- v0c --refutes--> v0a\n- v0a --supports--> v00\n- v00 --supports--> v09\n- v09 --refutes--> v0d\n- v0d --refutes--> v0e\n- v0e --refutes--> v16\n- v16 --refutes--> v02\n- v02 --supports--> v01\n- v01 --refutes--> v15\n- v15 --refutes--> v06\n- v06 --supports--> v14\n- v14 --refutes--> v07\n- v07 --supports--> v13\n- v13 --supports--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v0b\n- v07 --refutes--> v12\n- v0d --refutes--> v11\n- v0d --supports--> v03\n- v0e --refutes--> v04\n- v14 --refutes--> v10\n- v16 --supports--> v0f\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":"f059968ff171fe2e6fe809c7225248cb196d3c6e0834b4f0e371801a7e18dd9d","schema":"livegraph.boolean-propagation.v1","spec_hash":"936b954026bf59a52a1d8deb2a67a2b21a75a51a5f8b2502bc414e17e3e9d007","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-66a9567a753a4d4723e6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --refutes--> v13\n- v13 --refutes--> v01\n- v01 --supports--> v0f\n- v0f --supports--> v10\n- v10 --refutes--> v00\n- v00 --refutes--> v0b\n- v0b --supports--> v16\n- v16 --supports--> v14\n- v14 --refutes--> v02\n- v02 --supports--> v0c\n- v0c --refutes--> v0a\n- v0a --refutes--> v05\n- v05 --refutes--> v03\n- v03 --supports--> v07\n- v07 --supports--> v11\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v12\n- v05 --refutes--> v15\n- v09 --refutes--> v06\n- v0b --refutes--> v0d\n- v0c --refutes--> v0e\n- v10 --supports--> v04\n- v13 --refutes--> v08\n\n[VERIFIED ROOT]\nv09 = 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":"d92c2526dd825194cfba742890aafd765b52eecd9870d7d5996aa341930d5f21","schema":"livegraph.boolean-propagation.v1","spec_hash":"66a9567a753a4d4723e6444aad40e6501cd4c9378eab9685d2e28d9e377fe3fe","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-b8ee5a945764f1493a34","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --refutes--> v05\n- v05 --refutes--> v08\n- v08 --supports--> v03\n- v03 --supports--> v0e\n- v0e --refutes--> v10\n- v10 --supports--> v04\n- v04 --supports--> v00\n- v00 --supports--> v0b\n- v0b --supports--> v02\n- v02 --supports--> v07\n- v07 --supports--> v11\n- v11 --refutes--> v09\n- v09 --supports--> v16\n- v16 --supports--> v0d\n- v0d --supports--> v13\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v0a\n- v03 --supports--> v14\n- v09 --refutes--> v15\n- v11 --refutes--> v0f\n- v16 --refutes--> v0c\n- v16 --supports--> v06\n- v16 --supports--> v12\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":"101d9e7cd78c26efc242eab88a557ec5cd15f1108a883f949b9e5ec8c02a1c89","schema":"livegraph.boolean-propagation.v1","spec_hash":"b8ee5a945764f1493a341af9857e5ae63bd72f74ee2389aa4fcc66dce77e93e6","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-23624cc814bc585ebde2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --supports--> v06\n- v06 --refutes--> v0e\n- v0e --supports--> v0d\n- v0d --supports--> v03\n- v03 --supports--> v0a\n- v0a --supports--> v01\n- v01 --supports--> v09\n- v09 --refutes--> v11\n- v11 --refutes--> v13\n- v13 --supports--> v0c\n- v0c --refutes--> v15\n- v15 --supports--> v07\n- v07 --supports--> v08\n- v08 --refutes--> v00\n- v00 --supports--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v12\n- v04 --supports--> v02\n- v04 --supports--> v0b\n- v08 --refutes--> v14\n- v08 --supports--> v0f\n- v0d --refutes--> v10\n- v0d --supports--> 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":"8a84a77b9b9539c7a62e3eee0ffa4bd38b0baef48af1c1e8af7000b4abdb3e02","schema":"livegraph.boolean-propagation.v1","spec_hash":"23624cc814bc585ebde28cc28f1e248d38768da505b2ea6be0fecf0e4808437d","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-29af2a682aaf40fb3600","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --supports--> v05\n- v05 --supports--> v09\n- v09 --supports--> v16\n- v16 --refutes--> v0d\n- v0d --refutes--> v0c\n- v0c --supports--> v14\n- v14 --refutes--> v02\n- v02 --refutes--> v10\n- v10 --supports--> v0f\n- v0f --supports--> v00\n- v00 --refutes--> v03\n- v03 --refutes--> v0a\n- v0a --supports--> v06\n- v06 --refutes--> v04\n- v04 --supports--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v15\n- v03 --refutes--> v08\n- v0f --refutes--> v0b\n- v0f --refutes--> v12\n- v14 --refutes--> v11\n- v14 --supports--> v13\n- v16 --refutes--> v0e\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":"d0f782ec99e03fa5a60cb89639e40f0f1b1d7dcf3bc71d7dbf3c153372682bc7","schema":"livegraph.boolean-propagation.v1","spec_hash":"29af2a682aaf40fb36004d2dc4b27383df9c08c426c6e443c89f740562c76d60","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-ab38d3c2210d3d890e1c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --refutes--> v01\n- v01 --refutes--> v09\n- v09 --supports--> v0b\n- v0b --refutes--> v08\n- v08 --refutes--> v14\n- v14 --supports--> v0c\n- v0c --refutes--> v02\n- v02 --supports--> v13\n- v13 --supports--> v12\n- v12 --refutes--> v04\n- v04 --refutes--> v0d\n- v0d --supports--> v07\n- v07 --refutes--> v0e\n- v0e --refutes--> v03\n- v03 --supports--> v16\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v05\n- v02 --refutes--> v15\n- v07 --supports--> v11\n- v0c --supports--> v0f\n- v0d --refutes--> v06\n- v12 --supports--> v10\n- v14 --refutes--> v0a\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":"9176387bd3f0d9ada445d7e49ad460f1bd37ff8dfd4b69ce3c6d25198084ba0a","schema":"livegraph.boolean-propagation.v1","spec_hash":"ab38d3c2210d3d890e1c69ebf39e56bb773455132b9db8b02e1ad6553a0a22a7","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":15,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v16\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-c4ddb7da9dd74a53ac2a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0f --refutes--> v0c\n- v0c --refutes--> v11\n- v11 --refutes--> v0b\n- v0b --supports--> v02\n- v02 --supports--> v0e\n- v0e --refutes--> v0d\n- v0d --refutes--> v12\n- v12 --refutes--> v15\n- v15 --supports--> v08\n- v08 --refutes--> v01\n- v01 --supports--> v0a\n- v0a --supports--> v07\n- v07 --refutes--> v03\n- v03 --refutes--> v00\n- v00 --refutes--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v10\n- v07 --refutes--> v16\n- v08 --supports--> v14\n- v0c --supports--> v06\n- v0d --refutes--> v09\n- v11 --supports--> v13\n- v12 --refutes--> v05\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":"22c233d231bedd4efeea05d96b28e2944587bdf236bae773f62cc8c6822d141b","schema":"livegraph.boolean-propagation.v1","spec_hash":"c4ddb7da9dd74a53ac2a0d47202cc715349c31d8dfb90acd7863870577544549","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-974730b5ac5e68b92755","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v16 --refutes--> v15\n- v15 --refutes--> v10\n- v10 --refutes--> v04\n- v04 --supports--> v14\n- v14 --refutes--> v13\n- v13 --supports--> v02\n- v02 --refutes--> v12\n- v12 --supports--> v11\n- v11 --supports--> v08\n- v08 --refutes--> v0b\n- v0b --refutes--> v05\n- v05 --refutes--> v0a\n- v0a --supports--> v01\n- v01 --supports--> v0d\n- v0d --supports--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v03\n- v04 --refutes--> v07\n- v04 --supports--> v00\n- v0d --supports--> v0e\n- v11 --refutes--> v09\n- v14 --refutes--> v06\n- v16 --supports--> v0c\n\n[VERIFIED ROOT]\nv16 = 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":"3ed40f8a38a50e2210c9570ca17e9fea12cb2e174a554820867c09870a744293","schema":"livegraph.boolean-propagation.v1","spec_hash":"974730b5ac5e68b927551a6ebcc9ad51880bb4ec3494e9f5bf0c2bc8c1244490","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v16\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-819ece14c63c4c97d327","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v0a\n- v0a --refutes--> v08\n- v08 --refutes--> v16\n- v16 --refutes--> v06\n- v06 --supports--> v04\n- v04 --supports--> v0d\n- v0d --refutes--> v14\n- v14 --supports--> v0e\n- v0e --supports--> v05\n- v05 --refutes--> v0c\n- v0c --supports--> v13\n- v13 --refutes--> v0b\n- v0b --refutes--> v07\n- v07 --refutes--> v11\n- v11 --supports--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v02\n- v08 --refutes--> v10\n- v0c --refutes--> v12\n- v13 --refutes--> v15\n- v13 --supports--> v03\n- v14 --supports--> v01\n- v14 --supports--> v09\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":"4820f995f3f2050d497d1d0d28a5b3274a0c93cdb7532fb868bafe9342e819be","schema":"livegraph.boolean-propagation.v1","spec_hash":"819ece14c63c4c97d3276c5af328e071e2a5f8fdd05d3255f772b96747c9ba85","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-b7ccbac618488c641ea2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0f --supports--> v02\n- v02 --refutes--> v04\n- v04 --supports--> v0c\n- v0c --supports--> v03\n- v03 --supports--> v0e\n- v0e --supports--> v09\n- v09 --refutes--> v0b\n- v0b --refutes--> v08\n- v08 --refutes--> v15\n- v15 --supports--> v05\n- v05 --refutes--> v00\n- v00 --supports--> v11\n- v11 --supports--> v12\n- v12 --supports--> v0d\n- v0d --refutes--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v14\n- v09 --supports--> v0a\n- v0b --supports--> v10\n- v0d --refutes--> v07\n- v15 --refutes--> v13\n- v15 --refutes--> v16\n- v15 --supports--> v01\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":"cc72dbf1630e76f7214a7fb55102e94f7ea84a1f1b97aa73e6488fa5dad6fb99","schema":"livegraph.boolean-propagation.v1","spec_hash":"b7ccbac618488c641ea2b786962b1de8d9e0e0cf3ed025d5fe5ddb3e5ee1fb7f","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":15,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-121a9d82e6bb76480ddb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --refutes--> v07\n- v07 --refutes--> v13\n- v13 --refutes--> v11\n- v11 --supports--> v0f\n- v0f --supports--> v04\n- v04 --refutes--> v12\n- v12 --refutes--> v0e\n- v0e --refutes--> v00\n- v00 --supports--> v08\n- v08 --refutes--> v16\n- v16 --supports--> v0b\n- v0b --supports--> v01\n- v01 --refutes--> v02\n- v02 --supports--> v0d\n- v0d --refutes--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v07 --supports--> v05\n- v0b --supports--> v0c\n- v0e --refutes--> v15\n- v11 --refutes--> v14\n- v12 --refutes--> v06\n- v13 --refutes--> v03\n- v16 --supports--> 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":"04b8c2f61ce5629ae25b0761337c71916f7f4a3ac836bdc33f8c75f1a6493068","schema":"livegraph.boolean-propagation.v1","spec_hash":"121a9d82e6bb76480ddb1558b83d09c0951004884c950467071bf3a3b64671da","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-cf71176260115b1965de","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v11 --supports--> v05\n- v05 --refutes--> v10\n- v10 --refutes--> v0a\n- v0a --refutes--> v06\n- v06 --supports--> v02\n- v02 --supports--> v0e\n- v0e --refutes--> v0f\n- v0f --refutes--> v0b\n- v0b --supports--> v08\n- v08 --refutes--> v04\n- v04 --supports--> v0c\n- v0c --supports--> v07\n- v07 --refutes--> v03\n- v03 --supports--> v14\n- v14 --supports--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v16\n- v03 --refutes--> v15\n- v05 --supports--> v0d\n- v06 --supports--> v13\n- v08 --refutes--> v00\n- v0f --refutes--> v09\n- v11 --refutes--> v12\n\n[VERIFIED ROOT]\nv11 = 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":"edb1c4fac82d4c70a517ec2136ffe134da42190d6781de4093ecf374d3764886","schema":"livegraph.boolean-propagation.v1","spec_hash":"cf71176260115b1965de5fb0127c7f5f10a3e90b325024cfacb2a53ee9fee61b","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v14\"}],\"hop\":15,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-c36a122997f182a04f11","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v11 --refutes--> v03\n- v03 --refutes--> v0e\n- v0e --supports--> v07\n- v07 --supports--> v06\n- v06 --refutes--> v0f\n- v0f --supports--> v12\n- v12 --supports--> v05\n- v05 --refutes--> v00\n- v00 --refutes--> v09\n- v09 --refutes--> v02\n- v02 --refutes--> v16\n- v16 --refutes--> v15\n- v15 --refutes--> v10\n- v10 --refutes--> v13\n- v13 --supports--> v0c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v0b\n- v05 --supports--> v04\n- v07 --refutes--> v08\n- v09 --refutes--> v0a\n- v10 --supports--> v14\n- v11 --supports--> v0d\n- v13 --refutes--> v01\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":"b95a279289bc1d37c1ac5f5e9e67cc64c8814adfe482723df98d31ac4b192b07","schema":"livegraph.boolean-propagation.v1","spec_hash":"c36a122997f182a04f1118ec69b0aec93af063ef5f2fb7a23210483f618aee35","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":15,"item_id":"lggrpo1-cffe56c604ffa71c6ee2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v12 --refutes--> v07\n- v07 --refutes--> v0d\n- v0d --supports--> v16\n- v16 --supports--> v03\n- v03 --refutes--> v04\n- v04 --refutes--> v0c\n- v0c --refutes--> v0f\n- v0f --supports--> v0e\n- v0e --refutes--> v14\n- v14 --supports--> v00\n- v00 --refutes--> v02\n- v02 --refutes--> v0a\n- v0a --refutes--> v15\n- v15 --supports--> v13\n- v13 --refutes--> v10\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v11\n- v07 --supports--> v0b\n- v0c --supports--> v08\n- v0d --refutes--> v06\n- v0d --refutes--> v09\n- v13 --refutes--> v01\n- v15 --refutes--> v05\n\n[VERIFIED ROOT]\nv12 = 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":"6452e35d2b18e0c672ef0231bd1c80b64a05366aa851cb99adc1c4a94c0d0e88","schema":"livegraph.boolean-propagation.v1","spec_hash":"cffe56c604ffa71c6ee29e18071964f2b6ba54c5129c11dbb479b9981f71cec5","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v12\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":15,"item_id":"lggrpo1-b2fbba8205e93e9f210f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0d --refutes--> v16\n- v16 --supports--> v11\n- v11 --refutes--> v0e\n- v0e --refutes--> v0a\n- v0a --supports--> v05\n- v05 --supports--> v09\n- v09 --refutes--> v01\n- v01 --supports--> v15\n- v15 --refutes--> v02\n- v02 --refutes--> v07\n- v07 --supports--> v08\n- v08 --supports--> v06\n- v06 --refutes--> v03\n- v03 --refutes--> v14\n- v14 --supports--> v10\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v0f\n- v06 --refutes--> v0b\n- v08 --refutes--> v12\n- v11 --refutes--> v00\n- v11 --refutes--> v04\n- v14 --refutes--> v0c\n- v15 --refutes--> v13\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":"b2d680e1a83a080be93763c2de88edfd8630ab934f30d07da30b25189493c798","schema":"livegraph.boolean-propagation.v1","spec_hash":"b2fbba8205e93e9f210f790c3c19a5a793ffad6bdf2c4e140411a37a02636056","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":15,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-3e4e011bb47f880b294c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --refutes--> v08\n- v08 --refutes--> v09\n- v09 --refutes--> v0c\n- v0c --supports--> v11\n- v11 --refutes--> v01\n- v01 --supports--> v02\n- v02 --refutes--> v12\n- v12 --supports--> v03\n- v03 --refutes--> v07\n- v07 --refutes--> v0e\n- v0e --refutes--> v04\n- v04 --supports--> v06\n- v06 --refutes--> v0f\n- v0f --supports--> v18\n- v18 --supports--> v05\n- v05 --refutes--> v15\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v17\n- v02 --refutes--> v10\n- v02 --supports--> v16\n- v04 --supports--> v0d\n- v0e --supports--> v00\n- v0e --supports--> v14\n- v11 --refutes--> v0b\n- v18 --supports--> v13\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":"5fc157aaffed91edaa87fd17e478620aa86a21f29d3d34abc5a6a3b13c2f8530","schema":"livegraph.boolean-propagation.v1","spec_hash":"3e4e011bb47f880b294cdac0d20e2314f367cf68c0eb7e969f36e024f9ebbb6c","spec_json":"{\"edges\":[{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-7dea82e0dacf4ce42c27","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v17 --refutes--> v11\n- v11 --supports--> v16\n- v16 --refutes--> v18\n- v18 --supports--> v07\n- v07 --refutes--> v09\n- v09 --supports--> v01\n- v01 --refutes--> v0d\n- v0d --refutes--> v0e\n- v0e --refutes--> v10\n- v10 --refutes--> v00\n- v00 --supports--> v13\n- v13 --supports--> v0f\n- v0f --supports--> v0b\n- v0b --supports--> v04\n- v04 --supports--> v06\n- v06 --supports--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v14\n- v06 --refutes--> v0a\n- v09 --refutes--> v05\n- v09 --refutes--> v15\n- v0d --refutes--> v0c\n- v0e --supports--> v03\n- v10 --refutes--> v08\n- v16 --supports--> v12\n\n[VERIFIED ROOT]\nv17 = 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":"1273d410443b124b2e1bf27b8dc20a1131d0cca617173a8fe13a4672399ec63b","schema":"livegraph.boolean-propagation.v1","spec_hash":"7dea82e0dacf4ce42c2798d64fedab4aea2d229d21b08c2815a256d444b2ee9c","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v17\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-8898d2e5a759ce9420f4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --supports--> v0c\n- v0c --supports--> v12\n- v12 --supports--> v09\n- v09 --supports--> v17\n- v17 --supports--> v11\n- v11 --supports--> v15\n- v15 --refutes--> v08\n- v08 --refutes--> v06\n- v06 --refutes--> v14\n- v14 --refutes--> v03\n- v03 --refutes--> v00\n- v00 --refutes--> v0f\n- v0f --supports--> v05\n- v05 --supports--> v0d\n- v0d --refutes--> v16\n- v16 --supports--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v10\n- v06 --refutes--> v01\n- v08 --refutes--> v18\n- v0f --supports--> v0e\n- v14 --supports--> v0b\n- v15 --refutes--> v04\n- v16 --refutes--> v13\n- v17 --refutes--> v0a\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":"0a01f473254ec9eeb8de09d64c952d97215487dc013fa81750956fbf626c58e9","schema":"livegraph.boolean-propagation.v1","spec_hash":"8898d2e5a759ce9420f4ad84015ed3a971042be3b805a8d397845c078b92cd9e","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-6e26b3a0baa52d3efb6c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0e --supports--> v09\n- v09 --supports--> v00\n- v00 --supports--> v12\n- v12 --supports--> v0b\n- v0b --refutes--> v03\n- v03 --refutes--> v04\n- v04 --refutes--> v01\n- v01 --refutes--> v18\n- v18 --supports--> v05\n- v05 --refutes--> v08\n- v08 --supports--> v06\n- v06 --supports--> v14\n- v14 --refutes--> v0f\n- v0f --supports--> v0c\n- v0c --supports--> v0d\n- v0d --refutes--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v15\n- v01 --refutes--> v0a\n- v03 --refutes--> v07\n- v03 --refutes--> v13\n- v05 --supports--> v16\n- v08 --supports--> v17\n- v0d --refutes--> v11\n- v18 --refutes--> v10\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":"cc753248276080821ed8f1bcd5c3175276f16cd338c0632f3a61e8ef7704cc58","schema":"livegraph.boolean-propagation.v1","spec_hash":"6e26b3a0baa52d3efb6c6b83475c87e5effe691313c0f8f239d1d45df98e6436","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-6868240ed7f4bd70452b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v14 --refutes--> v07\n- v07 --refutes--> v0d\n- v0d --refutes--> v04\n- v04 --refutes--> v16\n- v16 --supports--> v0b\n- v0b --supports--> v01\n- v01 --refutes--> v08\n- v08 --refutes--> v0e\n- v0e --refutes--> v12\n- v12 --supports--> v09\n- v09 --supports--> v13\n- v13 --supports--> v06\n- v06 --refutes--> v03\n- v03 --refutes--> v00\n- v00 --supports--> v05\n- v05 --refutes--> v11\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0c\n- v01 --refutes--> v18\n- v01 --supports--> v0a\n- v05 --refutes--> v15\n- v07 --supports--> v02\n- v0b --supports--> v10\n- v0d --refutes--> v17\n- v0e --supports--> v0f\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":"cccf8bd264feba10a0b7290c0e3bcbed89c2ef9f3a6dc4df72db42403a48ee31","schema":"livegraph.boolean-propagation.v1","spec_hash":"6868240ed7f4bd70452b24fb0fac1cd8e457e57976920918d163aa10bffc62c0","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":16,\"root\":\"v14\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-08185d8110e57153cff9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --refutes--> v09\n- v09 --supports--> v03\n- v03 --supports--> v06\n- v06 --refutes--> v12\n- v12 --supports--> v0a\n- v0a --supports--> v0f\n- v0f --refutes--> v02\n- v02 --refutes--> v15\n- v15 --refutes--> v0d\n- v0d --refutes--> v0b\n- v0b --supports--> v17\n- v17 --refutes--> v13\n- v13 --refutes--> v11\n- v11 --refutes--> v01\n- v01 --refutes--> v14\n- v14 --refutes--> v16\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --supports--> v18\n- v07 --supports--> v04\n- v0a --refutes--> v05\n- v0a --supports--> v10\n- v0d --refutes--> v0c\n- v0f --refutes--> v00\n- v15 --refutes--> v0e\n- v15 --supports--> v08\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":"993c550e5ac85a175a7c9b15f50bc5829b109e52976864ec73de1bc85646dea0","schema":"livegraph.boolean-propagation.v1","spec_hash":"08185d8110e57153cff9d7200802cab3ab3fb119d9f9066dcfb71ce55f95cbe0","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v16\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-4b88fbd9682a4071906e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v14 --refutes--> v0a\n- v0a --supports--> v11\n- v11 --refutes--> v08\n- v08 --supports--> v05\n- v05 --supports--> v12\n- v12 --supports--> v0b\n- v0b --refutes--> v06\n- v06 --supports--> v15\n- v15 --supports--> v00\n- v00 --refutes--> v16\n- v16 --refutes--> v0d\n- v0d --supports--> v04\n- v04 --supports--> v0c\n- v0c --supports--> v17\n- v17 --refutes--> v02\n- v02 --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0f\n- v04 --refutes--> v10\n- v06 --refutes--> v07\n- v0c --refutes--> v09\n- v0d --refutes--> v13\n- v11 --refutes--> v0e\n- v11 --supports--> v18\n- v12 --supports--> v01\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":"bf5be27e564d998d909b9226f0ed37d6c33af3f9b0e774b8edec0e7a90a617dc","schema":"livegraph.boolean-propagation.v1","spec_hash":"4b88fbd9682a4071906eaba92eab1de0d381619fca27fa56baca0ef4e30db385","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v14\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-152b68f4a3a6d6536090","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v16 --refutes--> v0b\n- v0b --refutes--> v10\n- v10 --refutes--> v06\n- v06 --supports--> v08\n- v08 --refutes--> v18\n- v18 --supports--> v07\n- v07 --supports--> v0a\n- v0a --refutes--> v09\n- v09 --supports--> v11\n- v11 --refutes--> v0d\n- v0d --refutes--> v01\n- v01 --supports--> v04\n- v04 --supports--> v14\n- v14 --supports--> v0e\n- v0e --refutes--> v00\n- v00 --refutes--> v13\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v12\n- v01 --refutes--> v17\n- v04 --refutes--> v03\n- v08 --refutes--> v02\n- v09 --refutes--> v0f\n- v0b --supports--> v0c\n- v0d --supports--> v15\n- v0e --supports--> v05\n\n[VERIFIED ROOT]\nv16 = 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":"d9210988115c9178f6ca5f3ec7ce7f267161a7ecae89b4015386176db87c6e56","schema":"livegraph.boolean-propagation.v1","spec_hash":"152b68f4a3a6d65360904d3ee222fa659a89b1ed279ecffcfefac89568cd3d44","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v16\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-07c218df8704f353b8c7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0f --supports--> v14\n- v14 --refutes--> v0e\n- v0e --supports--> v01\n- v01 --refutes--> v16\n- v16 --supports--> v13\n- v13 --supports--> v06\n- v06 --refutes--> v05\n- v05 --refutes--> v07\n- v07 --supports--> v0c\n- v0c --refutes--> v08\n- v08 --supports--> v03\n- v03 --refutes--> v12\n- v12 --refutes--> v09\n- v09 --supports--> v0b\n- v0b --refutes--> v0a\n- v0a --refutes--> v10\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v07 --supports--> v0d\n- v07 --supports--> v18\n- v0a --refutes--> v00\n- v0b --refutes--> v17\n- v0c --refutes--> v02\n- v0c --supports--> v04\n- v13 --refutes--> v15\n- v14 --supports--> v11\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":"3542137b75e673dc7098e9a08473a4ebacbf63ddd9ddd60d9ab8147025f2bd94","schema":"livegraph.boolean-propagation.v1","spec_hash":"07c218df8704f353b8c7a3ab872ac753bc2df5ca6aa2353f8347e0680dfd5bfa","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v16\"}],\"hop\":16,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-7db8a90a7121da48b1d1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0f --refutes--> v0a\n- v0a --supports--> v02\n- v02 --supports--> v04\n- v04 --supports--> v10\n- v10 --supports--> v15\n- v15 --refutes--> v14\n- v14 --supports--> v0c\n- v0c --supports--> v00\n- v00 --refutes--> v18\n- v18 --supports--> v12\n- v12 --supports--> v09\n- v09 --supports--> v07\n- v07 --refutes--> v05\n- v05 --supports--> v06\n- v06 --supports--> v11\n- v11 --supports--> v17\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v13\n- v04 --supports--> v01\n- v07 --supports--> v0e\n- v09 --refutes--> v0d\n- v09 --supports--> v16\n- v0c --supports--> v0b\n- v11 --refutes--> v08\n- v15 --refutes--> v03\n\n[VERIFIED ROOT]\nv0f = 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":"ac101a46bd46abad5a2081f3024c820e8d9e782a69bcda9816820e1e1f7f137c","schema":"livegraph.boolean-propagation.v1","spec_hash":"7db8a90a7121da48b1d1ceed5129b302017f603f395ea78b4f7cda110f51146a","spec_json":"{\"edges\":[{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v17\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-861cac9405616c9bc250","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0d --supports--> v0c\n- v0c --supports--> v10\n- v10 --supports--> v05\n- v05 --supports--> v17\n- v17 --refutes--> v00\n- v00 --refutes--> v0e\n- v0e --supports--> v07\n- v07 --supports--> v11\n- v11 --refutes--> v06\n- v06 --supports--> v12\n- v12 --supports--> v09\n- v09 --refutes--> v0b\n- v0b --refutes--> v0f\n- v0f --supports--> v18\n- v18 --refutes--> v13\n- v13 --refutes--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v04\n- v07 --refutes--> v15\n- v0b --refutes--> v01\n- v0e --supports--> v02\n- v17 --refutes--> v14\n- v17 --supports--> v16\n- v18 --refutes--> v08\n- v18 --supports--> v03\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":"13c58642a4af0cdc439d3184f342bfb1792a0658ad97853aa3ed0f4f5afd05c0","schema":"livegraph.boolean-propagation.v1","spec_hash":"861cac9405616c9bc250ef576a75b8c68757da872b2e89256cab6da9f6b59349","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-bb50ba713587a8a10551","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v16 --supports--> v15\n- v15 --supports--> v0e\n- v0e --refutes--> v01\n- v01 --refutes--> v0a\n- v0a --refutes--> v0b\n- v0b --refutes--> v0f\n- v0f --supports--> v03\n- v03 --supports--> v0c\n- v0c --refutes--> v08\n- v08 --supports--> v12\n- v12 --refutes--> v07\n- v07 --supports--> v17\n- v17 --supports--> v02\n- v02 --supports--> v11\n- v11 --supports--> v05\n- v05 --supports--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v18\n- v03 --refutes--> v13\n- v07 --refutes--> v00\n- v0c --supports--> v04\n- v11 --refutes--> v10\n- v12 --supports--> v09\n- v15 --supports--> v06\n- v16 --refutes--> v14\n\n[VERIFIED ROOT]\nv16 = 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":"51cc4242b0f8898b7001695f17b2bf94b89e9b511322cb6a6df7c08bfaea1973","schema":"livegraph.boolean-propagation.v1","spec_hash":"bb50ba713587a8a105511fd79d01afafa80ff644aa864a7b1a5a0952bc7422d2","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v16\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-8264fb4439947a3d8eb4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --refutes--> v0e\n- v0e --supports--> v02\n- v02 --refutes--> v01\n- v01 --refutes--> v0b\n- v0b --refutes--> v00\n- v00 --refutes--> v11\n- v11 --refutes--> v06\n- v06 --refutes--> v14\n- v14 --supports--> v07\n- v07 --refutes--> v05\n- v05 --refutes--> v12\n- v12 --supports--> v10\n- v10 --supports--> v0a\n- v0a --supports--> v16\n- v16 --refutes--> v09\n- v09 --refutes--> v18\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v03\n- v06 --supports--> v13\n- v07 --refutes--> v17\n- v0a --supports--> v15\n- v0c --refutes--> v0f\n- v0c --supports--> v0d\n- v10 --refutes--> v08\n- v12 --supports--> v04\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":"ac4fc84090ee0639f044c937ca0632cda0caa0644865dc05bcbfef215fd6ddf4","schema":"livegraph.boolean-propagation.v1","spec_hash":"8264fb4439947a3d8eb4351347222d72ee3ac7b070576e5b14624c7f89d2703d","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":16,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v18\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-546d2376daf1ce876e2b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0d --refutes--> v0b\n- v0b --refutes--> v0c\n- v0c --refutes--> v0f\n- v0f --refutes--> v14\n- v14 --supports--> v11\n- v11 --supports--> v05\n- v05 --refutes--> v09\n- v09 --supports--> v01\n- v01 --supports--> v15\n- v15 --refutes--> v00\n- v00 --refutes--> v08\n- v08 --supports--> v07\n- v07 --refutes--> v18\n- v18 --supports--> v12\n- v12 --supports--> v03\n- v03 --refutes--> v10\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v13\n- v00 --supports--> v16\n- v05 --supports--> v06\n- v09 --supports--> v04\n- v0b --supports--> v02\n- v0d --refutes--> v0e\n- v11 --refutes--> v17\n- v14 --supports--> v0a\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":"4c51b12e401e652ce44e6905e354425ca6c1e6f2d5a98d9dcbebe3a8b7379e81","schema":"livegraph.boolean-propagation.v1","spec_hash":"546d2376daf1ce876e2bf4e351e513d38e8396fb99a2111bec00e64867ba1449","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-95281164bfc0149ae56d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v18 --supports--> v00\n- v00 --supports--> v03\n- v03 --supports--> v04\n- v04 --refutes--> v08\n- v08 --refutes--> v06\n- v06 --refutes--> v17\n- v17 --supports--> v0e\n- v0e --supports--> v0b\n- v0b --supports--> v02\n- v02 --refutes--> v0f\n- v0f --supports--> v0c\n- v0c --refutes--> v07\n- v07 --supports--> v12\n- v12 --refutes--> v14\n- v14 --supports--> v0a\n- v0a --refutes--> v11\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v16\n- v07 --refutes--> v15\n- v08 --refutes--> v01\n- v08 --refutes--> v09\n- v08 --supports--> v13\n- v0f --refutes--> v05\n- v12 --refutes--> v10\n- v12 --supports--> v0d\n\n[VERIFIED ROOT]\nv18 = 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":"41e20be435cb134d379dade49647b6a59c0a4e1fae721fb0a36fae943a4193df","schema":"livegraph.boolean-propagation.v1","spec_hash":"95281164bfc0149ae56d5e6af2975f34710dfad5437264836b14946331d8196d","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v18\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-f4fcc630c972fa37d41d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0e --refutes--> v07\n- v07 --supports--> v01\n- v01 --supports--> v08\n- v08 --refutes--> v04\n- v04 --supports--> v14\n- v14 --supports--> v0f\n- v0f --supports--> v18\n- v18 --refutes--> v17\n- v17 --supports--> v0b\n- v0b --supports--> v09\n- v09 --supports--> v10\n- v10 --supports--> v16\n- v16 --supports--> v11\n- v11 --supports--> v0d\n- v0d --refutes--> v0a\n- v0a --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v07 --supports--> v06\n- v08 --refutes--> v05\n- v09 --supports--> v03\n- v0d --refutes--> v0c\n- v0f --refutes--> v15\n- v10 --refutes--> v02\n- v14 --supports--> v13\n- v16 --supports--> v12\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":"a5cf7c9d5736152b022f212aa0e6135717de4a33a1e371342fad72e82bd098ac","schema":"livegraph.boolean-propagation.v1","spec_hash":"f4fcc630c972fa37d41db566389202a103da7bfd3dff5659d42f714231e1c917","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-0b79815b68ade4337adf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --refutes--> v03\n- v03 --supports--> v12\n- v12 --refutes--> v09\n- v09 --refutes--> v05\n- v05 --refutes--> v11\n- v11 --supports--> v10\n- v10 --refutes--> v0d\n- v0d --supports--> v07\n- v07 --supports--> v02\n- v02 --refutes--> v01\n- v01 --refutes--> v0f\n- v0f --supports--> v06\n- v06 --refutes--> v17\n- v17 --refutes--> v15\n- v15 --refutes--> v16\n- v16 --supports--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v00\n- v05 --refutes--> v0c\n- v09 --refutes--> v14\n- v09 --refutes--> v18\n- v0a --refutes--> v13\n- v0d --supports--> v0e\n- v0f --refutes--> v04\n- v11 --supports--> v08\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":"cfa0dc72651edb34f7e4af0f1a6b2686ea52eefab93262b488bc58ccfdafdf8b","schema":"livegraph.boolean-propagation.v1","spec_hash":"0b79815b68ade4337adf23497e58918735f317ef86f6e53a980d9803d9f7ea98","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-82f9fe66417ebb649170","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --supports--> v0c\n- v0c --supports--> v0b\n- v0b --refutes--> v05\n- v05 --supports--> v00\n- v00 --refutes--> v12\n- v12 --supports--> v13\n- v13 --supports--> v15\n- v15 --supports--> v17\n- v17 --refutes--> v09\n- v09 --refutes--> v16\n- v16 --supports--> v0e\n- v0e --refutes--> v01\n- v01 --refutes--> v04\n- v04 --refutes--> v0a\n- v0a --refutes--> v10\n- v10 --supports--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v14\n- v06 --supports--> v0d\n- v0c --refutes--> v11\n- v12 --refutes--> v18\n- v13 --supports--> v08\n- v16 --refutes--> v07\n- v16 --supports--> v03\n- v17 --supports--> v0f\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":"82101cddad741424252351b6cec52a567d7ca01837443dae976cdc0e3863802f","schema":"livegraph.boolean-propagation.v1","spec_hash":"82f9fe66417ebb649170a546200448b410e7205d6132547a949d6946d7fb7623","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-535e0fece3bfd746e55b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --supports--> v0f\n- v0f --supports--> v13\n- v13 --supports--> v12\n- v12 --refutes--> v02\n- v02 --refutes--> v14\n- v14 --refutes--> v01\n- v01 --refutes--> v04\n- v04 --refutes--> v0b\n- v0b --refutes--> v10\n- v10 --supports--> v05\n- v05 --supports--> v08\n- v08 --refutes--> v16\n- v16 --refutes--> v0c\n- v0c --supports--> v0e\n- v0e --refutes--> v0a\n- v0a --refutes--> v18\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v03\n- v02 --supports--> v17\n- v05 --supports--> v09\n- v08 --supports--> v0d\n- v0c --supports--> v15\n- v10 --refutes--> v11\n- v12 --supports--> v00\n- v13 --supports--> v06\n\n[VERIFIED ROOT]\nv07 = 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":"a2f32bb5d33b13ec1e0505209c69f7a8b631bd76f6fa44998e30015e8adcfe5e","schema":"livegraph.boolean-propagation.v1","spec_hash":"535e0fece3bfd746e55ba096d75021dcc65d6930dbf4a0f3ed7a072dc12ad9b1","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":16,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v18\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-64e0c9e51864c9b988f3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --supports--> v18\n- v18 --supports--> v10\n- v10 --refutes--> v14\n- v14 --supports--> v08\n- v08 --refutes--> v02\n- v02 --refutes--> v0c\n- v0c --refutes--> v0e\n- v0e --supports--> v03\n- v03 --refutes--> v01\n- v01 --refutes--> v0d\n- v0d --supports--> v07\n- v07 --supports--> v13\n- v13 --refutes--> v15\n- v15 --refutes--> v09\n- v09 --refutes--> v06\n- v06 --refutes--> v11\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v07 --refutes--> v0f\n- v08 --supports--> v12\n- v08 --supports--> v17\n- v09 --refutes--> v05\n- v09 --refutes--> v0b\n- v0c --supports--> v04\n- v0e --refutes--> v00\n- v0e --supports--> v16\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":"e11a0e3f38d18ef853ff5dc95516b0cc0c4459757fdcc1c47e380c02050cec88","schema":"livegraph.boolean-propagation.v1","spec_hash":"64e0c9e51864c9b988f35e2a7f3556cf4f3be6572d5108d37513f459508357af","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-5a063e545daa47033666","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0f --refutes--> v06\n- v06 --supports--> v10\n- v10 --supports--> v16\n- v16 --refutes--> v17\n- v17 --refutes--> v05\n- v05 --supports--> v01\n- v01 --supports--> v18\n- v18 --refutes--> v03\n- v03 --refutes--> v02\n- v02 --supports--> v0d\n- v0d --refutes--> v00\n- v00 --supports--> v04\n- v04 --refutes--> v0a\n- v0a --supports--> v09\n- v09 --refutes--> v0c\n- v0c --refutes--> v12\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v13\n- v01 --supports--> v0b\n- v02 --refutes--> v08\n- v02 --supports--> v07\n- v02 --supports--> v14\n- v04 --supports--> v11\n- v0f --refutes--> v0e\n- v10 --refutes--> v15\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":"8461222ef47e4f4586551a7c12a10b76d77d48f16d24b85247525fee884bb49b","schema":"livegraph.boolean-propagation.v1","spec_hash":"5a063e545daa470336666223b6f50cd77cb7ba00cc8e1f3dfac08e57426e9041","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-20ff8b301e5629ef99da","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v08\n- v08 --supports--> v0a\n- v0a --refutes--> v10\n- v10 --refutes--> v03\n- v03 --refutes--> v18\n- v18 --refutes--> v06\n- v06 --supports--> v02\n- v02 --supports--> v09\n- v09 --supports--> v17\n- v17 --refutes--> v05\n- v05 --supports--> v0e\n- v0e --supports--> v16\n- v16 --refutes--> v0c\n- v0c --refutes--> v0f\n- v0f --supports--> v14\n- v14 --supports--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v11\n- v05 --refutes--> v12\n- v08 --refutes--> v13\n- v09 --refutes--> v04\n- v10 --refutes--> v0b\n- v16 --refutes--> v0d\n- v17 --refutes--> v15\n- v17 --supports--> v07\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":"24854a95e3f9f266c28f30a187a6cce7410948b56f04f1fdb5130c4a83497a19","schema":"livegraph.boolean-propagation.v1","spec_hash":"20ff8b301e5629ef99daec5a3e52597e453138ee4e4be099522ec3eb1c989d71","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-263b12edd3dc24de1228","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v18 --supports--> v06\n- v06 --supports--> v00\n- v00 --refutes--> v0c\n- v0c --supports--> v09\n- v09 --refutes--> v11\n- v11 --supports--> v08\n- v08 --refutes--> v05\n- v05 --refutes--> v13\n- v13 --refutes--> v0e\n- v0e --refutes--> v12\n- v12 --refutes--> v10\n- v10 --supports--> v16\n- v16 --refutes--> v0f\n- v0f --supports--> v07\n- v07 --supports--> v01\n- v01 --refutes--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v14\n- v06 --refutes--> v04\n- v0c --refutes--> v02\n- v11 --refutes--> v0d\n- v13 --refutes--> v03\n- v13 --supports--> v17\n- v16 --refutes--> v0a\n- v18 --refutes--> v15\n\n[VERIFIED ROOT]\nv18 = 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":"beeb270b6257658da79034076bb915082f92da71ab84c066cddc69cbf77b508a","schema":"livegraph.boolean-propagation.v1","spec_hash":"263b12edd3dc24de12284212ddf1599674143dd9a1c3d163db5ee2d1799d2b04","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v18\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-c0852d4072dbe3a60975","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v18 --refutes--> v07\n- v07 --refutes--> v09\n- v09 --supports--> v0e\n- v0e --refutes--> v00\n- v00 --supports--> v08\n- v08 --refutes--> v11\n- v11 --refutes--> v0b\n- v0b --refutes--> v17\n- v17 --refutes--> v0f\n- v0f --supports--> v16\n- v16 --refutes--> v03\n- v03 --refutes--> v06\n- v06 --supports--> v0d\n- v0d --refutes--> v0a\n- v0a --refutes--> v02\n- v02 --supports--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v12\n- v07 --refutes--> v14\n- v09 --supports--> v04\n- v09 --supports--> v15\n- v0a --supports--> v0c\n- v0b --refutes--> v13\n- v0d --refutes--> v01\n- v16 --refutes--> v10\n\n[VERIFIED ROOT]\nv18 = 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":"c975bd8324e74a189569b1714f99ea5b0711a0c393cb11eb18d1e314941a971f","schema":"livegraph.boolean-propagation.v1","spec_hash":"c0852d4072dbe3a6097546a7f6ba0132aed2aeb23e703cfef76c934fbe7c9ece","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v18\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-ff1d23d699caead335ac","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --refutes--> v0a\n- v0a --supports--> v03\n- v03 --refutes--> v0d\n- v0d --supports--> v14\n- v14 --supports--> v06\n- v06 --refutes--> v00\n- v00 --refutes--> v02\n- v02 --supports--> v0b\n- v0b --refutes--> v0c\n- v0c --supports--> v15\n- v15 --supports--> v08\n- v08 --refutes--> v18\n- v18 --supports--> v01\n- v01 --refutes--> v11\n- v11 --supports--> v10\n- v10 --refutes--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v16\n- v02 --supports--> v17\n- v03 --refutes--> v13\n- v03 --supports--> v0f\n- v05 --supports--> v04\n- v0a --supports--> v07\n- v0a --supports--> v0e\n- v18 --supports--> v12\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":"28ff0756eb81ba8eed8e858a3b721cc25d92660472a153de3db5c95af15e7fa9","schema":"livegraph.boolean-propagation.v1","spec_hash":"ff1d23d699caead335ac9e979d56649ab270410c487f621e00d6985031768ea9","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-7b7dea422edf8f039860","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v10 --supports--> v07\n- v07 --refutes--> v06\n- v06 --refutes--> v08\n- v08 --supports--> v18\n- v18 --supports--> v0a\n- v0a --supports--> v03\n- v03 --refutes--> v13\n- v13 --refutes--> v02\n- v02 --refutes--> v05\n- v05 --refutes--> v04\n- v04 --refutes--> v17\n- v17 --supports--> v0b\n- v0b --supports--> v0e\n- v0e --supports--> v01\n- v01 --refutes--> v16\n- v16 --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v09\n- v03 --refutes--> v14\n- v06 --refutes--> v0c\n- v06 --refutes--> v0f\n- v07 --supports--> v12\n- v0a --refutes--> v15\n- v10 --supports--> v11\n- v18 --supports--> v0d\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":"f6e0cab64af7c37cf3068058e3725ed811e2e1f52f3d5d71974cd3db1fe4c4cc","schema":"livegraph.boolean-propagation.v1","spec_hash":"7b7dea422edf8f039860fc8c73bd27de955a941f519eaa25ffdf806fb06b94bc","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-769cec5fbbd34c66f74d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --supports--> v11\n- v11 --refutes--> v13\n- v13 --refutes--> v0c\n- v0c --supports--> v0a\n- v0a --refutes--> v16\n- v16 --refutes--> v04\n- v04 --refutes--> v05\n- v05 --refutes--> v03\n- v03 --supports--> v08\n- v08 --refutes--> v14\n- v14 --refutes--> v10\n- v10 --supports--> v0f\n- v0f --refutes--> v0e\n- v0e --supports--> v12\n- v12 --refutes--> v15\n- v15 --refutes--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v0d\n- v09 --supports--> v06\n- v0a --supports--> v00\n- v10 --refutes--> v01\n- v13 --supports--> v17\n- v14 --refutes--> v0b\n- v14 --supports--> v18\n- v16 --refutes--> 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":"27500222539b11620264a7a4c806b52ec62bc1e54a5d37c3fc4998d2f7fdcbd9","schema":"livegraph.boolean-propagation.v1","spec_hash":"769cec5fbbd34c66f74d9a00e7778fdc989f1c6f5ba1d520516b64b1d99604b7","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":16,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-f99a8f8cac2b8c514993","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v16 --supports--> v02\n- v02 --refutes--> v06\n- v06 --refutes--> v12\n- v12 --supports--> v0e\n- v0e --refutes--> v0b\n- v0b --supports--> v14\n- v14 --supports--> v13\n- v13 --supports--> v17\n- v17 --supports--> v05\n- v05 --refutes--> v03\n- v03 --supports--> v0c\n- v0c --refutes--> v07\n- v07 --refutes--> v04\n- v04 --refutes--> v00\n- v00 --supports--> v08\n- v08 --refutes--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v0a\n- v05 --refutes--> v10\n- v05 --refutes--> v11\n- v05 --supports--> v15\n- v07 --supports--> v0f\n- v0e --refutes--> v01\n- v14 --supports--> v09\n- v14 --supports--> v18\n\n[VERIFIED ROOT]\nv16 = 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":"b362dad99ea782d8227d956a343eb0ebb327ff94f99aa5677540e86ae27ff983","schema":"livegraph.boolean-propagation.v1","spec_hash":"f99a8f8cac2b8c5149932189a111ae48f5b89cc25fd6aa55d816003370104f5e","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v16\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-c67b9549c3acbf0db4cb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0e --supports--> v11\n- v11 --supports--> v08\n- v08 --refutes--> v0a\n- v0a --supports--> v02\n- v02 --refutes--> v0c\n- v0c --refutes--> v10\n- v10 --supports--> v05\n- v05 --refutes--> v14\n- v14 --refutes--> v18\n- v18 --supports--> v00\n- v00 --refutes--> v04\n- v04 --refutes--> v16\n- v16 --refutes--> v17\n- v17 --refutes--> v07\n- v07 --supports--> v03\n- v03 --refutes--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v0f\n- v04 --supports--> v13\n- v05 --supports--> v0b\n- v08 --refutes--> v01\n- v0e --refutes--> v09\n- v10 --supports--> v12\n- v16 --supports--> v0d\n- v17 --supports--> v15\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":"a826b89bf32cca59f9a78b66f01a7735fdbf658bcef3fd4a217a8b9478c17c50","schema":"livegraph.boolean-propagation.v1","spec_hash":"c67b9549c3acbf0db4cbdeae1d7599f75f87ae1345ce712749aa9742713c2419","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-d77748e15061cae1e1dc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0e --refutes--> v01\n- v01 --supports--> v12\n- v12 --supports--> v08\n- v08 --refutes--> v15\n- v15 --supports--> v06\n- v06 --supports--> v05\n- v05 --refutes--> v0c\n- v0c --refutes--> v11\n- v11 --refutes--> v09\n- v09 --supports--> v02\n- v02 --refutes--> v0b\n- v0b --refutes--> v14\n- v14 --refutes--> v18\n- v18 --supports--> v16\n- v16 --supports--> v0a\n- v0a --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v04\n- v01 --refutes--> v17\n- v05 --refutes--> v07\n- v05 --refutes--> v10\n- v06 --supports--> v00\n- v08 --refutes--> v0d\n- v0b --refutes--> v0f\n- v16 --refutes--> v13\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":"d47d2199823a0ced5a70fb3462e89a569e744940ed52f8ff0b15d26c52d72add","schema":"livegraph.boolean-propagation.v1","spec_hash":"d77748e15061cae1e1dc870845f64d685ae717655bb1a69fd6391fb3b3cbcf7f","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-9bec80c19702a0530c7a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v18 --supports--> v10\n- v10 --refutes--> v0c\n- v0c --refutes--> v07\n- v07 --refutes--> v09\n- v09 --supports--> v04\n- v04 --refutes--> v0a\n- v0a --supports--> v02\n- v02 --supports--> v16\n- v16 --supports--> v14\n- v14 --supports--> v0e\n- v0e --refutes--> v05\n- v05 --refutes--> v01\n- v01 --supports--> v17\n- v17 --refutes--> v08\n- v08 --supports--> v0d\n- v0d --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v06\n- v04 --supports--> v0b\n- v05 --refutes--> v03\n- v09 --refutes--> v0f\n- v09 --supports--> v13\n- v0e --refutes--> v15\n- v17 --supports--> v12\n- v18 --refutes--> v11\n\n[VERIFIED ROOT]\nv18 = 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":"cde14f14d32876f9633d93ef98d44982a0c8132851f7cf6ef65488d11bb489dc","schema":"livegraph.boolean-propagation.v1","spec_hash":"9bec80c19702a0530c7a4d42edf4d2c8ea2d80c3b16d432ded7ae02a026123c6","spec_json":"{\"edges\":[{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v18\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-d88a08b8ec03f855d9c6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --refutes--> v0e\n- v0e --refutes--> v14\n- v14 --supports--> v12\n- v12 --refutes--> v05\n- v05 --refutes--> v0a\n- v0a --refutes--> v16\n- v16 --refutes--> v02\n- v02 --refutes--> v06\n- v06 --refutes--> v01\n- v01 --refutes--> v15\n- v15 --refutes--> v0f\n- v0f --refutes--> v0c\n- v0c --supports--> v17\n- v17 --supports--> v18\n- v18 --supports--> v0d\n- v0d --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --supports--> v11\n- v0e --supports--> v10\n- v0f --supports--> v08\n- v12 --refutes--> v04\n- v12 --supports--> v0b\n- v14 --refutes--> v07\n- v15 --refutes--> v03\n- v17 --supports--> v13\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":"e3b5a919be873487493bd31783d84b1a0fa495c77e5c192d72910071988cffcc","schema":"livegraph.boolean-propagation.v1","spec_hash":"d88a08b8ec03f855d9c6cdd10b6108d3d9d1a36fb20da4425e6aca12ce1d7d57","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-7195288dc70c9617edb3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0e --refutes--> v0f\n- v0f --refutes--> v0d\n- v0d --supports--> v0c\n- v0c --supports--> v02\n- v02 --supports--> v00\n- v00 --supports--> v18\n- v18 --supports--> v09\n- v09 --refutes--> v12\n- v12 --supports--> v06\n- v06 --supports--> v16\n- v16 --supports--> v15\n- v15 --supports--> v04\n- v04 --refutes--> v08\n- v08 --supports--> v03\n- v03 --refutes--> v10\n- v10 --refutes--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v07\n- v00 --supports--> v0a\n- v03 --refutes--> v14\n- v04 --refutes--> v17\n- v08 --supports--> v05\n- v10 --refutes--> v11\n- v18 --refutes--> v01\n- v18 --refutes--> v13\n\n[VERIFIED ROOT]\nv0e = 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":"ec6fb5d64835a7deed5e16aa365e331c5ff5b91f498d8cc4cbe4d4fd1163dc9c","schema":"livegraph.boolean-propagation.v1","spec_hash":"7195288dc70c9617edb33f321a95bd5dc56b7f65ef98b6db0dd549f40255a9ec","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-0bef672e819f26ea8ce1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --supports--> v16\n- v16 --supports--> v07\n- v07 --refutes--> v03\n- v03 --supports--> v11\n- v11 --refutes--> v10\n- v10 --supports--> v02\n- v02 --refutes--> v05\n- v05 --refutes--> v12\n- v12 --refutes--> v0e\n- v0e --supports--> v08\n- v08 --supports--> v04\n- v04 --refutes--> v15\n- v15 --refutes--> v13\n- v13 --supports--> v0d\n- v0d --refutes--> v17\n- v17 --refutes--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v09\n- v05 --supports--> v14\n- v07 --refutes--> v06\n- v10 --refutes--> v0f\n- v11 --refutes--> v18\n- v12 --refutes--> v01\n- v12 --refutes--> v0c\n- v13 --supports--> v00\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":"126f802d100cdbac19a2eae6341e96b7533983d27d7bc8c3be6a8e92008f56b9","schema":"livegraph.boolean-propagation.v1","spec_hash":"0bef672e819f26ea8ce140336cde4bf6e12cb969a3fa1ebd5c91fb9f7c7fc51a","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-30f2c120972681e99e02","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v17 --refutes--> v0a\n- v0a --refutes--> v14\n- v14 --refutes--> v0b\n- v0b --refutes--> v08\n- v08 --supports--> v05\n- v05 --supports--> v0c\n- v0c --refutes--> v06\n- v06 --supports--> v01\n- v01 --supports--> v18\n- v18 --refutes--> v12\n- v12 --supports--> v04\n- v04 --refutes--> v0f\n- v0f --refutes--> v10\n- v10 --refutes--> v07\n- v07 --refutes--> v0d\n- v0d --supports--> v11\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v02\n- v05 --refutes--> v13\n- v05 --supports--> v16\n- v0a --refutes--> v00\n- v0c --refutes--> v0e\n- v10 --refutes--> v03\n- v10 --refutes--> v15\n- v10 --supports--> v09\n\n[VERIFIED ROOT]\nv17 = 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":"56568270fda9f5d8c14dd5b1d798246fdbdc531c9f06566a2544c1b6b670f23e","schema":"livegraph.boolean-propagation.v1","spec_hash":"30f2c120972681e99e02ca7b4aed5a311d229290494eb28703f31c0c6e8309b1","spec_json":"{\"edges\":[{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v17\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-6aee08c7fae3110fa9bc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --refutes--> v07\n- v07 --supports--> v14\n- v14 --supports--> v0f\n- v0f --supports--> v04\n- v04 --refutes--> v12\n- v12 --supports--> v05\n- v05 --supports--> v17\n- v17 --refutes--> v11\n- v11 --supports--> v06\n- v06 --refutes--> v0b\n- v0b --supports--> v02\n- v02 --supports--> v08\n- v08 --refutes--> v09\n- v09 --refutes--> v16\n- v16 --supports--> v03\n- v03 --supports--> v15\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v10\n- v0b --refutes--> v18\n- v0f --refutes--> v0d\n- v0f --supports--> v0c\n- v0f --supports--> v13\n- v12 --supports--> v00\n- v17 --refutes--> v0a\n- v17 --supports--> v0e\n\n[VERIFIED ROOT]\nv01 = 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":"29b5b954fd426a32b2954c9767e4b1f8a179448c2e57be8f17b2ed42fe565c8f","schema":"livegraph.boolean-propagation.v1","spec_hash":"6aee08c7fae3110fa9bc281619b7d8bc4890db7ada18667617db91f32654f2e5","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-5cf29f13558b8c1bd63b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v15 --refutes--> v04\n- v04 --refutes--> v05\n- v05 --supports--> v16\n- v16 --supports--> v01\n- v01 --refutes--> v0f\n- v0f --refutes--> v0a\n- v0a --supports--> v07\n- v07 --refutes--> v13\n- v13 --supports--> v00\n- v00 --refutes--> v10\n- v10 --refutes--> v0b\n- v0b --supports--> v17\n- v17 --supports--> v14\n- v14 --refutes--> v08\n- v08 --supports--> v0c\n- v0c --refutes--> v0e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v0d\n- v04 --supports--> v03\n- v05 --supports--> v18\n- v07 --refutes--> v09\n- v0b --supports--> v02\n- v10 --supports--> v06\n- v15 --refutes--> v11\n- v17 --refutes--> v12\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":"6f4345a9612b0eb3cd7356dcb8f9605e53e4a0886f33e04e80668e3c35a4d663","schema":"livegraph.boolean-propagation.v1","spec_hash":"5cf29f13558b8c1bd63ba15bcb8d258197bc466e932c990db59d9cf9ce95c0ab","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v15\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-2fe5a4d5d8dd1673c0f9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --supports--> v17\n- v17 --supports--> v14\n- v14 --refutes--> v0c\n- v0c --refutes--> v18\n- v18 --refutes--> v00\n- v00 --refutes--> v03\n- v03 --supports--> v06\n- v06 --refutes--> v04\n- v04 --refutes--> v15\n- v15 --supports--> v10\n- v10 --supports--> v0e\n- v0e --refutes--> v02\n- v02 --supports--> v12\n- v12 --refutes--> v0f\n- v0f --refutes--> v16\n- v16 --refutes--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v11\n- v03 --supports--> v09\n- v03 --supports--> v0d\n- v04 --supports--> v0a\n- v04 --supports--> v13\n- v06 --refutes--> v05\n- v0f --supports--> v01\n- v0f --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":"9ee6017fd08bba014253889583513957239ac2f230f4b76445fde590c9e9aa16","schema":"livegraph.boolean-propagation.v1","spec_hash":"2fe5a4d5d8dd1673c0f9d978085687f1afda2639b10a9ef70445c4bac41227a9","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-babf3f5267241de68781","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v10 --supports--> v09\n- v09 --refutes--> v02\n- v02 --supports--> v06\n- v06 --supports--> v14\n- v14 --supports--> v13\n- v13 --refutes--> v04\n- v04 --refutes--> v03\n- v03 --refutes--> v0d\n- v0d --refutes--> v01\n- v01 --supports--> v0e\n- v0e --refutes--> v15\n- v15 --supports--> v0c\n- v0c --supports--> v11\n- v11 --refutes--> v08\n- v08 --refutes--> v12\n- v12 --supports--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v00\n- v08 --refutes--> v16\n- v0c --supports--> v07\n- v0e --supports--> v05\n- v0e --supports--> v0a\n- v13 --refutes--> v18\n- v13 --supports--> v0b\n- v15 --supports--> v17\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":"e7676062ee58cfdfef25b7286e11368702f8672bb0af7edd790ea54fc1d2bb46","schema":"livegraph.boolean-propagation.v1","spec_hash":"babf3f5267241de68781f272acfc11222269cf81211298e5e0dc5fd1bc6472eb","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":16,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-7c6ebd6c18a970b44849","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v10 --refutes--> v05\n- v05 --refutes--> v15\n- v15 --refutes--> v0f\n- v0f --refutes--> v0e\n- v0e --supports--> v14\n- v14 --refutes--> v0b\n- v0b --supports--> v17\n- v17 --refutes--> v09\n- v09 --supports--> v08\n- v08 --supports--> v12\n- v12 --refutes--> v07\n- v07 --supports--> v11\n- v11 --supports--> v02\n- v02 --refutes--> v13\n- v13 --supports--> v0d\n- v0d --supports--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v0c\n- v08 --supports--> v03\n- v08 --supports--> v16\n- v0b --supports--> v18\n- v0f --refutes--> v00\n- v0f --supports--> v04\n- v10 --refutes--> v0a\n- v14 --refutes--> v06\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":"01d5f2209f031045e76fc3e6e264b4297ff21322532d30f15c0d6e50316ff5fa","schema":"livegraph.boolean-propagation.v1","spec_hash":"7c6ebd6c18a970b4484976a7dc8e7caf27f298daa09bc47fa61faffead687dae","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-420a0161ba5354f33a32","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v11 --supports--> v17\n- v17 --supports--> v15\n- v15 --refutes--> v02\n- v02 --refutes--> v0a\n- v0a --supports--> v07\n- v07 --supports--> v03\n- v03 --supports--> v14\n- v14 --supports--> v01\n- v01 --supports--> v0f\n- v0f --supports--> v10\n- v10 --supports--> v0c\n- v0c --supports--> v16\n- v16 --supports--> v0e\n- v0e --refutes--> v12\n- v12 --refutes--> v13\n- v13 --refutes--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v0a --refutes--> v18\n- v0c --refutes--> v09\n- v0c --supports--> v08\n- v0f --supports--> v0b\n- v11 --refutes--> v00\n- v13 --supports--> v04\n- v14 --supports--> v0d\n- v17 --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":"c8002ef81fec0f6d2a5df32692267cf4dbcb7b29dc88cbb8b5dfe997833dc88b","schema":"livegraph.boolean-propagation.v1","spec_hash":"420a0161ba5354f33a32abe253ba5090553f9123e3a7d3b6c6d0a40445c0f938","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-6caca9a3ac4e1321b03a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v13 --supports--> v08\n- v08 --refutes--> v0d\n- v0d --refutes--> v01\n- v01 --supports--> v0b\n- v0b --supports--> v05\n- v05 --refutes--> v02\n- v02 --supports--> v15\n- v15 --supports--> v10\n- v10 --refutes--> v14\n- v14 --supports--> v16\n- v16 --supports--> v0a\n- v0a --refutes--> v07\n- v07 --refutes--> v18\n- v18 --refutes--> v00\n- v00 --refutes--> v06\n- v06 --refutes--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v0f\n- v08 --refutes--> v12\n- v0a --refutes--> v0e\n- v0a --supports--> v04\n- v0d --refutes--> v03\n- v13 --supports--> v11\n- v14 --supports--> v17\n- v15 --refutes--> v0c\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":"3279576ff292d04bfb09d0dd1263c3cd4daa0a1f6532ed928f0badd559867e55","schema":"livegraph.boolean-propagation.v1","spec_hash":"6caca9a3ac4e1321b03acaab7c9393eb45743fd48d878ec23dd1681a16ae9d47","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v13\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-5be9f935b0f7531cc145","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v12 --supports--> v0d\n- v0d --refutes--> v16\n- v16 --supports--> v07\n- v07 --refutes--> v0c\n- v0c --supports--> v17\n- v17 --supports--> v00\n- v00 --supports--> v06\n- v06 --refutes--> v0b\n- v0b --supports--> v08\n- v08 --supports--> v09\n- v09 --supports--> v14\n- v14 --supports--> v02\n- v02 --refutes--> v03\n- v03 --supports--> v05\n- v05 --refutes--> v0e\n- v0e --supports--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v01\n- v07 --refutes--> v15\n- v08 --supports--> v0f\n- v0c --refutes--> v18\n- v0e --refutes--> v10\n- v16 --supports--> v04\n- v16 --supports--> v13\n- v17 --refutes--> v11\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":"f5be1030bcf4e2f67654f8f7ebc57f19b80d43d0d3d81cdc8821fab312ece93f","schema":"livegraph.boolean-propagation.v1","spec_hash":"5be9f935b0f7531cc145056c4e03d131e81c76b19fa9b43f58ebc74a592c8f5f","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-143405612ba5d9ed0e23","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --refutes--> v07\n- v07 --refutes--> v04\n- v04 --supports--> v13\n- v13 --refutes--> v11\n- v11 --refutes--> v01\n- v01 --supports--> v10\n- v10 --supports--> v08\n- v08 --refutes--> v02\n- v02 --refutes--> v03\n- v03 --supports--> v05\n- v05 --supports--> v0b\n- v0b --refutes--> v0c\n- v0c --refutes--> v16\n- v16 --refutes--> v12\n- v12 --refutes--> v14\n- v14 --supports--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v0d\n- v04 --refutes--> v06\n- v04 --refutes--> v18\n- v05 --refutes--> v0a\n- v07 --refutes--> v0f\n- v0b --refutes--> v15\n- v10 --supports--> v0e\n- v12 --refutes--> v17\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":"f67a0c7393e39e7aed065904f53ffc6ca54afbe1982a164a69d8f8fad451c3a0","schema":"livegraph.boolean-propagation.v1","spec_hash":"143405612ba5d9ed0e234de61bd396ff1b82274c4c04643580b2c48a911a105c","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":16,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-cdada920545eac0dfbc1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --supports--> v00\n- v00 --refutes--> v15\n- v15 --supports--> v12\n- v12 --refutes--> v0c\n- v0c --refutes--> v13\n- v13 --supports--> v14\n- v14 --refutes--> v0d\n- v0d --refutes--> v17\n- v17 --refutes--> v10\n- v10 --refutes--> v0e\n- v0e --refutes--> v07\n- v07 --supports--> v0b\n- v0b --refutes--> v09\n- v09 --refutes--> v05\n- v05 --refutes--> v18\n- v18 --refutes--> v16\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v0f\n- v07 --supports--> v0a\n- v09 --supports--> v11\n- v0d --refutes--> v04\n- v0d --refutes--> v08\n- v0e --refutes--> v06\n- v10 --refutes--> v03\n- v17 --refutes--> v02\n\n[VERIFIED ROOT]\nv01 = 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":"8f3f5a5429e6d39f4917c5f7bec5b3f1c6f9aa1fbd3b797e2ca28a25030787f1","schema":"livegraph.boolean-propagation.v1","spec_hash":"cdada920545eac0dfbc1793e43b515978ee076781384ab974d1674cc0cc6000f","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v16\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-079b3558ee68e7f1644d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v12 --refutes--> v00\n- v00 --refutes--> v03\n- v03 --supports--> v09\n- v09 --supports--> v0a\n- v0a --supports--> v10\n- v10 --supports--> v14\n- v14 --refutes--> v05\n- v05 --refutes--> v0c\n- v0c --supports--> v07\n- v07 --refutes--> v0f\n- v0f --supports--> v02\n- v02 --supports--> v16\n- v16 --refutes--> v0d\n- v0d --refutes--> v11\n- v11 --supports--> v15\n- v15 --refutes--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v01\n- v05 --supports--> v04\n- v07 --supports--> v18\n- v0d --supports--> v0e\n- v12 --refutes--> v13\n- v14 --supports--> v17\n- v15 --supports--> v06\n- v16 --refutes--> v0b\n\n[VERIFIED ROOT]\nv12 = 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":"20abe5b84345a70302dfb26d3dee297565e7736867383c410987c745d657cfb9","schema":"livegraph.boolean-propagation.v1","spec_hash":"079b3558ee68e7f1644d39745a7243d0adeb7f6fa5587b7ded8c68d9f3d92f58","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v16\"}],\"hop\":16,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-5e022be4eb623af1e312","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v03\n- v03 --refutes--> v02\n- v02 --refutes--> v07\n- v07 --supports--> v0a\n- v0a --supports--> v01\n- v01 --supports--> v08\n- v08 --refutes--> v0c\n- v0c --supports--> v0b\n- v0b --refutes--> v14\n- v14 --supports--> v0d\n- v0d --supports--> v16\n- v16 --supports--> v15\n- v15 --refutes--> v0f\n- v0f --refutes--> v17\n- v17 --refutes--> v10\n- v10 --supports--> v12\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v13\n- v0a --supports--> v04\n- v0f --refutes--> v0e\n- v0f --supports--> v09\n- v10 --refutes--> v18\n- v14 --supports--> v11\n- v16 --supports--> v00\n- v16 --supports--> v05\n\n[VERIFIED ROOT]\nv06 = 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":"fd0d69ce777f3026cccc8d70067b00e67d3ed82cb6baf450c951a755c950d3a3","schema":"livegraph.boolean-propagation.v1","spec_hash":"5e022be4eb623af1e312e4d6cf104cfd666be49089a01b716ab3a00dbdfbc005","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-53803d50df6ab147e9ed","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v17\n- v17 --refutes--> v13\n- v13 --supports--> v0f\n- v0f --refutes--> v0d\n- v0d --refutes--> v16\n- v16 --supports--> v18\n- v18 --refutes--> v08\n- v08 --supports--> v00\n- v00 --supports--> v12\n- v12 --supports--> v03\n- v03 --supports--> v0c\n- v0c --supports--> v11\n- v11 --refutes--> v0a\n- v0a --supports--> v01\n- v01 --refutes--> v0b\n- v0b --refutes--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v0e\n- v03 --supports--> v09\n- v0a --supports--> v10\n- v0c --refutes--> v05\n- v11 --supports--> v02\n- v11 --supports--> v15\n- v13 --refutes--> v07\n- v16 --supports--> v14\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":"f8d54ff42633dd38c395fae8c624a38526ca64fb7e17ea372e7fe74823f8d65b","schema":"livegraph.boolean-propagation.v1","spec_hash":"53803d50df6ab147e9eda997e4df7a72b3b11cb5aede437819f6e497ee86e96f","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-c11a5414784f54a5daae","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v14 --supports--> v07\n- v07 --supports--> v05\n- v05 --supports--> v04\n- v04 --refutes--> v03\n- v03 --refutes--> v0b\n- v0b --supports--> v10\n- v10 --supports--> v13\n- v13 --supports--> v16\n- v16 --supports--> v0d\n- v0d --refutes--> v00\n- v00 --supports--> v08\n- v08 --refutes--> v01\n- v01 --refutes--> v11\n- v11 --supports--> v18\n- v18 --refutes--> v0e\n- v0e --supports--> v12\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v0a\n- v01 --supports--> v17\n- v03 --refutes--> v09\n- v08 --refutes--> v02\n- v0d --supports--> v06\n- v13 --supports--> v0c\n- v14 --refutes--> v15\n- v16 --refutes--> v0f\n\n[VERIFIED ROOT]\nv14 = 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":"1fe0c1422b019b686c0bd6c17edfc7f678729eed3ab1013d8f0c54a369c914ce","schema":"livegraph.boolean-propagation.v1","spec_hash":"c11a5414784f54a5daaee67f15cb6441e2c3253d92ef669af8a5ad58d5c2f1a0","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v14\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-21fffc98be1d655d3657","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --refutes--> v10\n- v10 --refutes--> v11\n- v11 --refutes--> v13\n- v13 --refutes--> v16\n- v16 --supports--> v06\n- v06 --supports--> v14\n- v14 --supports--> v15\n- v15 --supports--> v18\n- v18 --refutes--> v0c\n- v0c --supports--> v04\n- v04 --supports--> v00\n- v00 --supports--> v0f\n- v0f --supports--> v01\n- v01 --refutes--> v08\n- v08 --supports--> v12\n- v12 --supports--> v0e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v03\n- v04 --refutes--> v0d\n- v06 --refutes--> v09\n- v07 --supports--> v05\n- v08 --refutes--> v0a\n- v0f --refutes--> v17\n- v10 --refutes--> v0b\n- v12 --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":"b421e1e8963434d76a56332fbbf4abd6b1effcd210685058a61e7655c62e3cf7","schema":"livegraph.boolean-propagation.v1","spec_hash":"21fffc98be1d655d36573133956e32751eeeb3e28fc6450d1f8001f55ebffc22","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-ae4fc3aa3686bc243c0f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --supports--> v0c\n- v0c --supports--> v14\n- v14 --supports--> v09\n- v09 --supports--> v04\n- v04 --refutes--> v02\n- v02 --supports--> v0b\n- v0b --supports--> v18\n- v18 --refutes--> v15\n- v15 --supports--> v10\n- v10 --supports--> v06\n- v06 --refutes--> v05\n- v05 --supports--> v12\n- v12 --supports--> v00\n- v00 --supports--> v08\n- v08 --supports--> v11\n- v11 --supports--> v0e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0d\n- v06 --supports--> v01\n- v07 --refutes--> v13\n- v07 --refutes--> v16\n- v07 --supports--> v0f\n- v0b --supports--> v17\n- v15 --refutes--> v0a\n- v15 --supports--> v03\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":"36a539ca304acdd0d30dc80a123330c9e66ad2b60fba72810dcd895008903cd1","schema":"livegraph.boolean-propagation.v1","spec_hash":"ae4fc3aa3686bc243c0feb66631f97f394253a3852df0f775039754d48bbc8a7","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-c16e4fc96dcfa9e0e5b0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --supports--> v09\n- v09 --refutes--> v05\n- v05 --refutes--> v0b\n- v0b --supports--> v03\n- v03 --refutes--> v02\n- v02 --supports--> v12\n- v12 --supports--> v0f\n- v0f --refutes--> v18\n- v18 --supports--> v15\n- v15 --refutes--> v17\n- v17 --supports--> v10\n- v10 --refutes--> v11\n- v11 --refutes--> v08\n- v08 --refutes--> v06\n- v06 --refutes--> v01\n- v01 --refutes--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v00\n- v01 --supports--> v0c\n- v08 --refutes--> v0a\n- v08 --refutes--> v16\n- v09 --supports--> v14\n- v10 --supports--> v13\n- v15 --supports--> v0d\n- v18 --supports--> v0e\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":"0a85b4fa13299cd7f4b38542040a5e02f3eac9ec9aed2f511d9929fc4860238f","schema":"livegraph.boolean-propagation.v1","spec_hash":"c16e4fc96dcfa9e0e5b0579b84dc78da413a40204010cdd041aae85e448f9914","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-3943aa91b7b5ff9f40f6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --refutes--> v03\n- v03 --refutes--> v14\n- v14 --supports--> v09\n- v09 --refutes--> v08\n- v08 --supports--> v13\n- v13 --refutes--> v0c\n- v0c --refutes--> v17\n- v17 --supports--> v16\n- v16 --refutes--> v10\n- v10 --supports--> v04\n- v04 --refutes--> v15\n- v15 --supports--> v11\n- v11 --supports--> v01\n- v01 --refutes--> v0e\n- v0e --refutes--> v0b\n- v0b --refutes--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v12\n- v08 --supports--> v06\n- v0e --refutes--> v0a\n- v10 --refutes--> v02\n- v10 --refutes--> v0f\n- v10 --supports--> v18\n- v11 --refutes--> v0d\n- v16 --supports--> v05\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":"62b92d12f22bd922b9ae76095e04b485d26f1eb63fd54d9a799d9e11d9e14e9e","schema":"livegraph.boolean-propagation.v1","spec_hash":"3943aa91b7b5ff9f40f6ab242291c2e5d25dfb2eba2316a616f69d6eb0c84f29","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-5d98f02883cf34c700d9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0f --refutes--> v13\n- v13 --refutes--> v17\n- v17 --supports--> v15\n- v15 --refutes--> v0b\n- v0b --refutes--> v16\n- v16 --refutes--> v0d\n- v0d --supports--> v08\n- v08 --supports--> v01\n- v01 --supports--> v07\n- v07 --refutes--> v04\n- v04 --supports--> v0a\n- v0a --supports--> v09\n- v09 --refutes--> v02\n- v02 --supports--> v10\n- v10 --refutes--> v0c\n- v0c --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v00\n- v02 --supports--> v06\n- v07 --refutes--> v11\n- v07 --supports--> v0e\n- v0a --supports--> v14\n- v0b --refutes--> v05\n- v0d --supports--> v18\n- v16 --refutes--> v12\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":"614307a5f1b190dc31a05551d589a666f64a00685a19d49a7bdffd96245174fd","schema":"livegraph.boolean-propagation.v1","spec_hash":"5d98f02883cf34c700d965b04e774bdce01b3f72c6fb8fe8c2779e9600eb386e","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-8b97cdbffa272abc1a57","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0f --refutes--> v00\n- v00 --refutes--> v0b\n- v0b --supports--> v15\n- v15 --refutes--> v07\n- v07 --refutes--> v08\n- v08 --refutes--> v0e\n- v0e --refutes--> v11\n- v11 --supports--> v17\n- v17 --refutes--> v04\n- v04 --supports--> v09\n- v09 --refutes--> v0a\n- v0a --refutes--> v14\n- v14 --supports--> v12\n- v12 --refutes--> v13\n- v13 --supports--> v0c\n- v0c --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v0d\n- v04 --supports--> v06\n- v07 --supports--> v18\n- v09 --supports--> v16\n- v11 --refutes--> v02\n- v13 --supports--> v05\n- v15 --refutes--> v10\n- v17 --refutes--> v01\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":"435c9fe8dcda63cef4c59a81c2c310394588f4000444d3b8eac79f9bf361a9e8","schema":"livegraph.boolean-propagation.v1","spec_hash":"8b97cdbffa272abc1a575bdadb176220ef0974c22ff4ff6a70ceab8c88294580","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-0cc6827da777b03055f2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --refutes--> v10\n- v10 --supports--> v03\n- v03 --refutes--> v0a\n- v0a --supports--> v15\n- v15 --refutes--> v14\n- v14 --refutes--> v0e\n- v0e --refutes--> v11\n- v11 --refutes--> v06\n- v06 --supports--> v05\n- v05 --refutes--> v16\n- v16 --refutes--> v0f\n- v0f --refutes--> v18\n- v18 --supports--> v0b\n- v0b --supports--> v0d\n- v0d --supports--> v17\n- v17 --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v08\n- v05 --supports--> v01\n- v0a --refutes--> v02\n- v0f --refutes--> v09\n- v14 --supports--> v0c\n- v17 --refutes--> v04\n- v17 --refutes--> v12\n- v18 --supports--> v13\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":"36ac392b0bf934aa5151cc99ceeb394c92bb01b6f8f70f0b7de194a0b3c60cb7","schema":"livegraph.boolean-propagation.v1","spec_hash":"0cc6827da777b03055f2b676052c8abdeab3e33e3b817017a01e9c915b8cf999","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-684b7065c7c412369af0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v18 --supports--> v12\n- v12 --refutes--> v06\n- v06 --refutes--> v08\n- v08 --refutes--> v15\n- v15 --supports--> v11\n- v11 --refutes--> v05\n- v05 --refutes--> v0d\n- v0d --refutes--> v0f\n- v0f --refutes--> v03\n- v03 --refutes--> v04\n- v04 --refutes--> v07\n- v07 --supports--> v10\n- v10 --supports--> v14\n- v14 --supports--> v0e\n- v0e --supports--> v09\n- v09 --supports--> v0c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v00\n- v03 --refutes--> v0b\n- v05 --refutes--> v17\n- v09 --supports--> v0a\n- v11 --refutes--> v13\n- v12 --supports--> v02\n- v12 --supports--> v16\n- v14 --supports--> v01\n\n[VERIFIED ROOT]\nv18 = 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":"8f39468ee7c893afe06861f7f26a7f0aff69a7ccade1edf8d4187234f9965237","schema":"livegraph.boolean-propagation.v1","spec_hash":"684b7065c7c412369af071ae2ae78c6478eecff9c5b187f1bf50f102fb5db0b4","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v18\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-bcd749dae54597bc8858","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --refutes--> v04\n- v04 --refutes--> v07\n- v07 --refutes--> v0e\n- v0e --refutes--> v06\n- v06 --supports--> v0d\n- v0d --supports--> v0b\n- v0b --supports--> v16\n- v16 --refutes--> v00\n- v00 --supports--> v10\n- v10 --refutes--> v03\n- v03 --supports--> v18\n- v18 --supports--> v0f\n- v0f --supports--> v11\n- v11 --refutes--> v08\n- v08 --supports--> v02\n- v02 --refutes--> v15\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v14\n- v07 --refutes--> v09\n- v08 --supports--> v0c\n- v0a --supports--> v17\n- v0b --supports--> v05\n- v0f --refutes--> v12\n- v0f --supports--> v01\n- v10 --refutes--> v13\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":"750ed52e2246725217ebca45ea0ef93d59c92c5ec2a9cc1431af6ee40ee0ae5e","schema":"livegraph.boolean-propagation.v1","spec_hash":"bcd749dae54597bc8858e4920700703c9b4feaa3c6315699ad84bf6de7d57887","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-096cefa6120171b8c367","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v18 --supports--> v0e\n- v0e --supports--> v11\n- v11 --refutes--> v10\n- v10 --supports--> v07\n- v07 --supports--> v0c\n- v0c --refutes--> v05\n- v05 --refutes--> v0b\n- v0b --supports--> v13\n- v13 --supports--> v01\n- v01 --refutes--> v04\n- v04 --refutes--> v00\n- v00 --refutes--> v0f\n- v0f --supports--> v0a\n- v0a --refutes--> v06\n- v06 --refutes--> v16\n- v16 --refutes--> v12\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v03\n- v06 --refutes--> v15\n- v07 --supports--> v0d\n- v0a --refutes--> v17\n- v0e --supports--> v08\n- v0f --refutes--> v09\n- v13 --supports--> v14\n- v16 --refutes--> v02\n\n[VERIFIED ROOT]\nv18 = 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":"3e0e8fd10f7287602b3ae36840a9ad69da01ba1ad7babafa6cf5f220b01d2532","schema":"livegraph.boolean-propagation.v1","spec_hash":"096cefa6120171b8c367ab8a3e49ea85d495b8c1f2f10956e615c34cb02239df","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v18\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-37c4ba202150995d28f9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v11 --supports--> v13\n- v13 --refutes--> v0c\n- v0c --refutes--> v06\n- v06 --refutes--> v04\n- v04 --supports--> v07\n- v07 --refutes--> v12\n- v12 --refutes--> v0b\n- v0b --supports--> v17\n- v17 --refutes--> v09\n- v09 --refutes--> v18\n- v18 --supports--> v05\n- v05 --refutes--> v03\n- v03 --refutes--> v0d\n- v0d --supports--> v0e\n- v0e --supports--> v0a\n- v0a --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v15\n- v05 --supports--> v02\n- v06 --refutes--> v08\n- v09 --supports--> v0f\n- v11 --refutes--> v10\n- v17 --supports--> v01\n- v17 --supports--> v14\n- v17 --supports--> v16\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":"03de0bbfc544f02264d19661a5044a778cbd0a4ea358d15360fc0b1d42cd0e64","schema":"livegraph.boolean-propagation.v1","spec_hash":"37c4ba202150995d28f948ed43054efbd2a20a979ebec9b8787dbb895c1be2b7","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-332ca52e8a446286677c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --refutes--> v10\n- v10 --supports--> v13\n- v13 --supports--> v0d\n- v0d --supports--> v06\n- v06 --supports--> v07\n- v07 --supports--> v04\n- v04 --supports--> v05\n- v05 --supports--> v18\n- v18 --refutes--> v09\n- v09 --refutes--> v0e\n- v0e --refutes--> v08\n- v08 --refutes--> v02\n- v02 --refutes--> v00\n- v00 --refutes--> v16\n- v16 --refutes--> v11\n- v11 --supports--> v12\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v17\n- v01 --refutes--> v0b\n- v02 --supports--> v0a\n- v05 --refutes--> v03\n- v06 --supports--> v15\n- v0e --refutes--> v0c\n- v0e --supports--> v14\n- v16 --refutes--> v0f\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":"43b56379569135a0efaec0311b4b43d82cbd32f29f7d9774ebf756b028cdcf1f","schema":"livegraph.boolean-propagation.v1","spec_hash":"332ca52e8a446286677c68d43faa44cbbf5794196c0563a6b17e891cf08d5929","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-7961f793ffba5e225681","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v12 --refutes--> v0a\n- v0a --supports--> v13\n- v13 --supports--> v02\n- v02 --supports--> v0b\n- v0b --supports--> v11\n- v11 --supports--> v10\n- v10 --refutes--> v14\n- v14 --refutes--> v06\n- v06 --refutes--> v16\n- v16 --refutes--> v0c\n- v0c --supports--> v00\n- v00 --supports--> v07\n- v07 --refutes--> v17\n- v17 --supports--> v03\n- v03 --supports--> v0f\n- v0f --refutes--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v04\n- v03 --refutes--> v0d\n- v06 --supports--> v08\n- v0c --supports--> v05\n- v0f --supports--> v15\n- v0f --supports--> v18\n- v10 --refutes--> v0e\n- v17 --refutes--> v09\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":"93451c5d873406f0c812eb47918e777abdc169b4f74963827c5a5ff6d317669d","schema":"livegraph.boolean-propagation.v1","spec_hash":"7961f793ffba5e2256819f8dfc1b38dd16e13913d8fe4eefc699f117e8648468","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v17\"}],\"hop\":16,\"root\":\"v12\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":16,"item_id":"lggrpo1-f55d06048fb119387dde","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v11 --refutes--> v0f\n- v0f --supports--> v06\n- v06 --supports--> v10\n- v10 --refutes--> v00\n- v00 --supports--> v03\n- v03 --refutes--> v04\n- v04 --supports--> v0e\n- v0e --supports--> v0b\n- v0b --supports--> v0c\n- v0c --supports--> v13\n- v13 --supports--> v09\n- v09 --refutes--> v01\n- v01 --refutes--> v17\n- v17 --refutes--> v0d\n- v0d --supports--> v18\n- v18 --supports--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v07\n- v01 --supports--> v05\n- v03 --refutes--> v0a\n- v0b --refutes--> v12\n- v0e --refutes--> v16\n- v10 --supports--> v08\n- v13 --supports--> v14\n- v18 --supports--> v15\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":"2fc481e6a6de1f9f50356c0f8fecf86432fee8d0830690ab8235b7469de2e0d4","schema":"livegraph.boolean-propagation.v1","spec_hash":"f55d06048fb119387ddeefff4af02e4ce93238f9f7eda4013109bb2a4cf7dec3","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":16,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_id","view":"ordered"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":16,"item_id":"lggrpo1-68698c84f28b861c0450","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --supports--> v01\n- v01 --refutes--> v09\n- v09 --refutes--> v0f\n- v0f --refutes--> v05\n- v05 --refutes--> v02\n- v02 --supports--> v0d\n- v0d --refutes--> v15\n- v15 --refutes--> v00\n- v00 --supports--> v0a\n- v0a --refutes--> v0e\n- v0e --supports--> v12\n- v12 --refutes--> v11\n- v11 --supports--> v0b\n- v0b --supports--> v07\n- v07 --supports--> v04\n- v04 --refutes--> v18\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v08\n- v03 --supports--> v13\n- v05 --supports--> v10\n- v07 --supports--> v06\n- v07 --supports--> v14\n- v07 --supports--> v17\n- v0d --refutes--> v16\n- v15 --supports--> v0c\n\n[VERIFIED ROOT]\nv03 = 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":"a8c009fac83f93bff8e99f3109433214c20676767c2d36e366f99b107b2272a5","schema":"livegraph.boolean-propagation.v1","spec_hash":"68698c84f28b861c045066a51a14441d9cd776a2e5ab2391588f8ab4e363a283","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v15\"}],\"hop\":16,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v18\"}","split":"eval_id","view":"ordered"}